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Limit theorems for Markov-modulated and reflected diffusion processes
Huang, G.
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Download date: 21 Aug 2020
Limit Theorems forMarkov-modulated and Reflected
Diffusion Processes
Gang Huang
Limit Th
eorems for M
arkov-modulated and Reflected D
iffusion Processes Gang H
uang
sample paths of a Markov-modulated geometric Brownian motion with small noise and its modulating Markov chain with rapid switching
Limit theorems forMarkov-modulated and re�ected
di�usion processes
Cover design: Proefschri�maken.nl || Uitgeverij BOXPressPrinted by: Proefschri�maken.nl || Uitgeverij BOXPress
Limit theorems forMarkov-modulated and re�ected
di�usion processes
ACADEMISCH PROEFSCHRIFT
ter verkrijging van de graad van doctor
aan de Universiteit van Amsterdam
op gezag van de Rector Magni�cus
prof. dr. D.C. van den Boom
ten overstaan van een door het College voor Promoties ingestelde
commissie, in het openbaar te verdedigen in de Agnietenkapel
op woensdag 17 juni 2015, te 14:00 uur
door Gang Huang
geboren te Shandong, China
PromotiecommissiePromotor: prof. dr. M.R.H. Mandjes (Universiteit van Amsterdam)
Copromotor: dr. P.J.C. Spreij (Universiteit van Amsterdam)
Overige leden:dr. A.J. van Es (Universiteit van Amsterdam)Prof. dr. A.J. Homburg (Universiteit van Amsterdam/VU Amsterdam)Prof. dr. R. Nunez-�eija (Universiteit van Amsterdam/CWI)dr. A.A.N. Ridder (VU Amsterdam)dr. J.A.M van der Weide (TU Del�)Prof. dr. J.H. van Zanten (Universiteit van Amsterdam)Prof. dr. A.P. Zwart (CWI/TU/e)
Faculteit: Faculteit der Natuurwetenschappen, Wiskunde en Informatica
Preface
According to the Doctorate Regulations of the University of Amsterdam, thefollowing list and explanation is provided. �e main body of this thesis con-sists of the following three papers wri�en by the author and his two super-visors.
[1] Huang, G., Mandjes, M. and Spreij, P., 2014. Weak convergence of Markov-modulated di�usion processes with rapid switching, Statistics & ProbabilityLe�ers, 86, 74–79.
[2] Huang, G., Mandjes, M. and Spreij, P., 2015. Large deviations for Markov-modulated di�usion processes with rapid switching, submi�ed.
[3] Huang, G., Mandjes, M. and Spreij, P., 2014. Limit theorems for re�ectedOrnstein-Uhlenbeck processes, Statistica Neerlandica, 68, 25–42.
�e topics and frameworks are results of intensive discussions betweenthe author and his supervisors. A large majority of the work was done bythe author. Nevertheless, this thesis would not have been completed withoutthe inspiration, consultation and kind support of the author’s supervisors.
I would like to take this opportunity to express my in�nite thanks toMichel and Peter. Also, from the bo�om of my heart, I am very grateful toeveryone at the KdV Institute for Mathematics and the University of Ams-terdam who helped me in various respects.
Finally, I would like to thank dr. A.J. van Es, Prof. dr. A.J. Homburg, Prof.dr. R. Nunez-�eija, dr. A.A.N. Ridder, dr. J.A.M van der Weide, Prof. dr.
5
J.H. van Zanten, Prof. dr. A.P. Zwart for kindly being members of my thesiscommi�ee and carefully reading my thesis.
6
Conventions
We list several symbols and conventions which are frequently used in thisthesis. R is the set of real numbers andR+ := [0,1). A spaceX is said to bePolish if it is a complete, separable and metrizable topological space equippedwith its Borel �-algebra B(X). Let CA(E) denote the space of E-valuedcontinuous functions on A with the uniform norm kfk1 := supt2A |f(t)|and the metric d(f, g) := kf � gk1. If E = R, we write CA for simplicity.
A stochastic basis is de�ned to be a probability space (⌦,F ,P) equippedwith a �ltration {Ft}t2R+ . It is called complete ifF isP-complete, {Ft}t2R+
is right continuous and if F0 contains all the P-null sets of F . Sometimes a�ltration is denoted byFt for simplicity. LetX be a random variable de�nedon (⌦,F , {Ft}t2R+ ,P) taking values in (X,B(X)). �e distribution (or law)of X on X is denoted by P �X�1. A stochastic process is o�en denoted byXt.
We also use the following short-hand notations frequently.
SDE stochastic di�erential equationLDP large deviations principleOU Ornstein-UhlenbeckROU re�ected Ornstein-UhlenbeckDROU doubly re�ected Ornstein-UhlenbeckODE ordinary di�erential equationCLT central limit theoremFCLT functional central limit theorem
7
8
Contents
Preface 5
Conventions 7
Overview 11
1 Preliminaries 191.1 Weak convergence . . . . . . . . . . . . . . . . . . . . . . . . 191.2 Large deviations . . . . . . . . . . . . . . . . . . . . . . . . . 211.3 �e stochastic processes studied in this thesis . . . . . . . . 27
2 Weak convergence of Markov-modulated di�usion processeswith rapid switching 312.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.2 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3 Proof of tightness . . . . . . . . . . . . . . . . . . . . . . . . 342.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3 Large deviations forMarkov-modulated di�usionprocesseswithrapid switching 433.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.3 Exponential tightness . . . . . . . . . . . . . . . . . . . . . . 493.4 Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . 553.5 �e upper bound for the local LDP . . . . . . . . . . . . . . . 60
9
3.6 �e lower bound for the local LDP . . . . . . . . . . . . . . . 723.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4 Limit theorems for re�ected Ornstein-Uhlenbeck processes 914.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.2 Transient asymptotics for OU processes . . . . . . . . . . . . 934.3 LDPs for re�ected di�usion processes . . . . . . . . . . . . . 984.4 Transient asymptotics for ROU processes . . . . . . . . . . . 1014.5 Transient asymptotics for DROU processes . . . . . . . . . . 1034.6 CLTs and FCLTs for the loss and idle processes . . . . . . . . 104
Bibliography 111
Summary 117
Samenvatting 119
10
Overview
�e scope
In this thesis, we consider limit theorems of two types: one focuses on weakconvergence of stochastic processes, and the other one concerns large devi-ations for stochastic processes. Both of them have deep roots in probabilitytheory.
Consider a system with some random forces, e.g., a stochastic process.One branch of research studies under what situations such system displaysspeci�c regularities. For instance, ergodic theory explains that the long timebehavior of a system is governed by an invariant measure, if such measureexists. Another obvious example of this branch concerns central limit theo-rems. �e investigation of weak convergence in this thesis falls in this cat-egory. �e focus is on establishing the limiting probability measure of a se-quence of probability measures in the weak topology under a certain scaling(time, space, etc.).
A second branch of research aims at characterizing probabilities of a sys-tem behaving abnormally, and understanding features of this system when arare event happens. Generalized from calculating tail probabilities, the the-ory of large deviations is one of the most important �elds of this kind. Itconstitutes a uni�ed framework to study a sequence of rare-event probabili-ties which vanishes to zero exponentially. Moreover, a key task in the theoryof large deviations is obtaining a rate function, which quanti�es the rate ofthe exponential decay in the underlying probability model.
We study the above mentioned limit theorems for two variants of one-dimensional di�usion processes. One is theMarkov-modulated di�usion pro-
11
cess, and the other one is the re�ected di�usion process, with a focus onthe re�ected Ornstein-Uhlenbeck process. �ey inherit analytical tractabil-ity from di�usion processes, and meanwhile they possess distinctive featureswhich make them very a�ractive in various applications. �e term ‘di�usionprocesses’ has a rather restrictive meaning in this thesis. It refers to Ito’s dif-fusions, which are solutions of the following stochastic di�erential equation(SDE):
dSt = b(St)dt+ �(St)dBt,
where Bt is the standard Brownian motion. It is a Markov process with al-most surely continuous sample paths. �e part b(St)dt, which is called thedri� term, is used to model the general trend of the di�usion process. �eother part, i.e. �(St)dBt, which is named the di�usion term, represents therandom �uctuation associated with the di�usion process.
As we can see, di�usion processes have only one random source, i.e., aBrownian motion. In contrast, the Markov-modulated di�usion process hasan extra random source which is a Markov chain. It is de�ned as a solutionto the following SDE
dMt = b(Mt, Xt)dt+ �(Mt, Xt)dBt,
where Bt is a standard Brownian motion and Xt is a �nite-state Markovchain which is independent of Bt. �e Markov chain Xt is sometimes re-ferred to as the background process. Its main feature is that the coe�cientsof the dri� and di�usion terms depend on the states of the background pro-cess. Also, the pair (Mt, Xt) is a Markov process.
We aim to discover how the dynamics of the Markov chainXt a�ect thebehavior of the Markov-modulated di�usion process. When the jumps ofXt are sparse, the moments of jumps can o�en be seen clearly from a sam-ple path of Mt, which is in fact a concatenation of pieces of sample pathsof di�erent non-modulated di�usion processes. (See Figure 2.1 at the end ofChapter 2.) In contrast, when Xt jumps extremely fast among its states, thebehavior of Mt is harder to precisely describe. �is has motivated us to rig-orously study Markov-modulated di�usion processes with rapidly switchingbackground processes from the perspectives of weak convergence and largedeviations. �e results we obtained have considerably deepened our under-standing of the limiting behavior of Markov-modulated di�usion processes.
12
�e Ornstein-Uhlenbeck (OU) process is a popular di�usion process inphysics and �nance due to its mean-reverting property. It is real-valued, butin many models (e.g. queueing models) the stochastic process is requiredto stay positive or in a speci�c domain. �en re�ected Ornstein-Uhlenbeck(ROU) and doubly re�ectedOrnstein-Uhlenbeck (DROU) processes, which areanother variant of one-dimensional di�usion processes, become natural sub-stitutes. �ey are de�ned as OU processes which have instantaneous re�ec-tion at one boundary or two boundaries, respectively. �ey can be used invarious applications since their sample paths are restricted to the half-line oran interval. Our study of them is motivated by queueing theory, but severalother applications can be thought of, too. We analyze the transient asymp-totics of OU, ROU and DROU processes. We also investigate the limitingbehavior of DROU processes at the boundaries.
�e main results
So far we have introduced the scope of this thesis. Next we present anoverview of our main results, as well as the corresponding explanations ofthe approaches used. �ere are four main results in this thesis, which are alllimit theorems. �e �rst two, which are presented in Chapter 2 and Chapter3 respectively, concern the Markov-modulated di�usion process. �e othertwo, which are stated in Chapter 4, are about (doubly) re�ected Ornstein-Uhlenbeck processes.
We study the Markov-modulated di�usion process with a focus on situa-tions when the Markov chain is rapidly switching. �is particular switchingregime is achieved by scaling the transition intensity matrixQ ofXt toQ/✏,where ✏ is a small strictly positive number. We denote X✏
t a sequence of d-state ergodic Markov chains with the transition intensity matrixQ/✏ and theergodic distribution ⇡ = (⇡1, · · · ,⇡d). If the expected number of jumps perunit time is y for Xt, then the time-scaling entails that it is y/✏ for X✏
t . Itmeans that the Markov chain jumps extremely fast among its states when✏! 0.
Our �rst main result proves that a sequence of Markov-modulated di�u-
13
sion processes with rapid switching
d
˜M ✏t = b(X✏
t , ˜M ✏t )dt+ �(X✏
t , ˜M ✏t )dB
✏t
converges weakly to an ordinary (i.e. non-modulated) di�usion process
d
ˆMt =ˆb( ˆMt)dt+ �( ˆMt)dBt,
where
ˆb(x) :=dX
i=1
b(i, x)⇡i, �(x) :=
dX
i=1
�2(i, x)⇡i
!1/2
.
�is can be seen as an averaging behavior. Weak convergence results are usu-ally proved in two parts: convergence of �nite dimensional distributions andtightness of stochastic processes. �e �rst part in our case has been provedin Skorohod [51]. We prove its tightness property based on Aldous’ tight-ness criterion. �is result shows that, when the background chain switchesrapidly, it is hard to distinguish a Markov-modulated di�usion process froma non-modulated di�usion process.
Our next main result, in contrast, indicates that no such property car-ries over to the result of large deviations for Markov-modulated di�usionprocesses with rapid switching. �e impact of a fast switching backgroundchain does appear in the small noise asymptotics of the Markov-modulateddi�usion process, as shown in the LDPs. More precisely, we study the largedeviations for the following SDE:
(1) dM ✏t = b(X✏
t ,M✏t )dt+
p✏�(X✏
t ,M✏t )dBt.
Since M ✏t evolves in the random environment of X✏
t , we need to design acoupling to separate the e�ects of the vanishing of the di�usion term andthe fast varying of the Markov chain, but at the same time to keep track ofboth of them. Since the scaling Q to Q/✏ is equivalent to speeding up timeby a factor ✏�1, one could informally say thatX✏
t relates to a faster time scalethanM ✏
t , and therefore essentially exhibits stationary behavior ‘around’ this
14
speci�c t. �en it is custom to consider the occupation measure ⌫✏ of theMarkov chain X✏
t de�ned as
⌫✏(!; t, i) =
Z t
01{X✏
s
(!)=i}ds.
As its name suggests, ⌫✏(·;T, i)measures the timeX✏t spends in state i during
the time interval [0, T ]. We look into the paired process (M ✏, ⌫✏), whose jointdistribution is then a coupling of the marginal distributions of M ✏ and ⌫✏.�e pair (M ✏, ⌫✏) is called a coupling according to the standard terminologyof probability theory. (See for example Villani [56].)
Our second main result is the joint sample-path LDP for the coupling(M ✏, ⌫✏) on their image spaceCT ⇥MT . �e rate function is obtained as thesum of
˜IT (⌫) :=
Z T
0sup
u2U
"�
dX
i=1
(Qu)(i)
u(i)K⌫(s, i)
#ds
and
IT (', ⌫) :=
8<
:
1
2
Z T
0
['0t � ˆb(⌫,'t)]
2
�2(⌫,'t)dt if ' 2 HT ,
1 otherwise.
�eir precise de�nitions will be given in Chapter 3. Informed readers willimmediately recognize that these rate functions are variants of those for dif-fusion processes, as given in e.g. Freidlin and Wentzell [16], and for occupa-tion measures of Markov processes, as given in e.g. Donsker and Varadhan[9]. �ose rate functions are introduced in Section 1.2 for readers’ conve-nience. We remark that the result in Donsker and Varadhan [9] relates to⌫1(!; t, ·)/t for t large, but our statement is for the sample paths of ⌫✏ forsmall ✏.
One method of proving the LDP for a family of probability measures on ametric space, as was introduced in the seminal papers of Liptser and Pukhal-skii [39] and Liptser [37], is to �rst prove exponential tightness, and then thelocal LDP (precise de�nitions of these notions will be given in Chapter 1).Our work by and large follows this approach. Importantly, the model con-sidered in Liptser [37] is similar to ours, in that it also studies the SDE (1),but in the setup of Liptser [37] the process X✏
t is another di�usion process
15
(rather than a �nite-state Markov chain). It means that we can roughly fol-low the structure of the proof presented in [37] (we also rely on the methodof stochastic exponentials, for example), but there are crucial di�erences atmany places. For instance, as we point out below, there are several noveltiesthat have the potential of being used in other se�ings, too.
One of the methodological novelties is the following. We explore a niceconnection between regularity properties of the rate function ˜IT (⌫) in theLDP for (M ✏, ⌫✏) and a dense subset of the image space MT of ⌫✏. On thisdense subset, the optimizer of the integrand of ˜IT (⌫) is in�nitely di�eren-tiable. �is eliminates many di�culties in the computation and leads us to�rst prove the local LDP on a dense subset of CT ⇥MT . We then extend thelocal LDP toCT ⇥MT by continuity properties of the rate functions IT (', ⌫)and ˜IT (⌫).
�e LDPs for each componentM ✏ and ⌫✏ are then derived as corollariesfrom our second main result by the contraction principle. (See�eorem 1.2.7in Chapter 1 for details.) �e small noise LDP for the Markov-modulateddi�usion processes (which is M ✏ alone) is also studied in a newly publishedpaper by He and Yin [22] in a se�ing of multi-dimensional processes andtime-depending transition intensity matrices. In our corresponding result,which is Corollary 3.2.2, the rate function for M ✏ is decomposed into twoparts that allow an appealing interpretation: the �rst part corresponds tothe rare behavior of the background process X✏, whereas the second partcorresponds to the rare behavior of M ✏ (conditional on the rare behavior ofX✏). �e rate function in He and Yin [22] is less explicit, in that it is expressedin terms of an H-functional in which the aforementioned two parts cannotbe distinguished. �e sample-path LDP for occupation measures of rapidswitching Markov chain (which is ⌫✏ alone) is obtained in �eorem 5.1 inHe et al. [23]. �e rate function, which is also expressed in terms of anH-functional, coincides with the rate function in our LDP for ⌫✏ (Corollary3.2.3) when the transition intensity matrix is time-homogeneous. However,focusing on obtaining the LDP for the Markov-modulated di�usion processtogether with the background process, our aim and approach in this thesisare entirely di�erent from theirs.
Our third main result provides insight into transient rare-event probabil-ities of ROU and DROU processes. We explicitly compute the decay rate of
16
the transient probability to reach a given extreme level
lim
✏!0✏ logP(Y ✏
T > b | Y ✏0 = x)
for the ROU process with re�ection at 0
dY ✏t = (↵� �Y ✏
t )dt+p✏�dBt + dL✏
t,
with ✏ > 0 typically small. �e transient distribution (at time T > 0, for anyinitial value x > 0) of the OU process is explicitly known and has a Normaldistribution. We lack such result for the ROU process. As an aside, we notethat the stationary distribution of ROU process is known in Ward and Glynn[58], which is a truncated Normal distribution.
�e computation is carried out into three steps. Firstly, we compute theabove decay rate for the OU process relying on the sample-path LDP fordi�usion processes. Meanwhile, we also identify the associated most likelypaths. �is idea is used for computing blocking probabilities of the Erlangqueue in Shwartz and Weiss [49]. Secondly, we derive explicit expressionsof the rate function in the LDP for re�ected di�usion processes. �irdly, weapply the above result together with the non-negativity of the optimizingpath of the OU process to obtain the decay rate for the ROU process. By thesame method, we also compute the decay rate for the DROU process. �eresults of OU, ROU and DROU processes are the same thanks to the mean-reverting property of OU processes.
Our fourth main result focuses on properties of the loss process Ut andthe idle process Lt of DROU processes
dZt = (↵� �Zt)dt+ �dBt + dLt � dUt.
With h(·) being a twice continuously di�erentiable real function, we applyIto’s formula on h(Zt) and require h(·) to satisfy certain ordinary di�erentialequations (ODEs) and speci�c initial and boundary conditions in order toconstruct martingales related to Ut and Lt. �is approach is a well-knowntechnique in stochastic calculus used inmany papers, e.g. Liptser [38], Zhangand Glynn [64]. �e presence of Zt in the dri� term leads to ODEs withnonconstant coe�cients, which seriously complicates the derivation of exact
17
solutions. We prove central limit theorems for centered and scaled Ut andLt as t ! 1 by the above approach. We then prove the centered and scaledUnt and Lnt converge weakly to a standard Brownian motion as n ! 1 byShiryaev’s convergence of triplets.
As an aside, we mention that various properties of Markov-modulatedOrnstein-Uhlenbeck processes have been studied by the author and his co-authors in Huang et al. [25]. �ey have a strong focus on explicit computa-tions for transient and stationary moments, and do not have a direct relationto the results presented in this monograph. Hence they are not included inthis thesis.
We close this section by describing the organization of this thesis whichconsists of four chapters. Chapter 1 contains an introduction to basic notionsand theorems of weak convergence and large deviations. We also preciselyde�ne the stochastic processes used in this thesis. Chapter 2, which statesour �rst main result, is devoted to studying the weak convergence of a se-quence of Markov-modulated di�usion processes when the Markov chainis ergodic and rapidly switching. �e content of this chapter has been pub-lished as Huang et al. [26]. Chapter 3, which contains our secondmain result,continues the study of Markov-modulated di�usion processes with rapidlyswitching in the context of large deviations. �is chapter is based on Huanget al. [28]. Chapter 4 explores transient properties of ROU and DROU pro-cesses. �is chapter, which includes our third and fourth main results, hasappeared as Huang et al. [27].
18
Chapter 1
Preliminaries
In this chapter, we lay the theoretical foundations for the rest of the thesis.We do not intend to give a complete introduction to weak convergence andlarge deviations. Instead, we only state a set of de�nitions and theoremsthat are relevant to the results presented in this thesis. We also give precisede�nitions of the stochastic processes used in this thesis.
1.1 Weak convergence
�is section introduces basic notions of weak convergence of probabilitymeasures. For a full treatment, readers can consult classic books of Del-lacherie and Meyer [6], Billingsley [4] and Jacod and Shiryaev [31]. We re-strict our exposition to probability measures on a Polish space X.
LetCb(X) denote the space of real-valued continuous and bounded func-tions onX. Let P(X) be the space of all probability measures on (X,B(X)).�e weak topology (or topology of weak convergence) on P(X) is de�nedas the coarsest topology for which the mappings µ 2 P(X) !
RX fdµ
are continuous for all f 2 Cb(X). �en a sequence of probability measuresµn 2 P(X) is said to converges weakly (or in the weak topology) to a prob-ability measure µ 2 P(X) if and only if
RX fdµn !
RX fdµ as n ! 1, for
all f 2 Cb(X). �e weak topology on P(X) also can be generated by thesets Uf,x,� = {⌫ 2 P(X) :
��RX fd⌫ � x
�� < �} where f 2 Cb(X), � > 0
19
and x 2 R. It is well-known that P(X) is a Polish space if X is Polish.�e Levy-Prohorov metric is one of the metrics compatible with the weaktopology.
Stochastic processes Xnt , Xt de�ned on (⌦
n,Fn, {Fnt }t2R+ ,Pn
) and(⌦,F , {Ft}t2R+ ,P) can be viewed as random variables taking values in X.For example, X can be CR+ . A sequence ofXn
t is said to converge weakly toXt with respect to X if the sequence of probability measures Pn � (Xn
)
�1
converges weakly to P �X�1 in P(X).�e acclaimed Prohorov’s theorem says that a sequence {Pn} in P(X)
is relatively compact for the weak topology if and only if it is tight. Recallthat a sequence {Pn} in P(X) is called tight if for every ✏ > 0 there existsa compact subset K ⇢ X such that Pn(X \ K) 6 ✏ for all Pn. In termsof stochastic processes, an essential way to prove Xn
t converges weakly toXt is to verify the tightness of Pn � (Xn
)
�1 and weak convergence of the�nite-dimensional distributions of Xn
t to those of Xt.Weak convergence of �nite-dimensional distributions is usually proved
by the method of characteristic functions. If Xnt and Xt are Markov pro-
cesses, it also can be proved by convergence of the generators of Xnt to the
generator of Xt. A convenient way to prove tightness is applying Aldous’tightness criterion which we use in Chapter 2.
An extremely rich theory of weak convergence of stochastic processesconcerns semimartingales. Here we present the simplest form of weak con-vergence theorem for one-dimensional semimartingales. Let Xn
t be a se-quence of semimartingales de�ned on (⌦,F , {Fn
t }t2R+ ,P) such thatXn0 =
0. Each one admits the Levy-Ito decomposition (or canonical decomposition)
Xnt = Bn
t +Xnct +
Z t
0
Z
|x|61xd(µn � ⌫n) +
Z t
0
Z
|x|>1xdµn,
where Bnt is a predictable process of locally bounded variation; Xnc
t is thecontinuous local martingale component of Xn
t ; µn is the jump measure ofXn
t and ⌫n is the compensator of µn. Let ↵nt = Bn
t �P
0<s6t�Bns which
is the continuous part of Bnt . Let �nt denote the quadratic variation process
hXncit.We denoteDR+ the space of cadlag functions onR+ with Skorohod’s J1-
topology and its related Borel �-algebra. �en DR+ becomes a Polish space.
20
�e following theorem in Shiryaev [48] shows that weak convergence ofXnt
can be determined by convergence of its triplet of predictable characteristics(Bn
t ,�nt , ⌫
n).
�eorem 1.1.1 Suppose Xt is a continuous Gaussian martingale de�ned on(⌦,F , {Ft}t2R+ ,P). Xn
t converges weakly to Xt with respect to DR+ if thefollowing holds as n ! 1 for each t 2 R+ and ✏ 2 (0, 1]
(A)
Z t
0
Z
|x|>✏⌫n(ds, dx)
P! 0,
(B) sup
0<s6t
�����↵ns +
X
0<u6s
Z
|x|6✏x⌫n({u}, dx)
�����P! 0,
(C)�nt +
Z t
0
Z
|x|<✏x2⌫n(ds, dx)�
X
0<s6t
Z
|x|6✏x⌫n({s}, dx)
!2P! hXit.
It is used in the last section of Chapter 4 for a continuous semimartingale.�en it simpli�es to verify (B) and (C) without terms of ⌫n.
1.2 Large deviationsLarge deviations is a prominent and vast research subject inmodern probabil-ity theory. Interested readers can �nd the general theory of large deviationsin many brilliant books. To name a few, Varadhan [54], den Hollander [24],Dembo and Zeitouni [7], Freidlin and Wentzell [16], Deuschel and Stroock[8]. �ere is also a nice recent survey paper by Varadhan [55]. Another sur-vey paper by Ellis [13] discusses the strong link between the theory of largedeviations and statistical mechanics.
Let us be given a Polish space X with a metric ⇢. A function I(·) : X ![0,1] is called a rate function if there exists x 2 X such that I(x) < 1,and for every 0 6 c < 1, the set {x : I(x) 6 c} is a compact set in X.Here we follow the terminology of Varadhan [54]. It is also called a goodrate function, e.g., in Dembo and Zeitouni [7]. A rate function is obviouslylower semi-continuous. �e central theme in large deviations is to obtain thefollowing large deviations principle for a sequence of probability measures.
21
De�nition 1.2.1 A family of probability measures P✏ on (X,B(X)) is said toobey the LDP with a rate function I(·) if:(1) For every closed set F ⇢ X, lim sup✏!0 ✏ logP✏
(F ) 6 � infx2F I(x).(2) For every open set O ⇢ X, lim inf✏!0 ✏ logP✏
(O) > � infx2O I(x).
�ere is an equivalent de�nition of a LDP which uses action functional.�at o�en appears in the literature on dynamical systems and di�erentialequations. Interested readers can �nd it in Freidlin and Wentzell [16].
A sequence of X-valued stochastic processes X✏t de�ned on stochastic
bases (⌦✏,F ✏, {F ✏t }t2R+ ,P✏
) is said to obey the LDP if P✏ � (X✏)
�1 obeysthe LDP. Large deviations is analogous to weak convergence in the sense thata LDP can be proved by proving exponential tightness in conjunction with alocal LDP. Let us give the de�nitions of those two concepts.
De�nition 1.2.2 (Deuschel and Stroock [8]) A family of probability measuresP✏ on (X,B(X)) is said to be exponentially tight, if for every L < 1, thereexists a compact setKL ⇢ X such that
lim sup
✏!0✏ logP✏
(X \KL) 6 �L.
�eorem (P) in Puhalskii [46] shows the connection between exponentialtightness and a LDP. It states that a family of probability measures P✏ on(X,B(X)) is exponentially tight if and only if any subsequence P✏0 of P✏
contains a further subsequence P✏00 such that P✏00 obeys the LDP with somerate function.
De�nition 1.2.3 (Puhalskii [46], Liptser and Puhalskii [37]) A family of prob-ability measures P✏ on (X,B(X)) is said to obey the local LDP with a functionI(·) if for every x 2 X,
(1.1) lim sup
�!0lim sup
✏!0✏ logP✏
({y 2 X : ⇢(x, y) 6 �}) 6 �I(x),
(1.2) lim inf
�!0lim inf
✏!0✏ logP✏
({y 2 X : ⇢(x, y) 6 �}) > �I(x).
One connection between the local LDP and the LDP is as follows.
22
�eorem 1.2.4 If a sequence of probability measures P✏ on (X,B(X)) satis-�es the LDP with a rate function I , then it satis�es the local LDP with the ratefunction I .
Proof We show the derivation of (1.1) only. Inequality (1.2) can be treatedsimilarly. We de�ne �(s) = {x : I(x) 6 s}. By the property of therate function, we know �(s) is compact for every s > 0. Let x 2 X. IfI(x) = 0, (1.1) is valid automatically. Suppose I(x) > 0. For any � > 0
such that I(x) � � > 0, the compact set �(I(x) � �) does not contain x.Also, their distance ⇢(x,�(I(x)� �)) > 0.We choose a positive � such that� < ⇢(x,�(I(x)� �))/2. �en we have
P✏({y 2 X : ⇢(x, y) 6 �}) 6 P✏
({y 2 X : ⇢(y,�(I(x)� �)) > �}).
We denote {y 2 X : ⇢(y,�(I(x) � �)) > �} by F . �en (2) in De�nition1.2.1 implies lim sup✏!0 ✏ logP✏
(F ) 6 � infy2F I(y). Since ⇢ is continuousin y and � is arbitrary,
lim sup
�!0lim sup
✏!0✏ logP✏
(F ) 6 �I(x).
Hence, (1.1) is proved. ⇤
�e above theorem and its proof are adapted from�eorem 3.3.5 in Frei-dlin and Wentzell [16]. It indicates that a LDP implies its corresponding lo-cal LDP. Also, a sequence of probability measures is exponentially tight ifit obeys the LDP. �e following theorem shows that the converse is true aswell. �e theorem and proof are taken from Liptser and Puhalskii [37].
�eorem 1.2.5 If a family of probability measures P✏ on (X,B(X)) is expo-nentially tight and obeys the local LDP with a function I , then it obeys the LDPwith the rate function I .
Proof By�eorem (P) in Puhalskii [46], if P✏ is exponentially tight, any sub-sequence P✏0 of P✏ contains a further subsequence P✏00 such that P✏00 obeysthe LDP with a rate function ˜I . By �eorem 1.2.4, P✏00 obeys the local LDP
23
with the rate function ˜I . Since we have the following relations
�I(x) 6 lim inf
�!0lim inf
✏!0✏ logP✏
({y 2 X : ⇢(x, y) 6 �})
6 lim inf
�!0lim inf
✏00!0✏00 logP✏00
({y 2 X : ⇢(x, y) 6 �})
6 lim sup
�!0lim sup
✏00!0✏00 logP✏00
({y 2 X : ⇢(x, y) 6 �})
6 lim sup
�!0lim sup
✏!0✏ logP✏
({y 2 X : ⇢(x, y) 6 �}) 6 �I(x),
we conclude ˜I(x) = I(x) and hence I(x) is a rate function.It is le� to prove (1) and (2) in De�nition 1.2.1. We only show the proof
of (2), (1) is proved in the same way. For an open set O ⇢ X, we can choosea subsequence P✏0 of P✏ such that
lim inf
✏!0✏ logP✏
(O) = lim
✏0!0✏0 logP✏0
(O).
Again by�eorem (P) in Puhalskii [46], there exists a subsequence P✏00 of P✏0
such that it obeys the LDP with a rate function ˜I , and ˜I = I . �en by (2) inDe�nition 1.2.1,
lim
✏00!0✏00 logP✏00
(O) > � inf
x2O˜I(x).
Hence we havelim inf
✏!0✏ logP✏
(O) > � inf
x2OI(x),
which proves (1). ⇤
�e following Lemma, which is Lemma 1.4 in Borovkov and Mogulskiı[5], shows that a local LDP on a dense subset of X is enough for the valida-tion of the local LDP on X, provided the rate function possesses a continuityproperty. Its proof is modi�ed from Lemma 1.4 in Borovkov and Mogulskiı[5] to �t our de�nition of the local LDP.
Lemma 1.2.6 Let ˜X be a dense subset of X.(i) If for every x 2 X there exists a sequence xn 2 ˜X converging to x andlim infn!1 I(xn) > I(x), then the ful�llment of (1.1) for all x 2 ˜X impliesthe same for all x 2 X.
24
(ii) If for every x 2 X with I(x) < 1 there exists a sequence xn 2 ˜X con-verging to x and lim supn!1 I(xn) 6 I(x), then the ful�llment of (1.2) forall x 2 ˜X implies the same for all x 2 X.
Proof For every x 2 X, there exists a sequence xn 2 ˜X such that xn ! xas n ! 1. For a �xed � > 0, we can choose all n big enough such that⇢(x, xn) < �.
(i) By the triangle inequality, ⇢(y, xn) 6 ⇢(y, x) + ⇢(x, xn). �en wehave
P✏({y 2 X : ⇢(x, y) 6 �}) 6 P✏
({y 2 X : ⇢(y, xn)) 6 2�}).
Since (1.1) is ful�lled for all xn,
lim sup
�!0lim sup
✏!0✏ logP✏
({y 2 X : ⇢(x, y) 6 �}) 6 �I(xn) 6 � lim inf
n!1I(xn).
Since lim infn!1 I(xn) > I(x), the desired result holds.(ii) �e triangle inequality implies that ⇢(x, y) 6 ⇢(x, xn) + ⇢(xn, y).
�en we have
P✏({y 2 X : ⇢(x, y) 6 2�}) > P✏
({y 2 X : ⇢(y, xn)) 6 �}).
Similarly, the assumption lim supn!1 I(xn) 6 I(x) implies
lim inf
�!0lim inf
✏!0✏ logP✏
({y 2 X : ⇢(x, y) 6 2�}) > �I(xn) > �I(x).
⇤
O�en a LDP can be derived from a known LDP by the following contrac-tion principle. �e theorem and its proof are taken from Dembo and Zeitouni[7]. We follow the convention that inf(;) = 1.
�eorem 1.2.7 Let X,Y be two Polish spaces. Let P✏ on (X,B(X)) be a se-quence of probability measures obeying the LDP with a rate function I(·). Forany continuous function F : X ! Y, the sequence of probability measuresP✏ �F�1 on (Y,B(Y)) obeys the LDP with a rate function J(y) = inf{I(x) :F (x) = y}, 8y 2 Y.
25
Proof We �rst prove that J is a rate function. Since I is a rate function, whichis lower semi-continuous, for any y 2 F (X), the in�mum in the de�nitionof J is obtained at some point in X. �en the level sets of J are
�J(s) = {y 2 Y : J(y) 6 s} = {F (x), x 2 X : I(x) 6 s} = F (�I(s)),
where �I(s) = {x 2 X : I(x) 6 s} is the level set of I . Since �I(s)is compact and F is continuous, �J(s) is also compact. Hence, J is a ratefunction.
Next we prove P✏ � F�1 obeys the LDP with the rate function J(y). LetO be a subset ofY, the continuity property of F implies that F�1
(O) is openor close if F is open or close. �en the following relation
inf
y2OJ(y) = inf
x2F�1(O)I(x)
and the LDP for P✏ imply the desired result. ⇤
A�er the general introduction to the theory of large deviations, we pro-vide two fundamental examples next. A celebrated result in Donsker andVaradhan [9] concerns the long time behavior of occupation measures ofMarkov processes. In particular, the rate function in the LDP for occupationmeasures of �nite-state continuous-time Markov chains is
I(⌫) = sup
u>0
"�
dX
i=1
(Qu)(i)
u(i)⌫(i)
#.
In the above formula, u(i) > 0 for all i. And Q is the transition intensitymatrix and (Qu)(i) =
Pdj=1Qiju(j). We refer readers to den Hollander
[24] for details.Another classical topic, initiated by Freidlin and Wentzell [16], inves-
tigates sample-path large deviations for the following di�usion with smallnoise
dX✏t = b(X✏
t )dt+p✏�(X✏
t )dBt.
Let HT be the Cameron-Martin space on [0, T ] of functions ' 2 CT suchthat '(t) =
R t0 '
0(s)ds and
R T0 ('0
(s))2ds < 1. With certain assumptions
26
which can be found in Dembo and Zeitouni [7], the rate function is
I(') =
8<
:
1
2
Z T
0
['0t � b('t)]
2
�2('t)dt if ' 2 HT ,
1 otherwise.
�ose two rate functions are very well-known. We list them here for thepurpose of comparison with the rate functions in Chapter 3. We also useI(') in the case of OU processes in Chapter 4.
1.3 �e stochastic processes studied in this thesis�roughout this section, a complete stochastic basis (⌦,F , {Ft}t2R+ ,P) is�xed. Let Bt denote a standard Brownian motion de�ned on ⌦ and adaptedto {Ft}t2R+ .
Markov-modulated di�usion processesLet Xt be a �nite-state time-homogeneous Markov chain de�ned on ⌦ withtransition intensity matrix Q and state space S := {1, · · · , d} for somed 2 N. Also, Xt is independent of the standard Brownian motion Bt. �efollowing SDE
Mt = M0 +
Z t
0b(Xs,Ms)ds+
Z t
0�(Xs,Ms)dBs,
has a unique solutionMt onR+ under the assumptions that b(·, ·) and �(·, ·)are locally Liptschitz continuous and grow not faster than linearly, i.e., forany r < 1, there is a positive constantKr such that
|b(i, x)� b(i, y)|+ |�(i, x)� �(i, y)| 6 Kr|x� y|, 8i 2 S, |x|, |y| 6 r;
and there exists a positive constant C such that
|b(i, x)|+ |�(i, x)| 6 C(1 + |x|), 8i 2 S, x 2 R.
�e solutionMt is called the Markov-modulated di�usion process. Its prop-erties have been intensively studied in the literature. Readers can consult
27
excellent books by Skorohod [51], Yin and Zhu [63] and Mao and Yuan [42].�e concept of Markov modulation is also known as regime switching. �eMarkov chainXt is o�en referred to as the background process, or the mod-ulating Markov chain.
Markov-modulated di�usion processes have proven to be particularlyuseful in �nance, as they are able to capture the transition of various eco-nomic states. See for instance the option pricing models of Yao et al. [60]and Jobert and Rogers [32] using Markov-modulated geometric Brownianmotions, and the term structure model of Ellio� and Siu [12] using Markov-modulated versions of theOrnstein-Uhlenbeck andCox-Ingersoll-Ross types.�is kind of models allows the volatility and rate of returns to be random,and it turns out that parameters can be relatively easily calibrated on realdata.
Re�ected di�usion processesWe give the de�nition of one-dimensional SDEswith re�ecting boundary
conditions. Let D� 2 R be an open interval, @D and D denote its boundaryand closure. Let ⌫(x) denote the function giving the inward normal at x 2@D, i.e. ⌫(x) = 1 if x is a �nite le� endpoint of D and ⌫(x) = �1 if xis a �nite right endpoint of D. We assume that b(·) and �(·) are uniformlyLipschitz continuous. �en there exists a unique solution (Xt, ⇠t) 2 D ⇥ Rto the following equations
Xt = x0 +R t0 b(Xs)ds+
R t0 �(Xs)dBs + ⇠t, x0 2 D,(1.3)
|⇠|t =R t0 1@D(Xs)d|⇠|s, ⇠t =
R t0 ⌫(Xs)d|⇠|s.
Here |⇠|t denotes the total variation of ⇠t by time t. �e process Xt is calledthe re�ected di�usion process with domain D�. Both Xt and ⇠t are contin-uous processes. And ⇠t is of locally bounded variation and changes valuesonly when Xt 2 @D.
Existence and uniqueness of solutions to re�ected SDE’s is closely relatedto the deterministic Skorohod problem. It was �rst proved by Skorohod in[50] when the domain is the half-line. Tanaka in [53] generalized the resultto convex domains in Rd, and Lions and Sznitman in [36] further extendedit to admissible domains.
28
Ornstein-Uhlenbeck processes�e OU process is de�ned as the unique solution to the SDE:
dXt = (↵� �Xt)dt+ �dBt, X0 = x 2 R,
where ↵ 2 R, �,� > 0. �e choice � > 0 is only made for de�niteness,from a distributional point of view nothing changes if it is replaced with��.�e OU process is mean-reverting towards the value ↵/�. It allows a highdegree of analytical tractability since it is a Gaussian process. As a result,it has wide-spread use in a broad range of application domains, such as �-nance, life sciences, and operations research. For example, it is known as theVasicek model in �nance to model short rates.
Re�ected Ornstein-Uhlenbeck processesIn this thesis, the ROU process is de�ned to have re�ection at a lower
boundary 0. It satis�es the following SDE where we additionally assume↵ > 0,
dYt = (↵� �Yt)dt+ �dBt + dLt, Y0 = x > 0,
whereLt could be interpreted as an idle process. More precisely,Lt is de�nedas the minimal nondecreasing process such that Yt > 0 for t > 0. It holdsthat
R[0,T ] 1{Yt
>0}dLt = 0 for any T > 0.Ward and Glynn [57, 58, 59] show that the ROU process can be used to
approximate the number-in-system processes in M/M/1 and GI/GI/1 queueswith reneging under scalings and the DROU process can be seen as an ap-proximation of the associated �nite-bu�er queue. Srikant andWhi� [52] alsoshow that the number-in-system process in a GI/M/n loss model can be ap-proximated by ROU processes. For other applications, we refer to e.g. theintroduction in Giorno et al. [18] and references therein.
Doubly-re�ected Ornstein-Uhlenbeck processesLikewise, DROU processes with re�ection at two boundaries can be con-
structed through the SDE
dZt = (↵� �Zt)dt+ �dBt + dLt � dUt, Z0 = x 2 [0, d],
29
whereR[0,T ] 1{Zt
>0}dLt = 0 and Ut is the loss process at the boundary d,i.e.,
R[0,T ] 1{Zt
<d}dUt = 0 for any T > 0. In the context of queues with�nite-capacity, Ut is the continuous analog to the cumulative amount of lossover [0, t], and that explains why we refer to it as the ‘loss process’.
30
Chapter 2
Weak convergence ofMarkov-modulated di�usionprocesses with rapidswitching
2.1 Introduction
In this chapter we study the limiting behavior of the Markov-modulated dif-fusion process when the modulating Markov chain is rapidly switching andergodic. Let Xt be a �nite-state time-homogeneous Markov chain with atransition intensity matrix Q and a state space S := {1, · · · , d} for somed 2 N. We speed up the process Xt by a factor n, and consider the result-ing system as n ! 1. In more concrete terms, it means that we replace thetransition rate matrix Q by nQ, and we denote Xn
t the sequence of rapidlyswitchingMarkov chains. We let ⇡ be the ergodic distribution correspondingtoXt, and hence also toXn
t for every n 2 N. Each processMnt is de�ned on
a complete stochastic basis (⌦n,Fn, {Fnt }t2R+ ,Pn
) through the SDE
(2.1) Mnt = m0 +
Z t
0b(Xn
s ,Mns )ds+
Z t
0�(Xn
s ,Mns )dB
ns , Xn
0 = x0,
31
with x0 2 S and Bnt a Brownian motion. �e Markov chain Xn
t is inde-pendent of the Brownian motion Bn
t . In order to make the notation in thesequel not unnecessarily heavy, we drop the dependence on n everywherewhere it doesn’t lead to confusion, e.g. we simply use P for probability, E forexpectation and B for Brownian motion.
Processes with rapid switching have been studied in great detail by Sko-rohod [51], where it is noted that his analysis covers models that are moregeneral than the one introduced above. In particular, �eorem 2.8 in Skoro-hod [51] implies that, as n ! 1, the sequence of the Markov-modulateddi�usion processes {Mn
t }t2[0,T ] converges, in terms of �nite-dimensionaldistributions, to a (non-modulated) di�usion process { ˆMt}t2[0,T ] whose co-e�cients are obtained by averaging the coe�cients of Mn
t with respect tothe ergodic distribution ⇡.
�ere is also a large body of literature devoted to asymptotic behav-ior of Markov-modulated (di�usion) processes. For example, Yin and Zhou[62] and Nguyen and Yin [43] studied the weak convergence of Markov-modulated random sequences where the modulating Markov chains and themodulated processes are both indexed by integers. �eir continuous-timeinterpolations under time scaling are proved to converge weakly to switch-ing di�usion processes. �e result was applied to construct asymptoticallyoptimal portfolios for discrete-time models of Markowitz’s mean-varianceportfolio selection problem in Zhou and Yin [65] and Yin and Zhou [62].
�e main result in this chapter is a proof of the tightness of {Mnt }t2[0,T ].
�at, combined with convergence of �nite-dimensional distributions men-tioned above, implies the weak convergence of {Mn
t }t2[0,T ] to { ˆMt}t2[0,T ].In our proof, Aldous’ tightness criterion in Aldous [1] is the main ingredi-ent, in combination with a technique which is used to prove the exponentialtightness of two scaled di�usions in Liptser [37].
2.2 Main result
We �rst introduce some notation. For an arbitrary stochastic processHt, wede�ne H⇤
t := sups6t |Hs|. We denote DR+ the space of cadlag functions onR+ with Borel �-algebra, equipped with Skorohod’s J1-topology. We then
32
impose some assumptions on the coe�cients of (2.1).
(A.I) Local Lipschitz continuity: for any r < 1, there is a positive constantKr such that
|b(i, x)� b(i, y)|+ |�(i, x)� �(i, y)| 6 Kr|x� y|, 8i 2 S, |x|, |y| 6 r.
(A.II) Linear growth: there exists a positive constant C such that
|b(i, x)|+ |�(i, x)| 6 C(1 + |x|), 8i 2 S, x 2 R.
(A.III) Xnt is ergodic with the ergodic distribution ⇡ = (⇡1, ...,⇡d) for every
n 2 N.Assumptions (A.I) and (A.II) guarantee a unique solutionMn
t of (2.1), andthe following estimate for �xed T > 0 and n 2 N:
(2.2) E(Mn⇤T )
� 6 K(T, �) < 1, 8� > 0;
cf. Yin and Zhu [63]. �e following result is our main result of weak conver-gence.
�eorem 2.2.1 If Assumptions (A.I)-(A.III) hold, then as n ! 1, Mnt con-
verges weakly with respect to C[0,T ] to ˆMt, which is the solution of the SDE
ˆMt = m0 +
Z t
0
ˆb( ˆMs)ds+
Z t
0�( ˆMs)dBs,
where
ˆb(x) :=dX
i=1
b(i, x)⇡i, �(x) :=
dX
i=1
�2(i, x)⇡i
!1/2
.
As we mentioned in Section 1.1, proofs of this type of weak conver-gence results typically consist of two parts: (i) proving that all the �nite-dimensional convergence of {Mn
t }t2[0,T ] to { ˆMt}t2[0,T ], and (ii) proving thetightness ofMn
t inC[0,T ]. Recall from Section 2.1 that the �rst part is a conse-quence of�eorem 2.8 in Skorohod [51] whose proof is based on the methodof generators. It needs to be pointed out that in Skorohod [51] the term con-vergence in distribution is to be understood as �nite-dimensional convergence(cf. page 77 of Skorohod [51]). Our contribution, as given in the next section,is the proof of the tightness part.
33
2.3 Proof of tightness�e proof of tightness relies on the following theorem, to which we refer asAldous’ tightness criterion, see Aldous [1], or �eorem 6.3.1 in Liptser andShiryaev [40] . For T > 0, let �T (Fn
t ) denote the family of stopping timesadapted to Fn
t taking values in [0, T ].
�eorem 2.3.1 LetHn= {Hn
t }t>0 be a sequence inDR+ de�ned on a stochas-tic basis (⌦,F ,Fn
t ,P). If (i) for all T > 0,
lim
K!1lim sup
n!1P✓sup
t6T|Hn
t | > K
◆= 0,
and (ii) for all T > 0 and ⌘ > 0,
lim
�#0lim sup
n!1sup
⌧2�T
(Fn
t
)P✓sup
t6�|Hn
⌧+t �Hn⌧ | > ⌘
◆= 0,
then the sequence Hn is tight.
It is well-known that Skorohod’s J1-topology and the uniform topologyare equivalent on the space of continuous functions. SinceMn
t is a sequenceof continuous processes, {Mn
t }t2[0,T ] is tight in C[0,T ] if we prove that itsatis�es the above two conditions for any T > 0. We start with introducingsome additional notation. Let ⌧ 2 �T (Fn
t ). �en we de�ne
Cnt :=
Z t
0�(Xn
s ,Mns )dBs,
Y nt :=
Z ⌧+t
⌧�(Xn
s ,Mns )dBs,
Qnt :=
Z ⌧+t
⌧b(Xn
s ,Mns )ds.
We �rst establish an auxiliary result.
Lemma 2.3.2 For all T > 0, ⌘ > 0, and K > 0,
lim
�#0lim sup
n!1sup
⌧2�T
(Fn
t
)P✓sup
t6�|Y n
t | > ⌘,Mn⇤T+1 6 K
◆= 0.
34
Proof By (A.II) and estimate (2.2), for any �xed n and all T > 0,
EZ T
0�2(Xn
s ,Mns )ds 6 E
Z T
0C2
⇥2 + 2|Mn
s |2⇤ds
6 2C2T + 2T (K(T, 2))2 < 1.
Conclude thatCnt is a square-integrable continuous martingale. Also, Y n
t is acontinuous martingale adapted to (Fn
⌧+t)t>0 due to�eorem 4.7.1 in Liptserand Shiryaev [40]. We therefore �nd that
Znt := exp
✓�Y n
t � �2
2
hY nit◆, � 2 R,
is a positive local martingale, and hence a supermartingale. So EZn� 6 1 for
any stopping time �. Take � := inf{t > 0 : |Y nt | > ⌘}. It follows that
P✓sup
t6�|Y n
t | > ⌘,Mn⇤T+1 6 K
◆= P
�|Y n
� | > ⌘,� 6 �,Mn⇤T+1 6 K
�
= P(Y n� > ⌘,� 6 �,Mn⇤
T+1 6 K) + P(Y n� 6 �⌘,� 6 �,Mn⇤
T+1 6 K).
When Mn⇤T+1 6 K , 0 6 � 6 � 6 1 and ⌧ 2 �T (Fn
t ), Assumption (A.II)implies
hY ni� =
Z ⌧+�
⌧�2(Xn
s ,Mns )ds 6 C2
(1 +K)
2�.
Denote CK := C2(1 +K)
2, we have for any � > 0,
exp
✓�⌘ � �2
2
CK�
◆1(Y n
� > ⌘,� 6 �,Mn⇤T+1 6 K)
6 Zn� 1(Y n
� > ⌘,� 6 �,Mn⇤T+1 6 K),
almost surely. As this holds for any � > 0, we take the supremum of the le�-hand side in the previous display. Recalling EZn
� 6 1, taking expectationsyields
exp
✓sup
�>0
✓�⌘ � �2
2
CK�
◆◆P(Y n
� > ⌘,� 6 �,Mn⇤T+1 6 K)
6 EZn�1(Y
n� > ⌘,� 6 �,Mn⇤
T+1 6 K) 6 1.
35
Upon rewriting the above equation, we obtain
P(Y n� > ⌘,� 6 �,Mn⇤
T+1 6 K)
6 exp
✓� sup
�>0
✓�⌘ � �2
2
CK�
◆◆= exp
✓� ⌘2
2CK�
◆.
It thus follows that, for all T > 0, ⌘ > 0, and K > 0,
lim
�#0lim sup
n!1sup
⌧2�T
(Fn
t
)P(Y n
� > ⌘,� 6 �,Mn⇤T+1 6 K) = 0.
Moreover,
lim
�#0lim sup
n!1sup
⌧2�T
(Fn
t
)P(Y n
� 6 �⌘,� 6 �,Mn⇤T+1 6 K) = 0
is proved in the same way by considering � < 0. We have thus establishedthe claim. ⇤
We are now ready to prove the tightness claim.
Proposition 2.3.3 For all T > 0, the sequence {Mnt }t2[0,T ] is tight in C[0,T ].
Proof Firstly, we verify the condition (i) of�eorem 2.3.1. For any T > 0,
sup
t6T|Mn
t | 6 |m0|+ sup
t6T
Z t
0|b(Xn
s ,Mns )|ds+ sup
t6T
����Z t
0�(Xn
s ,Mns )dBs
���� ,
almost surely. By (A.II),
Mn⇤T 6 |m0|+ C
Z T
0(1 +Mn⇤
s )ds+ Cn⇤T ,
almost surely. Since |m0|+CT +Cn⇤T is nonnegative and non-decreasing in
T , Gronwall’s inequality implies
(2.3) Mn⇤T 6 eCT
[|m0|+ CT + Cn⇤T ] ,
36
almost surely. Now de�ne jK := K exp(�CT ) � |m0| � CT . �en (2.3)entails that for su�ciently large K such that jK > 0,
P(Mn⇤T > K) 6 P(Cn⇤
T > jK) 6 j�2K E(Cn⇤
T )
2,
using Chebyshev’s inequality. By (A.II) again, we have
�2(Xns ,M
ns ) 6 C2
(1 +Mn⇤s )
2.
Since (2.3) remains valid with replacing T by s for any s 6 T , we �nd
�2(Xns ,M
ns ) 6 C2
[1 + eCs(|m0|+ Cs+ Cn⇤
s )]
2
6 C2[1 + eCT
(|m0|+ CT + Cn⇤s )]
2
6 C2[2
�1 + eCT |m0|+ eCTCT
�2+ 2e2CT
(Cn⇤s )
2]
6 LT [1 + (Cn⇤s )
2],
where LT is a positive constant not depending onK . Applying Doob’s max-imal inequality on Cn
t and plugging in the above estimate, we obtain
E(Cn⇤T )
2 6 4E✓Z T
0�2(Xn
s ,Mns )ds
◆6 4LTT + 4LT
Z T
0E(Cn⇤
s )
2ds
6 4LTT e4LT
T .
�e last inequality follows from Gronwall’s inequality. Hence, for all T > 0,
(2.4) lim
K!1lim sup
n!1P (Mn⇤
T > K) = 0.
Secondly, we verify part (ii) of �eorem 2.3.1. To this end, note that forarbitrary T > 0, � 6 1, and stopping time ⌧ 2 �T (Fn
t ),
P✓sup
t6�|Mn
⌧+t �Mn⌧ | > ⌘
◆
6 P✓sup
t6�|Mn
⌧+t �Mn⌧ | > ⌘,Mn⇤
T+1 6 K
◆+ P(Mn⇤
T+1 > K)
6 P⇣Qn⇤
� > ⌘
2
,Mn⇤T+1 6 K
⌘+ P
⇣Y n⇤� > ⌘
2
,Mn⇤T+1 6 K
⌘(2.5)
+ P(Mn⇤T+1 > K).
37
We show that the three terms in the right-hand side of (2.5) converge to 0in the right way. We start with the �rst term. By virtue of Chebyshev’sinequality and (A.II),
P⇣Qn⇤
� > ⌘
2
,Mn⇤T+1 6 K
⌘
6 P✓Z ⌧+�
⌧|b(Xn
s ,Mns )|ds >
⌘
2
,Mn⇤T+1 6 K
◆
6 4
⌘2E✓Z ⌧+�
⌧|b(Xn
s ,Mns )|ds
◆2
1(Mn⇤T+1 6 K)
6 4
⌘2E✓Z ⌧+�
⌧C(1 + |Mn
s |)ds◆2
1(Mn⇤T+1 6 K)
6 4
⌘2C2
(1 +K)
2�2.
As a result,
lim
�#0lim sup
n!1sup
⌧2�T
(Fn
t
)P⇣Qn⇤
� > ⌘
2
,Mn⇤T+1 6 K
⌘= 0.
�e second term in the right-hand side of (2.5) converges to 0 in the desiredway as well; by Lemma 2.3.2,
lim
�#0lim sup
n!1sup
⌧2�T
(Fn
t
)P⇣Y n⇤� > ⌘
2
,Mn⇤T+1 6 K
⌘= 0.
�e desired convergence of the third term on the right-hand side of (2.5)follows directly from (2.4). Upon combining the above, we have found that,for all T > 0 and ⌘ > 0,
lim
�#0lim sup
n!1sup
⌧2�T
(Fn
t
)P✓sup
t6�|Mn
⌧+t �Mn⌧ | > ⌘
◆= 0.
We have proven the claim. ⇤
38
2.4 Examples
�e sequence ofMarkov-modulatedOrnstein-Uhlenbeck processes {Mnt }t>0
with rapid switching is de�ned as
Mnt = m0 +
Z t
0(↵(Xn
s )� �(Xns )M
ns ) ds+
Z t
0�(Xn
s )dBs,
where ↵, �,� are functions on S and n 2 N. By �eorem 2.2.1, it convergesweakly with respect to C[0,T ] to the (non-modulated) Ornstein-Uhlenbeckprocess { ˆMt}t>0, characterized by
ˆMt = m0 +
Z t
0
⇣↵� � ˆMs
⌘ds+
Z t
0�dBs,
where
↵ :=
dX
i=1
↵(i)⇡i, � :=
dX
i=1
�(i)⇡i, � =
dX
i=1
�2(i)⇡i
!1/2
.
A second example is that of a sequence of Markov-modulated geometricBrownianmotions under rapid switching. �e processes {Sn
t }t>0 are de�nedby, for n 2 N,
Snt = s0 +
Z t
0µ(Xn
s )Sns ds+
Z t
0�(Xn
s )Sns dBs.
�en �eorem 2.2.1 implies that the weak limit of the {Snt }t>0, as n ! 1,
is a (non-modulated) geometric Brownian motion { ˆSt}t>0, which is de�nedas
ˆSt = s0 +
Z t
0µ ˆSsds+
Z t
0� ˆSsdBs,
where
µ :=
dX
i=1
µ(i)⇡i, � :=
dX
i=1
�2(i)⇡i
!1/2
.
39
Next we simulate sample paths of Markov-modulated geometric Brown-ian motions to build up a bit intuition. LetXn
t be a Markov chain with statespace {0, 1, 2}. When n = 1, the transition intensity matrix is
Q =
2
4�5 2 3
1 �2 1
5 5 �10
3
5 .
When n = 100 and n = 10, 000, the transition intensity matrices are 100Qand 10, 000Q respectively. We de�ne µ(0) = �(0) = �2, µ(1) = �(1) =
0.2, µ(2) = �(2) = 2. �en the following �gures are sample paths of S1t ,
S100t , S10,000
t on [0, 1] respectively. �e simulation code is modi�ed from theone in Iacus [29].
Time
S
0.0 0.2 0.4 0.6 0.8 1.0
23
45
6
Figure 2.1: a sample path of S1t on [0, 1]
From Figure 2.1, transitions of states of the Markov chain and the cor-responding di�usions are obvious. �ose become di�cult to distinguish inFigure 2.2. �e sample path of S10000
t looks verymuch like the one of an ordi-nary di�usion in Figure 2.3. �is illustrates our main result from an intuitiveperspective. On the other hand, it also raises a question: how do we dis-
40
Time
S1
0.0 0.2 0.4 0.6 0.8 1.0
24
68
Figure 2.2: a sample path of S100t on [0, 1]
Time
S2
0.0 0.2 0.4 0.6 0.8 1.0
46
810
1214
16
Figure 2.3: a sample path of S10000t on [0, 1]
41
tinguish a Markov-modulated di�usion process from an ordinary di�usionwhen the background chain switches rapidly?
�is question will be partially answered in the next chapter, where westudy large deviations, by exploring the impact of a fast switching back-ground chain on the small noise asymptotics of Markov-modulated di�usionprocesses. �e rate function in the LDP for the Markov-modulated di�usionprocess is di�erent from the one of the ordinary di�usion process.
42
Chapter 3
Large deviations forMarkov-modulated di�usionprocesses with rapidswitching
3.1 Introduction
�e se�ing in this chapter is a complete stochastic basis (⌦,F , {Ft}t2R+ ,P).Like in the previous chapter, we denoteXt a �nite-state time-homogeneousMarkov chain with a transition intensity matrix Q and a state space S :=
{1, · · · , d} for some d 2 N. For a strictly positive ✏, we scaleQ toQ/✏ =: Q✏,and denote byX✏
t the Markov chain with this transition intensity matrixQ✏.We also make a scaling of the function �(· , · ) to
p✏�(· , · ) in the Markov-
modulated di�usion. �e resulting process M ✏t is de�ned as the unique so-
lution to
(3.1) M ✏t = M ✏
0 +
Z t
0b(X✏
s,M✏s)ds+
p✏
Z t
0�(X✏
s,M✏s)dBs,
where Bt is a standard Brownian motion which is independent of X✏t . �e
Markov chain X✏t is independent of the Brownian motion Bt for all ✏.
43
In this chapter, we investigate how rapid-switching behavior of X✏t af-
fects the small-noise asymptotics of X✏t -modulated di�usion processes on
the interval [0, T ] (for any �xed strictly positive T ). For simplicity, we willassume throughout this chapter that M ✏
0 ⌘ 0, whereas X✏0 starts at an arbi-
trary x 2 S, for all ✏. When we write e.g. E[M ✏t ], this is to be understood as
the expectation ofM ✏t with the above initial conditions. We also assume that
there exist i, x such that �(i, x) 6= 0.Recall from our reasoning in Overview at the beginning of the thesis, the
main idea is to construct a coupling (M ✏, ⌫✏) to separate the e�ects of thevanishing of the di�usion term and the fast varying of the Markov chain, butat the same time to keep track of both of them. As mentioned in Overview,⌫✏ denotes the occupation measure ofX✏
t , which is de�ned on ⌦⇥ [0, T ]⇥Sas
(3.2) ⌫✏(!; t, i) =
Z t
01{X✏
s
(!)=i}ds.
Moreover, we can use the derivative of ⌫✏(t) to gauge the in�nitesimal changeof the occupation measure of X✏
t , at any t 2 [0, T ]. �en the coupling(M ✏, ⌫✏) is the main object studied in this chapter.
A well-known result in Donsker and Varadhan [9] concerns the LDP for⌫1(!; t, ·)/t as t ! 1, i.e., the LDP of the fraction of time spent in the indi-vidual states of the background process. In contrast, we consider the sample-path LDP for ⌫✏ on [0, T ] as ✏ ! 0. More precisely, we de�ne the imagespace MT of ⌫✏ restricted on [0, T ] as the space of functions ⌫ on [0, T ]⇥ Ssatisfying ⌫(t, i) =
R t0 K⌫(s, i)ds, where
Pdi=1K⌫(s, i) = 1, K⌫(s, i) > 0
for every i 2 S, s 2 [0, T ], andK⌫(s, i) being Borel measurable with respectto s; K⌫ is referred to as the kernel of ⌫. �e metric onMT is de�ned as
dT (µ, ⌫) = sup
06t6T,i2S
����Z t
0Kµ(s, i)ds�
Z t
0K⌫(s, i)ds
���� .
We can also view MT as a subset of C[0,T ](Rd) which is the space of Rd-
valued continuous functions on [0, T ]. In addition, the metric dT on MT isthe same as the uniform metric on C[0,T ](Rd
) restricted toMT .We also de�ne CT as the image space of M ✏, which is the space of
functions f 2 C[0,T ](R) and f(0) = 0 equipped with the uniform metric
44
⇢T (f, g) := sup06t6T |f(t)�g(t)|.�e product metric ⇢T ⇥dT onCT ⇥MT
is de�ned for (', ⌫), ('0, ⌫ 0) 2 CT ⇥MT by
(⇢T ⇥ dT )((', ⌫), ('0, ⌫ 0)) := ⇢T (','
0) + dT (⌫, ⌫
0).
We denote by B(CT ⇥MT ) the Borel �-algebra generated by the topologyinduced by ⇢T ⇥ dT .
�e main result of this chapter is the joint sample-path LDP for (M ✏, ⌫✏)onCT ⇥MT . �e associated (joint) large deviations rate function is obtainedin quite an explicit form. It is actually the sum of two expressions that weintroduce later, viz. (3.4), i.e., the rate function IT (', ⌫) corresponding toM ✏,and (3.3), i.e., the rate function ˜IT (⌫) corresponding to ⌫✏.
As mentioned in Overview and Section 1.2, the proof which consists ofproving exponential tightness and the local LDP was inspired by (and someparts resemble) Liptser [37]. Recall that the di�usion process is modulated byanother di�usion process in Liptser [37], but our modulating process is the�nite-state Markov chain. Also recall from Overview that a key to prove thelocal LDP is the relationship among the rate function ˜IT (⌫), a dense subset ofMT and the continuous properties of ˜IT (⌫) and IT {', ⌫}. We give the detailsin Section 3.4. Another pivotal role in proving the local LDP is the followingstochastic exponential which is directly related to the Markov chainX✏
t andits rate function ˜IT (⌫) (as given in (3.3)):
u(t,X✏t )
u(0, X✏0)
exp
�Z t
0
@@su(s,X
✏s) + (Q✏ u)(s,X✏
s)
u(s,X✏s)
ds
!, u 2 U,
where U denote the space of functions on [0, T ]⇥ S being continuously dif-ferentiable on [0, T ] and infs2[0,T ],i2S u(s, i) > 0. We follow the conventionthat (Q✏u)(s, i) =
Pdj=1Q
✏ij u(s, j), for i 2 S. See Section 3.5 for more
details.�e LDPs for each componentM ✏ and ⌫✏ are then derived as corollaries
from the joint LDP for (M ✏, ⌫✏) in the standard way, i.e., by an applicationof the contraction principle (�eorem 1.2.7).
Next we impose some assumptions on the SDE (3.1). It is noted that (A.1)(‘Lipschitz continuity’) implies (A.2) (‘linear growth’). We chose to include(A.2) as well for ease of reference in later sections.
45
(A.1) Lipschitz continuity: there is a positive constantK such that
|b(i, x)� b(i, y)|+ |�(i, x)��(i, y)| 6 K|x� y|, 8i 2 S, x, y 2 R.
(A.2) Linear growth: there exists a positive constantK (which might be dif-ferent from theK used in (A.1)) such that
|b(i, x)|+ |�(i, x)| 6 K(1 + |x|), 8i 2 S, x 2 R.
(A.3) Irreducibility: the o�-diagonal entries of the transition intensity matrixQ are strictly positive. Hence, the Markov chain X✏
t is irreducible forall ✏ and has an invariant probability measure ⇡ = (⇡(1), · · · ,⇡(d)).
Now we introduce some extra notation and function spaces. For an arbi-trary stochastic process or a function Yt, we denote the running maximumprocess by Y ⇤
t := sups6t |Ys|. For a semimartingale Yt such that Y0 = 0,its stochastic exponential is de�ned as a semimartingale E (Y )t which is theunique strong solution to
E (Y )t = 1 +
Z t
0E (Y )s�dYs.
Recall that we denote HT the Cameron-Martin space of functions ' 2 CT
such that '(t) =R t0 '
0(s)ds and '0 is square-integrable on [0, T ]. We call '0
the derivative of '.�e large deviations analysis forMarkov-modulation stochastic processes
is currently an active research �eld. Besides the previouslymentioned papersof He et al. [23] and He and Yin [22] in Overview, we list a few more. Guillin[19] proved the averaging principle (moderate deviations) of Equation (3.1)where X✏
t is an exponentially ergodic Markov process and b,� are boundedfunctions. He and Yin [21] studied the moderate-deviations behavior of M ✏
t
in the SDE (3.1), where � ⌘ 0 and X✏t is a non-homogeneous Markov chain
with two time-scales. Lasry and Lions [34] and Fournie et al. [15] consid-ered large deviations for the hi�ing times of Markov-modulated di�usionprocesses with rapid switching.
�e structure of this chapter is as follows. In Section 3.2, we state themain result and explain the steps of its proof. In Section 3.3, exponential
46
tightness of (M ✏, ⌫✏) is veri�ed. We identify a dense subset of CT ⇥MT inSection 3.4, and explore regularity properties of the rate function on it. �eupper bound and lower bound of the local LDP for (M ✏, ⌫✏) are proved inSections 3.5 and 3.6 respectively. We present a number of technical lemmasin Section 3.7.
3.2 Main resultsWe �rst introduce the de�nitions of the rate functions involved in the mainresult. �e rate function corresponding to ⌫✏ is de�ned as
(3.3) ˜IT (⌫) :=
Z T
0sup
u2U
"�
dX
i=1
(Qu)(i)
u(i)K⌫(s, i)
#ds, ⌫ 2 MT ,
where we recall the notation (Qu)(i) =
Pdj=1Qiju(j), for i 2 S, and U
denotes the set of d-dimensional component-wise strictly positive vectors.We now de�ne the rate function corresponding to M ✏. For any (', ⌫) 2CT ⇥MT , we de�ne
(3.4) IT (', ⌫) :=
8<
:
1
2
Z T
0
['0t � ˆb(⌫,'t)]
2
�2(⌫,'t)dt if ' 2 HT ,
1 otherwise.
where
ˆb(⌫, x) :=dX
i=1
b(i, x)K⌫(t, i), �(⌫, x) :=
dX
i=1
�2(i, x)K⌫(t, i)
!1/2
.
In the above formulae, we follow the conventions that 0/0 = 0 and n/0 =
1, for all n > 0. When we �x a time T , (M ✏, ⌫✏) is understood as a jointprocess restricted on [0, T ]. Let P � (M ✏, ⌫✏)�1 denote P((M ✏, ⌫✏) 2 ·),which is a family of probability measures on (CT ⇥ MT ,B(CT ⇥ MT )).Also, P � (M ✏
)
�1 and P � (⌫✏)�1 are families of probability measures on(CT ,B(CT )) and (MT ,B(MT )) respectively. �e following theorem is ourmain result which states the joint sample-path LDP of (M ✏, ⌫✏) on [0, T ], as✏! 0.
47
�eorem 3.2.1 For every T > 0, the family P � (M ✏, ⌫✏)�1 obeys the LDP in(CT ⇥MT , ⇢T ⇥ dT ) with the rate function
LT (', ⌫) = IT (', ⌫) + ˜IT (⌫).
Proof �e proof relies on applying�eorem 1.2.5. We �rst need to prove theexponential tightness of P � (M ✏, ⌫✏)�1 on (CT ⇥MT ,B(CT ⇥MT )), i.e.,for every L < 1, there exists a compact setKL ⇢ CT ⇥MT such that
lim sup
✏!0✏ logP ((M ✏, ⌫✏) 2 CT ⇥MT \KL) 6 �L.
It is obvious thatP�(M ✏, ⌫✏)�1 is exponentially tight if so areP�(M ✏)
�1 andP � (⌫✏)�1. As we mentioned earlier, MT is a subset of C[0,T ](Rd
). For any⌫ 2 MT , its derivative K⌫(s, i) is bounded by 1. �en all ⌫ 2 MT have thesame Lipschitz constant, and hence MT is equicontinuous. It is easily seenthatMT is bounded and closed. �en the Arzela-Ascoli theorem implies thatMT is compact. �e exponential tightness of P � (⌫✏)�1 is satis�ed sincewe can take KL = MT . Exponential tightness of P � (M ✏
)
�1 is veri�ed inProposition 3.3.3 below.
Secondly, we proceed to prove that P � (M ✏, ⌫✏)�1 obeys the local LDPwith the rate function LT (', ⌫). �at is, for every (', ⌫) 2 CT ⇥ MT , weneed to obtain the upper bound
lim sup
�!0lim sup
✏!0✏ logP(⇢T (M ✏,') + dT (⌫
✏, ⌫) 6 �) 6 �LT (', ⌫),
and the lower bound
lim inf
�!0lim inf
✏!0✏ logP(⇢T (M ✏,') + dT (⌫
✏, ⌫) 6 �) > �LT (', ⌫).
�e core of the proof is proving the local LDP on a dense subset ofCT ⇥MT .�e upper bound is validated in Proposition 3.5.4. �e lower bound is �rstproved in Proposition 3.6.3 given the condition infi,x �2(i, x) > 0. �en thecondition is li�ed in Proposition 3.6.5 by a perturbation argument. ⇤
�e LDP for P � (M ✏)
�1 only is then derived from �eorem 3.2.1 by thecontraction principle (�eorem 1.2.7). Recall that we follow the conventionthat inf(;) = 1.
48
Corollary 3.2.2 �e family P � (M ✏)
�1 obeys the LDP with the rate functioninf⌫2M
T
LT (', ⌫).
At an intuitive level, ˜IT (⌫) can be interpreted as the ‘cost’ of forcing ⌫✏to behave like ⌫ on [0, T ]. �e other term IT (', ⌫), can be seen as the ‘cost’of the sample paths of M ✏ being close to ' conditional on ⌫✏ behaving like ⌫on [0, T ]. �en inf⌫2M LT (', ⌫) indicates the minimal ‘cost’ of the samplepaths of M ✏ being close to ' on [0, T ].
Suppose F is a set of sample paths of M ✏ and is a closed subset of CT .We can also interpret Corollary 3.2.2 as the concentration of the probabil-ity P � (M ✏
)
�1(F ) on the ‘most likely path’ arg inf'2F (inf⌫2M
T
[IT (', ⌫)+˜IT (⌫)]). So there are two sources contributing to the large deviations be-havior of M ✏: IT (', ⌫) represents the contribution resulting from the smallnoise, and ˜IT (⌫) represents the one from the rapid switching of the modu-lating Markov chain.
Again by the contraction principle, P�(⌫✏)�1 obeys the LDP in (MT , dT )with the rate function inf'2C
T
IT (', ⌫) + ˜IT (⌫). Since b(i, ·) is Lipschitzcontinuous, there exists a solution ' 2 HT of '0
t =
ˆb(⌫,'t) for all ⌫ 2MT . It immediately follows that inf'2C
T
IT (', ⌫) = 0. Hence, we have thefollowing corollary.
Corollary 3.2.3 �e family P � (⌫✏)�1 obeys the LDP in (MT , dT ) with therate function ˜IT (⌫).
3.3 Exponential tightness
We show the exponential tightness of P�(M ✏)
�1 by Aldous-Pukhalskii-typesu�cient conditions, as dealt with in e.g. Aldous [1], Liptser and Pukhalskii[39]. �e following criterion for exponential tightness in CT , as well as anauxiliary lemma, are adapted from�eorem 3.1 and Lemma 3.1 in Liptser andPukhalskii [39]. �ey consider cadlag processes, but our se�ing is continuousprocesses. Let �T (Ft) denote the family of stopping times adapted to Ft
taking values in [0, T ].
�eorem 3.3.1 Let Y ✏t : (⌦, {Ft}t6T ,P) ! CT . If
49
(i)lim
K0!1lim sup
✏!0✏ logP
�Y ✏⇤T > K 0�
= �1,
(ii)
lim
�!0lim sup
✏!0✏ log sup
⌧2�T
(Ft
)P✓sup
t6�|Y ✏
⌧+t � Y ✏⌧ | > ⌘
◆= �1, 8⌘ > 0,
then P � (Y ✏)
�1 is exponentially tight.
Lemma 3.3.2 Let Y = (Yt)t>0 be a continuous semimartingale with Y0 = 0.LetD denote the part corresponding to a predictable process of locally boundedvariation, and V the part corresponding to the quadratic variation of the localmartingale. Assume that for T > 0 there exists a convex functionH(�),� 2 Rwith H(0) = 0 and such that for all � 2 R and t 6 T
�Dt + �2Vt/2 6 tH(�⇠), a.s.,
where ⇠ is a nonnegative random variable de�ned on the same probability spaceas Y . �en, for all c > 0 and ⌘ > 0,
P(Y ⇤T > ⌘) 6 P(⇠ > c) + exp
⇢� sup
�2R[�⌘ � TH(�c)]
�.
We are now ready to prove the exponential tightness claim. �e tech-nique borrows elements from Liptser [37].
Proposition 3.3.3 For every T > 0, the family P � (M ✏)
�1 is exponentiallytight on (CT ,B(CT )).
Proof Firstly, we verify the condition (i) of�eorem 3.3.1 for the processM ✏⇤T .
For any T > 0, evidently,
M ✏⇤T 6
Z T
0|b(X✏
s,M✏s)|ds+ sup
t6T
����p✏
Z t
0�(X✏
s,M✏s)dBs
���� , a.s..
50
We denote C✏t :=
p✏R t0 �(X
✏s,M
✏s)dBs. By (A.2),
M ✏⇤T 6 K
Z T
0(1 +M ✏⇤
s )ds+ C✏⇤T = KT + C✏⇤
T +K
Z T
0M ✏⇤
s ds, a.s..
SinceKT+C✏⇤T is nonnegative and non-decreasing in T , Gronwall’s inequal-
ity implies
(3.5) M ✏⇤T 6 eKT
[KT + C✏⇤T ] , a.s..
Now de�ne jK0:= K 0
exp(�KT ) � KT . �en (3.5) entails that for su�-ciently largeK 0 such that jK0 > 0,
P(M ✏⇤T > K 0
) 6 P(C✏⇤T > jK0
) 6 j�1/✏K0 E
h(C✏⇤
T )
1/✏i,
using Chebyshev’s inequality. We thus conclude
(3.6) ✏ logP(M ✏⇤T > K 0
) 6 � log jK0+ ✏ logE
h(C✏⇤
T )
1/✏i.
We assume that 1/✏ > 2 in the rest of the proof (justi�ed by the factthat we consider the limit ✏! 0). Since C✏
t is a local martingale, the process|C✏
t |1/✏ has a unique Doob-Meyer decomposition; let ˇC✏t denote the unique
predictable increasing process in this decomposition. Applying a local mar-tingale maximal inequality (see e.g. Liptser and Shiryaev [40,�m. 1.9.2]) toC✏t , we have for the running maximum process that
(3.7) Eh(C✏⇤
T )
1/✏i6✓
1
1� ✏
◆1/✏
E⇥ˇC✏T
⇤.
In order to obtain an explicit expression for ˇC✏t , we apply Ito’s formula
to |C✏t |1/✏. �is means that, for any t 2 [0, T ],
|C✏t |1/✏ =
1p✏
Z t
0|C✏
s|1/✏�1sign(C✏s)�(X
✏s,M
✏s)dBs
+
1� ✏
2✏
Z t
0|C✏
s|1/✏�2�2(X✏s,M
✏s)ds.
51
We notice that the �rst part is a local martingale and the second part is apredictable increasing process. As a consequence,
(3.8) ˇC✏T =
1� ✏
2✏
Z T
0|C✏
s|1/✏�2�2(X✏s,M
✏s)ds.
Invoking (A.2) again, we have that �2(X✏s,M
✏s) 6 K2
(1+M ✏⇤s )
2. Since (3.5)remains valid when replacing T by s, for any s 6 T , we �nd
|C✏s|1/✏�2�2(X✏
s,M✏s) 6 (C✏⇤
s )
1/✏�2K2⇥1 + eKs
(Ks+ C✏⇤s )
⇤2
6 (C✏⇤s )
1/✏�2K2⇥1 + eKT
(KT + C✏⇤s )
⇤2
6 (C✏⇤s )
1/✏�2K2[2(1 + eKTKT )2 + 2e2KT
(C✏⇤s )
2].
Let LT,K = 2K2max
�(1 + eKTKT )2, e2KT
. �en
|C✏s|1/✏�2�2(X✏
s,M✏s) 6 (C✏⇤
s )
1/✏�2LT,K
⇥1 + (C✏⇤
s )
2⇤
6 L0T,K
h1 + (C✏⇤
s )
1/✏i,
where L ⌘ L0T,K is a positive constant not depending onK 0 (nor ✏). We plug
(3.8) and the above estimate into (3.7), so as to obtain
E[(C✏⇤T )
1/✏] 6
✓1
1� ✏
◆1/✏
E1� ✏
2✏
Z T
0|C✏
s|1/✏�2�2(X✏s,M
✏s)ds
�
6✓
1
1� ✏
◆1/✏�1 L
2✏
✓T +
Z T
0Eh(C✏⇤
s )
1/✏ids
◆
6✓
1
1� ✏
◆1/✏�1 LT
2✏exp
"✓1
1� ✏
◆1/✏�1 LT
2✏
#,
the last inequality following from Gronwall’s inequality. Now observe that(1� ✏)1�1/✏ is decreasing on ✏ 2 [0, 12), with limiting value e as ✏! 0. As aresult, we have the following upper bound on the exponential decay rate of
52
E[(C✏⇤T )
1/✏]:
lim sup
✏!0✏ logE
h(C✏⇤
T )
1/✏i
6 lim sup
✏!0
"✏ log
✓1
1� ✏
◆1/✏�1
+ ✏ logLT
2✏+
✓1
1� ✏
◆1/✏�1 LT
2
#
6 eLT
2
< 1.
Hence, by (3.6), for all T > 0, condition (i) of �eorem 3.3.1 follows for theprocessM ✏⇤
T :
(3.9) lim
K0!1lim sup
✏!0✏ logP
�M ✏⇤
T > K 0�= �1.
Secondly, we verify condition (ii) of�eorem 3.3.1. To this end, note thatfor arbitrary T > 0, � 6 1, and stopping time ⌧ 2 �T (Ft),
P✓sup
t6�|M ✏
⌧+t �M ✏⌧ | > ⌘
◆(3.10)
6 P✓sup
t6�(M ✏
⌧+t �M ✏⌧ ) > ⌘
◆+ P
✓sup
t6�(M ✏
⌧ �M ✏⌧+t) > ⌘
◆.
We can see thatM ✏⌧+t�M ✏
⌧ is a semimartingale with respect to the �ltration{F⌧+t}t>0. For any ⌧ 2 �T (Ft), we denote
D✏t :=
Z ⌧+t
⌧b(X✏
s,M✏s)ds, V ✏
t := ✏
Z ⌧+t
⌧�2(X✏
s,M✏s)ds.
By (A.2), we have, for all � 2 R, t 6 � 6 1 and ⌧ 6 T ,
�D✏t +
�2
2
V ✏t 6 |�|K(1 +M ✏⇤
T+1)t+�2✏
2
K2(1 +M ✏⇤
T+1)2t, a.s..
We de�ne
H(�) := |�|+ �2✏
2
, ⇠ := K(1 +M ✏⇤T+1).
53
�en M ✏⌧+t � M ✏
⌧ satis�es the conditions of Lemma 3.3.2 and, for all c >0, ⌘ > 0,
P✓sup
t6�(M ✏
⌧+t �M ✏⌧ ) > ⌘
◆6 P(⇠ > c) + exp
⇢� sup
�2R[�⌘ � �H(�c)]
�.
Since P(⇠ > c) = P(M ✏⇤T+1 > c/K � 1), it follows that
P✓sup
t6�(M ✏
⌧+t �M ✏⌧ ) > ⌘
◆
6 2max
✓P⇣M ✏⇤
T+1 >c
K� 1
⌘, exp
⇢� sup
�2R[�⌘ � �H(�c)]
�◆.
�e supremum of �⌘ � �H(�c) can be explicitly evaluated:
sup
�2R[�⌘ � �H(�c)]
= sup
�2R
�⌘ � �|�|c� �
�2c2✏
2
�
=
1
✏sup
�>0
(⌘✏� �c✏)�� �c2✏2
2
�2�=
(⌘ � �c)2
2✏�c2.
As a consequence, for all positive c,
lim
�!0lim sup
✏!0✏ log exp
⇢� sup
�2R[�⌘ � �H(�c)]
�= lim
�!0�(⌘ � �c)2
2�c2= �1.
It is concluded that for any ⌧ 2 �T (Ft) and c > 0,
lim
�!0lim sup
✏!0✏ logP
✓sup
t6�(M ✏
⌧+t �M ✏⌧ ) > ⌘
◆
6 lim sup
✏!0✏ logP
⇣M ✏⇤
T+1 >c
K� 1
⌘.
By (3.9), we know
lim
c!1lim sup
✏!0✏ logP
⇣M ✏⇤
T+1 >c
K� 1
⌘= �1.
54
It implies
lim
�!0lim sup
✏!0✏ log sup
⌧2�T
(Ft
)P✓sup
t6�(M ✏
⌧+t �M ✏⌧ ) > ⌘
◆
6 inf
c>0lim sup
✏!0✏ logP
⇣M ✏⇤
T+1 >c
K� 1
⌘= �1.
Moreover, the claim
lim
�!0lim sup
✏!0✏ log sup
⌧2�T
(Ft
)P✓sup
t6�(M ✏
⌧ �M ✏⌧+t) > ⌘
◆= �1
is proved in the same way. �us the desired claim follows from (3.10). ⇤
3.4 Auxiliary resultsWe �rst identify a dense subset of CT ⇥ MT which substantially simpli�esthe proof of the local LDP in the next two sections. Let M+
T be the subset ofMT such that K⌫(s, i) > 0, 8s 2 [0, T ], i 2 S, and let M++
T be the subset ofM+
T such thatK⌫(·, i) 2 C1[0,T ], 8i 2 S.
Lemma 3.4.1 M++T is dense in MT .
Proof We prove the claim in two steps. Firstly, we show that M++T is dense
in M+T . We begin by introducing the standard molli�er J(x) on R, i.e.,
J(x) :=
8<
:k exp
✓1
|x|2 � 1
◆if |x| < 1,
0 if |x| > 1,
where k > 0 is selected so thatRR J(x)dx = 1. For each ⌘ > 0, we de�ne
J⌘(x) := ⌘�1J(x/⌘). For any ⌫ inM+T , we extend the domain ofK⌫(·, i) to
(�1, T + 1), as follows:
K⌫(s, i) :=
8<
:
K⌫(s, i) if s 2 [0, T ],K⌫(0, i) if s 2 (�1, 0),K⌫(1, i) if s 2 (T, T + 1).
55
Since K⌫(·, i) is integrable on (�1, T + 1) for each i 2 S, we can de�ne itsmolli�cation as
K⌘⌫ (s, i) =
Z ⌘
�⌘J⌘(y)K⌫(s� y, i)dy, for s 2 (�1 + ⌘, T + 1� ⌘), ⌘ < 1.
By�eorem C.6 in Evans [14],K⌘⌫ (·, i) is smooth on (�1+⌘, T +1�⌘) and
K⌘⌫ (·, i) ! K⌫(·, i) almost everywhere as ⌘ ! 0.Next we proceed to show that ⌫⌘ with the kernel K⌘
⌫ (s, i) is an elementof M++
T . It is clear that K⌘⌫ (·, i) 2 C1
[0,T ] when restricted to [0, T ], and inadditionwe haveK⌘
⌫ (s, i) > 0, for all s 2 [0, T ] and i 2 S. As a consequence,we only need to prove that
Pdi=1K
⌘⌫ (s, i) = 1, for all s 2 [0, T ]. For any
s 2 [0, T ] and y 2 [�⌘, ⌘], it holds thatPd
i=1K⌫(s� y, i) = 1 since s� y 2(�1, T + 1). We thus have that
dX
i=1
K⌘⌫ (s, i) =
Z ⌘
�⌘J⌘(y)
dX
i=1
K⌫(s� y, i)dy =
Z ⌘
�⌘J⌘(y)dy = 1.
For any t 6 T, i 2 S, due to the fact that K⌘⌫ (·, i) ! K⌫(·, i) on (0, t)
almost everywhere as ⌘ ! 0 andK⌘⌫ (s, i) 6 1, it holds that
Z t
0K⌘
⌫ (s, i)ds !Z t
0K⌫(s, i)ds, as ⌘ ! 0,
appealing to the dominated convergence theorem. SinceR t0 K
⌘⌫ (s, i)ds is
increasing in t,R t0 K⌫(s, i)ds is continuous in t and d is �nite, we obtain the
following uniform convergence:
dT (⌫⌘, ⌫) = sup
t2[0,T ],i2S
����Z t
0K⌘
⌫ (s, i)ds�Z t
0K⌫(s, i)ds
���� ! 0, as ⌘ ! 0,
using Result 1.1.21 in Jacod [30].Secondly, we prove that M+
T is dense in MT . Noticing that K⌫(s, i) canbe 0 for some i, s, we de�ne (for any ⌫ 2 M) an ⌫⌘ 2 M+
T through
K⌘⌫ (s, i) :=
K⌫(s, i) + ⌘
1 + ⌘d,
56
⌘ > 0, for all i, s. Notice thatK⌘⌫ (·, i) also converge toK⌫(·, i) almost every-
where as ⌘ ! 0. As is directly veri�ed, dT (⌫⌘, ⌫) 6 T (⌘ + ⌘d)/(1 + ⌘d).�en the desired result holds. Consequently, M++
T is dense in MT by thetriangle inequality. ⇤
We then present a regularity property of the rate function ˜IT (⌫) onM++T .
Lemma 3.4.2 Fix s 2 [0, T ] and ⌫ 2 M++T . �en there is an optimizer u⇤(s, ·)
of
inf
u2U
"dX
i=1
(Qu)(i)
u(i)K⌫(s, i)
#
such that u⇤(·, i) 2 C1[0,T ], for all i 2 S, and u⇤ 2 U.
Proof As obviously
dX
i=1
(Qu)(i)
u(i)K⌫(s, i) =
dX
i=1
QiiK⌫(s, i) +dX
i=1
Pdj 6=iQiju(j)
u(i)K⌫(s, i),
the optimization problem essentially reduces to
inf
u2U
"dX
i=1
Pdj 6=iQiju(j)
u(i)K⌫(s, i)
#.
We let rji := u(j)/u(i), for i 6= j. Since rji = 1/rij , the optimizationproblem can be wri�en as a minimization over d(d� 1)/2 variables:
inf
rji
>0
dX
i=1
i�1X
j=1
hQijrjiK⌫(s, i) +Qjir
�1ji K⌫(s, j).
i
Observe that for any i, j, k the equality rijrjk = rik needs to hold, whichcorresponds to (d) := (d� 1)(d� 2)/2 constraints. We then perform thechange of variables xji := log rji, and denote by X = (x21, · · · , xd(d�1))
T
the d(d � 1)/2 variables. Le�ing K⌫(s) = (K⌫(s, 1), · · · ,K⌫(s, d))T, wetransform the above optimization problem into
inf
Xf(K⌫(s),X),
57
where
f(K⌫(s),X) :=
dX
i=1
i�1X
j=1
⇥Qije
xjiK⌫(s, i) +Qjie
�xjiK⌫(s, j)
⇤,
with (d� 1)(d� 2)/2 additional constraints to be imposed.�e gradient vector of f with respect toX is
DXf =
✓@f
@x21, · · · , @f
@xd(d�1)
◆,
and the corresponding Hessian matrix D2Xf is the diagonal matrix which
has entries of the form QijexjiK⌫(s, i) + Qjie�xjiK⌫(s, j) on its diagonal.
�e idea is now to split the vector X into X0 = (x21, · · · , xd1)T and X1
(where the la�er vector corresponds with the remaining (d) variables). Dueto the constraints, we have X1 = LX0 where L is a matrix of dimension (d)⇥(d�1). �e next step is to include the constraints into the optimizationequation f . It yields the following new optimization problem, on which noadditional constraints need to be imposed anymore:
inf
X0
ˆf(K⌫(s),X0), where ˆf(K⌫(s),X0) = f(K⌫(s), (X0,LX0)).
Observe that f is a globally strictly convex function of X on a convex do-main, and consequently ˆf is a strictly convex function of X0. Hence, thereis a unique minimizer X⇤
0(s) = (x⇤21(s), · · · , x⇤d1(s))T for any s 2 [0, T ].Since we have that both K⌫(s, i) > 0 and Qij > 0 for i 6= j, any entry ofX⇤
0(s) cannot be �1 or1. We thus conclude thatX⇤0(s) 2 Rd�1.
Let I be the (d� 1)-dimensional identity matrix. �en we de�ne
˜f(K⌫(s),X0) := DX0ˆf(K⌫(s),X0) = DXf(K⌫(s), (X0,LX0))
✓I
L
◆,
which is a smooth function on Rd ⇥ Rd�1 such that ˜f(K⌫(s),X⇤0(s)) = 0.
�e gradient matrix of ˜f with respect toX0 evaluated inX⇤0(s) is
G := D2X0
˜f(K⌫(s),X⇤0(s))
= (I LT)D2
Xf(K⌫(s), (X⇤0(s),LX⇤
0(s)))
✓I
L
◆.
58
Let |G| denote the determinant ofG. SinceH is a positive-de�nite diagonalmatrix and L is of full rank, we conclude that |G| 6= 0, for all s 2 (0, T ).
Hence, the implicit function theorem (cf.�eorem C.8 in Evans [14]) im-plies thatX⇤
0(s) is a smooth function ofK⌫(s): sinceK⌫(·, i) 2 C1[0,T ] for all
i 2 S, we conclude that X⇤0(s) 2 C1
[0,T ](Rd�1
). It also follows that the cor-responding minimizer in terms of the variables rij , say (r⇤21(s), · · · , r⇤d1(s)),is in C1
[0,T ]((0,1)
d�1). Recalling that rji = u(j)/u(i), we set
(u⇤(s, 1), u⇤(s, 2), · · · , u⇤(s, d)) ⌘ (1, r⇤21(s), · · · , r⇤d1(s))
on [0, T ]. �en (u⇤(s, 1), · · · , u⇤(s, d)) is an optimizer corresponding to
inf
u2U
"dX
i=1
(Qu)(i)
u(i)K⌫(s, i)
#,
and u⇤(·, i) 2 C1[0,T ] for all i 2 S. It is easily seen that u⇤(s, i) > 0. �en
infs2[0,T ],i2S u⇤(s, i) > 0 by continuity of u⇤ on [0, T ]. Hence, u⇤ 2 U. ⇤
�e following continuity property of the rate functions will be used inproving the upper and lower bounds.
Lemma 3.4.3 Let ⌫⌘, ⌫ 2 MT with kernelsK⌘⌫ andK⌫ such thatK
⌘⌫ (·, i) !
K⌫(·, i) a.e. as ⌘ ! 0 on [0, T ] for each i 2 S. �en
(i) ˜IT (⌫⌘) ! ˜IT (⌫) as ⌘ ! 0;
(ii) IT (', ⌫⌘) ! IT (', ⌫) as ⌘ ! 0, 8' 2 HT , if infi,x �2(i, x) > 0.
Proof (i) Let ⇢ be a d-dimensional vector such thatPd
i=1 ⇢(i) = 1 and ⇢(i) >0. By Lemma 4.22 in den Hollander [24],
� inf
u2U
"dX
i=1
(Qu)(i)
u(i)⇢(i)
#
is continuous in ⇢ and positive. Moreover, for all ⇢, realizing that theQii arenegative,
� inf
u2U
"dX
i=1
(Qu)(i)
u(i)⇢(i)
#6 �
dX
i=1
Qii
59
as a consequence of
� inf
u2U
"dX
i=1
(Qu)(i)
u(i)⇢(i)
#= �
dX
i=1
Qii⇢(i)� inf
u2U
"dX
i=1
Pdj 6=iQiju(j)
u(i)⇢(i)
#.
Hence,
sup
u2U
"�
dX
i=1
(Qu)(i)
u(i)K⌘
⌫ (s, i)
#! sup
u2U
"�
dX
i=1
(Qu)(i)
u(i)K⌫(s, i)
#,
as ⌘ ! 0, almost everywhere on [0, T ]. Also,
� inf
u2U
"dX
i=1
(Qu)(i)
u(i)K⌘
⌫ (s, i)
#6 �
dX
i=1
Qii
for all s. �en the desired result follows directly by applying the dominatedconvergence theorem.
(ii) When infi,x �2(i, x) > 0, it is easily seen by continuity that
'0t � ˆb(⌫⌘,'t)]
2
�2(⌫⌘,'t)! ['0
t � ˆb(⌫,'t)]2
�2(⌫,'t)
a.e. as ⌘ ! 0. Let �2 denote infi,x �2(i, x). For every ⌫ 2 MT , we have
['0t � ˆb(⌫,'t)]
2
�2(⌫,'t)6 |'0
t|2 + (
Pdi=1 |b(i,'t)|)2 + 2|'0
t|(Pd
i=1 |b(i,'t)|)�2
.
Since ' is absolutely continuous and b(i, x) is Lipschitz continuous in x,Pdi=1 |b(i,'t)| < b < 1 on [0, T ]. Hence, ['0
t � ˆb(⌫,'t)]2/�2(⌫,'t) 6
(|'0t|+ b)2/�2. Since '0 is square-integrable on [0, T ], IT (', ⌫⌘) ! IT (', ⌫)
as ⌘ ! 0 by again applying the dominated convergence theorem. ⇤
3.5 �e upper bound for the local LDP�is section considers the upper bound in the local LDP, whereas the nextsection concentrates on the corresponding lower bound. Recall that our aimis to establish
lim sup
�!0lim sup
✏!0✏ logP(⇢T (M ✏,') + dT (⌫
✏, ⌫) 6 �) 6 �LT (', ⌫),
60
with LT (', ⌫) as de�ned in Section 3.2. Our approach, which has a similarstructure as the one used in Liptser in [37], �nds an exponential upper boundon the probability P(⇢T (M ✏,') + dT (⌫✏, ⌫) 6 �) relying on the method ofstochastic exponentials. As it turns out, this bound should contain the ratefunction LT (', ⌫) as desired.
We start by introducing some additional notation. Let ST denote thespace of all step functions on [0, T ] of the form, for k 2 N and real numbers�0, · · · ,�k,
�(t) = �01{t=0}(t) +kX
i=0
�i1(ti
,ti+1](t), 0 = t0 < · · · < tk+1 = T.
For any ' 2 CT , we introduce the following notation
Z T
0�(s)d's :=
kX
i=0
�i['T^ti+1 � 'T^t
i
].
In the sequel we frequently use the process
N ✏t :=
1p✏
Z t
0�(s)�(X✏
s,M✏s)dBs, � 2 ST ,
which has the stochastic exponential
E (N ✏)t = exp
✓N ✏
t �1
2
hN ✏it◆,wherehN ✏it =
1
✏
Z t
0�2(s)�2(X✏
s,M✏s)ds.
Next we introduce a stochastic exponential associated with the occupa-tion measure ⌫✏. For any u(·, ·) 2 U,
ˆN ✏t = u(t,X✏
t )� u(0, X✏0)�
Z t
0
@
@su(s,X✏
s)ds�Z t
0(Q✏u)(s,X✏
s)ds
is a local martingale on [0, T ] due to Ito’s formula. We de�ne
˜N ✏t :=
Z t
0
1
u(s�, X✏s�)
d
ˆN ✏s .
61
�en
(3.11) E (
˜N ✏)t =
u(t,X✏t )
u(0, X✏0)
exp
�Z t
0
@@su(s,X
✏s) + (Q✏u)(s,X✏
s)
u(s,X✏s)
ds
!
is the stochastic exponential of ˜N ✏t . Indeed,
dE (
˜N ✏)t =
u(t,X✏t )
u(0, X✏0)
exp
�Z t
0
@@su(s,X
✏s) + (Q✏u)(s,X✏
s)
u(s,X✏s)
ds
!
⇥ �
@@tu(t,X
✏t ) + (Q✏u)(t,X✏
t )
u(t,X✏t )
dt
!
+ exp
�Z t
0
@@su(s,X
✏s) + (Q✏u)(s,X✏
s)
u(s,X✏s)
ds
!du(t,X✏
t )
u(0, X✏0)
=
E (
˜N ✏)t�
u(t�, X✏t�)
du(t,X✏
t )�@
@tu(t,X✏
t )dt� (Q✏u)(t,Xt)dt
�
=
E (
˜N ✏)t�
u(t�, X✏t�)
d
ˆN ✏t .
Since inft2[0,T ],i2S u(t, i) > 0, ˜N ✏t is a local martingale and its stochastic
exponential E (
˜N ✏)t is also a local martingale by �eorem 1.4.61 in Jacod
and Shiryaev [31]. �en E (
˜N ✏)t is a martingale since it is bounded.
We will also use this martingale property when applying a change ofmeasure in the next section. �e martingale E (
˜N ✏)t is an extension of the
exponential martingale studied by Palmowski and Rolski in [44].
Lemma 3.5.1 E (
˜N ✏)tE (N ✏
)t is a localmartingale, andE[E (
˜N ✏)tE (N ✏
)t] 61.
Proof By Pro�er [45,�eorem 2.38],
E (
˜N ✏)tE (N ✏
)t = E (
˜N ✏+N ✏
+ [
˜N ✏, N ✏])t,
where [ ˜N ✏, N ✏] denotes the quadratic covariation process. Since ˜N ✏
t is a purejump local martingale andN ✏
t is a continuous local martingale, [ ˜N ✏, N ✏] = 0.
62
�en E (
˜N ✏)tE (N ✏
)t is the stochastic exponential of the local martingale˜N ✏t + N ✏
t and a local martingale too. Since a positive local martingale isa supermartingale, E[E (
˜N ✏)tE (N ✏
)t] 6 E[E (
˜N ✏)0E (N ✏
)0] = 1. ⇤
�e above lemma evidently implies that
Eh1{⇢
T
(M✏,T ,')+dT
(⌫✏,⌫)6�}E (
˜N ✏)TE (N ✏
)T
i6 1.
According to the above inequality, we can �nd an exponential upper boundof
P(⇢T (M ✏,') + dT (⌫✏, ⌫) 6 �),
if we derive non-random exponential lower bounds of E (
˜N ✏)T and E (N ✏
)T
on the set {⇢T (M ✏,') + dT (⌫✏, ⌫) 6 �}. �e next two lemmas obtain theestimates of desired lower bounds. Lemma 3.5.2 focuses on E (N ✏
)T , whereasLemma 3.5.3 covers E (
˜N ✏)T .
Lemma 3.5.2 For every (', ⌫) 2 CT ⇥MT and every � 2 ST , � > 0, thereexists a positive constant K�,',T not depending on ✏ or � such that
E (N ✏)T > exp
⇢��✏K�,',T
+
1
✏
Z T
0�(s)d's �
Z T
0�(s)ˆb(⌫,'s)ds�
Z T
0
�2(s)
2
�2(⌫,'s)ds
��
on the set {⇢T (M ✏,') + dT (⌫✏, ⌫) 6 �}.
Proof It is �rst realized that, by (3.1), N ✏t can be rearranged as
N ✏t =
1
✏
Z t
0�(s)dM ✏
s �Z t
0�(s)b(X✏
s,M✏s)ds
�.
63
�en a straightforward computation yields that
N ✏T � 1
2
hN ✏iT
=
1
✏
Z T
0�(s)dM ✏
s �Z T
0�(s)b(X✏
s,M✏s)ds
�Z T
0
�2(s)
2
�2(X✏s,M
✏s)ds
�
=
1
✏
Z T
0�(s)dM ✏
s �Z T
0�(s)d's
�
�1
✏
Z T
0�(s)b(X✏
s,M✏s)ds�
Z T
0�(s)ˆb(⌫,'s)ds
�
�1
✏
Z T
0
�2(s)
2
�2(X✏s,M
✏s)ds�
Z T
0
�2(s)
2
�2(⌫,'s)ds
�
+
1
✏
Z T
0�(s)d's �
Z T
0�(s)ˆb(⌫,'s)ds�
Z T
0
�2(s)
2
�2(⌫,'s)ds
�.
As a consequence, we evidently have
N ✏T � 1
2
hN ✏iT
> 1
✏
Z T
0�(s)d's �
Z T
0�(s)ˆb(⌫,'s)ds�
Z T
0
�2(s)
2
�2(⌫,'s)ds
�
�1
✏
����Z T
0�(s)dM ✏
s �Z T
0�(s)d's
����
�1
✏
����Z T
0�(s)b(X✏
s,M✏s)ds�
Z T
0�(s)ˆb(⌫,'s)ds
����
�1
✏
����Z T
0
�2(s)
2
�2(X✏s,M
✏s)ds�
Z T
0
�2(s)
2
�2(⌫,'s)ds
���� a.s..
Hence, by repeated use of the triangle inequality, we�nd thatN ✏T�
12hN
✏iT >
64
✏�1G✏T a.s., where G✏
T is given by
Z T
0�(s)d's �
Z T
0�(s)ˆb(⌫,'s)ds�
Z T
0
�2(s)
2
�2(⌫,'s)ds
�
�����Z T
0�(s)dM ✏
s �Z T
0�(s)d's
����
�����Z T
0�(s)b(X✏
s,M✏s)� �(s)b(X✏
s,'s)ds
����
�����Z T
0�(s)b(X✏
s,'s)� �(s)ˆb(⌫,'s)ds
����
�����Z T
0
�2(s)
2
�2(X✏s,M
✏s)�
�2(s)
2
�2(X✏s,'s)ds
����
�����Z T
0
�2(s)
2
�2(X✏s,'s)�
�2(s)
2
�2(⌫,'s)ds
���� .
In the rest of the proof, all objects are considered on the set {⇢T (M ✏,')+dT (⌫✏, ⌫) 6 �}; we analyze all absolute values in the previous display sep-arately. Let us start with considering the �rst absolute value; we thus �ndthat����Z T
0�(s)dM ✏
s �Z T
0�(s)d's
����
=
������
kX
j=1
�j⇣M ✏
T^tj+1
� 'T^tj+1 � (M ✏
T^tj
� 'T^tj
)
⌘������6 2�⇤T �.
Now consider the second absolute value. �e Lipschitz condition (A.1) im-plies that
����Z T
0�(s)b(X✏
s,M✏s)� �(s)b(X✏
s,'s)ds
����
6Z T
0|�(s)|K|M ✏
s � 's|ds 6 �⇤T �KT.
65
For the fourth one, (A.1) also entails that����Z T
0
�2(s)
2
⇥�2(X✏
s,M✏s)� �2(X✏
s,'s)⇤ds
����
6 �2⇤T2
Z T
0K�|�(X✏
s,M✏s) + �(X✏
s,'s)|ds
Since ' is continuous on [0, T ], there exists a positive constant r such that'⇤T 6 r� �. It yields thatM ✏⇤
T 6 r on the set {⇢T (M ✏,')+ dT (⌫✏, ⌫) 6 �}.By the linear growth condition (A.2) and the above reasoning
|�(X✏s,M
✏s) + �(X✏
s,'s)| 6 K(1 + |M ✏s |) +K(1 + |'s|) 6 2K(1 + r).
We conclude that����Z T
0
�2(s)
2
⇥�2(X✏
s,M✏s)� �2(X✏
s,'s)⇤ds
���� 6 �2⇤T �K2(1 + r)T.
�en, concerning the third absolute value,����Z T
0�(s)b(X✏
s,'s)� �(s)ˆb(⌫,'s)ds
����
=
������
kX
j=0
Z tj+1
tj
�j [b(X✏s,'s)� ˆb(⌫,'s)]ds
������
6kX
j=0
�����
Z tj+1
tj
�j
dX
i=1
b(i,'s)[1{X✏
s
=i} �K⌫(s, i)]ds
�����
=
kX
j=0
�����
Z tj+1
tj
dX
i=1
fj(i, s)[1{X✏
s
=i} �K⌫(s, i)]ds
�����
6kX
j=0
dX
i=1
�����
Z tj+1
tj
fj(i, s)[1{X✏
s
=i} �K⌫(s, i)]ds
�����
where fj(i, s) := �jb(i,'s). Since b(i, ·) is Lipschitz continuous and 's
is absolutely continuous, fj(i, s) is of bounded variation. �en, by Lemma
66
3.7.1,
sup
i2S
�����
Z tj+1
tj
fj(i, s)[1{X✏
s
=i} �K⌫(s, i)]ds
����� 6 Cj�,
where Cj is a constant. We thus conclude that
����Z T
0�(s)b(X✏
s,'s)� �(s)ˆb(⌫,'s)ds
���� 6kX
j=0
Cj�d 6 C�.
A similar procedure yields for the last absolute value����Z T
0
�2(s)
2
�2(X✏s,'s)�
�2(s)
2
�2(⌫,'s)ds
���� 6 C 0�.
Upon collecting these inequalities, we �nd
E (N ✏)T > exp
⇢��✏K�,',T
+
1
✏
Z T
0�(s)d's �
Z T
0�(s)ˆb(⌫,'s)ds�
Z T
0
�2(s)
2
�2(⌫,'s)ds
��
where we denote
K�,',T := 2�⇤T + �⇤TKT + �2⇤T K2(1 + r)T + C + C 0,
which is a positive constant not depending on � or ✏. ⇤
Lemma 3.5.3 For every ⌫ 2 MT , every u 2 U and every �, � > 0, there existpositive constants Cu, C 0
u,Ku andKQ,u not depending on ✏ or � such that
E (
˜N ✏)T > Ku exp
✓�Cu�d� �d� C 0
uTd�1
✏KQ,u�d
�1
✏
Z T
0
dX
i=1
Qu(s, i)
u(s, i)K⌫(s, i)ds
!
on the set {⇢T (M ✏,') + dT (⌫✏, ⌫) 6 �}.
67
Proof First observe that
E (
˜N ✏)T
=
u(T,X✏T )
u(0, X✏0)
exp
�Z T
0
dX
i=1
@@su(s, i) + (Q✏u)(s, i)
u(s, i)1{X✏
s
=i}ds
!
=
u(T,X✏T )
u(0, X✏0)
exp
�
dX
i=1
Z T
0
@@su(s, i)
u(s, i)[1{X✏
s
=i} �K⌫(s, i)]ds
�dX
i=1
Z T
0
@@su(s, i)
u(s, i)K⌫(s, i)ds
!⇥
exp
�1
✏
dX
i=1
Z T
0
Qu(s, i)
u(s, i)[1{X✏
s
=i} �K⌫(s, i)]ds
�1
✏
dX
i=1
Z T
0
Qu(s, i)
u(s, i)K⌫(s, i)ds
!.
By the de�nition ofu andX✏0 = x, we have thatKu := mini,x u(T, i)/u(0, x)
is a positive constant. Hence E (
˜N ✏)T majorizes
Ku exp
�
dX
i=1
�����
Z T
0
@@su(s, i)
u(s, i)[1{X✏
s
=i} �K⌫(s, i)]ds
�����
�dX
i=1
�����
Z T
0
@@su(s, i)
u(s, i)K⌫(s, i)ds
�����
!⇥
exp
�1
✏
dX
i=1
����Z T
0
Qu(s, i)
u(s, i)[1{X✏
s
=i} �K⌫(s, i)]ds
����
�1
✏
dX
i=1
Z T
0
Qu(s, i)
u(s, i)K⌫(s, i)ds
!.
On the set {⇢T (M ✏,') + dT (⌫✏, ⌫) 6 �}, Lemma 3.7.1 implies that, for any� > 0, i 2 S,
�����
Z T
0
@@su(s, i)
u(s, i)[1{X✏
s
=i} �K⌫(s, i)]ds
����� 6 Cu� + �,
68
����Z T
0
Qu(s, i)
u(s, i)[1{X✏
s
=i} �K⌫(s, i)]ds
���� 6 KQ,u�, 8i 2 S.
Since @@su(s, i)/u(s, i) is continuous on [0, T ],
�����
Z T
0
@@su(s, i)
u(s, i)K⌫(s, i)ds
����� 6 C 0uT.
Hence,
E (
˜N ✏)T > Ku exp
✓�Cu�d� �d� C 0
uTd�1
✏KQ,u�d
�1
✏
Z T
0
dX
i=1
Qu(s, i)
u(s, i)K⌫(s, i)ds
!.
We have thus proven our claim. ⇤
Now we are ready to prove the upper bound in the local LDP.
Proposition 3.5.4 For every (', ⌫) 2 CT ⇥MT ,
lim sup
�!0lim sup
✏!0✏ logP(⇢T (M ✏,') + dT (⌫
✏, ⌫) 6 �) 6 �LT (', ⌫).
Proof Due to Lemma 3.4.1, CT ⇥M++T is dense in CT ⇥MT . We �rst prove
that the upper bound holds on CT ⇥ M++T . For every ⌫ 2 M++
T , it is animmediate implication of Lemma 3.4.2 that there is an optimizer u⇤(·, ·) of
inf
u2U
"dX
i=1
(Qu)(i)
u(i)K⌫(s, i)
#
such that u⇤ 2 U. We denote
E u⇤t =
u⇤(t,X✏t )
u⇤(0, X✏0)
exp
�Z t
0
@@su
⇤(s,X✏
s) + (Q✏u⇤)(s,X✏s)
u⇤(s,X✏s)
ds
!.
Lemma 3.5.1 implies that
(3.12) Eh1{⇢
T
(M✏,')+dT
(⌫✏,⌫)6�}Eu⇤T E (N ✏
)T
i6 1
69
for every � 2 ST . By virtue of Lemmas 3.5.2 and 3.5.3, we have a non-randomlower bound for E u⇤
T E (N ✏)T on the set {⇢T (M ✏,') + dT (⌫✏, ⌫) 6 �}.
Hence, (3.12) implies that, for all � 2 ST ,
P(⇢T (M ✏,') + dT (⌫✏, ⌫) 6 �)
6 1
Ku⇤exp
✓Cu⇤�d+ C 0
u⇤Td+1
✏KQ,u⇤�d
+
1
✏
Z T
0
dX
i=1
Qu⇤(s, i)
u⇤(s, i)K⌫(s, i)ds
!⇥ exp
⇢�
✏K�,',T
�1
✏
Z T
0�(s)d's �
Z T
0�(s)ˆb(⌫,'s)ds�
Z T
0
�2(s)
2
�2(⌫,'s)ds
��.
We observe thatZ T
0
dX
i=1
Qu⇤(s, i)
u⇤(s, i)K⌫(s, i)ds
= �Z T
0sup
u2U
"�
dX
i=1
(Qu)(i)
u(i)K⌫(s, i)
#ds = �˜IT (⌫).
It directly entails that, again for all � 2 ST ,
✏ logP(⇢T (M ✏,') + dT (⌫✏, ⌫) 6 �)(3.13)
6 �Z T
0�(s)d's �
Z T
0�(s)ˆb(⌫,'s)ds�
Z T
0
�2(s)
2
�2(⌫,'s)ds
�
+K�,',T � � ✏ logKu⇤+ ✏
�Cu⇤�d+ C 0
u⇤Td�+KQ,u⇤�d� ˜IT (⌫).
It is easily seen that all the terms with � or ✏ vanish as � ! 0, ✏! 0. Asa consequence we conclude, by minimizing the right hand-side over �, thatthe decay rate
lim sup
�!0lim sup
✏!0✏ logP(⇢T (M ✏,') + dT (⌫
✏, ⌫) 6 �) 6 �˜IT (⌫)
� sup
�2ST
Z T
0�(s)d's �
Z T
0�(s)ˆb(⌫,'s)ds�
Z T
0
�2(s)
2
�2(⌫,'s)ds
�.
70
Since b(i, x) and �(i, x) satisfy the linear growth condition (A.2), ˆb(⌫, x) and�(⌫, x) are of linear growth as well. �en Liptser and Pukhalskii [39, Lemma6.1] implies that
sup
�2ST
Z T
0�(s)d's �
Z T
0�(s)ˆb(⌫,'s)ds�
Z T
0
�2(s)
2
�2(⌫,'s)ds
�
=
8<
:
Z T
0sup
�2R
�'0
s � �ˆb(⌫,'s)��2
2
�2(⌫,'s)
�ds if ' 2 HT ,
1 otherwise.
For s 2 [0, T ] such that �2(⌫,'s) 6= 0 and ' 2 HT , it is well-known (cf.Fredlin and Wentzell [16], Liptser [37]) that
sup
�2R
�'0
s � �ˆb(⌫,'s)��2
2
�2(⌫,'s)
�=
['0s � ˆb(⌫,'s)]
2
2�2(⌫,'s).
For s 2 [0, T ] such that �2(⌫,'s) = 0 and ' 2 HT ,
sup
�2R
�'0
s � �ˆb(⌫,'s)��2
2
�2(⌫,'s)
�=
⇢0 if '0
s =ˆb(⌫,'s),
1 otherwise.
Hence, with the conventions 0/0 = 0 and n/0 = 1 (for all n > 0) being inforce,Z T
0sup
�2R
�'0
s � �ˆb(⌫,'s)��2
2
�2(⌫,'s)
�ds =
1
2
Z T
0
['0t � ˆb(⌫,'t)]
2
�2(⌫,'t)dt
if ' 2 HT .Hence the lower bound for the dense subsetCT ⇥M++
T is established. Inconsideration of Lemma 1.2.6, the upper bound is proved for CT ⇥MT if wecan show lim inf⌘!0
˜IT (⌫⌘) > ˜IT (⌫) and lim inf⌘!0 IT (', ⌫⌘) > IT (', ⌫)when ⌫⌘ 2 M++
T ! ⌫ 2 MT . From the proof of Lemma 3.4.1, we realizethat the above convergence happens in the way thatK⌘
⌫ (·, i) ! K⌫(·, i) a.e..�en by Lemma 3.4.3, ˜IT (⌫) satis�es the �rst inequality. We denote
F�(', ⌫) =
Z T
0�(s)d's �
Z T
0�(s)ˆb(⌫,'s)ds�
Z T
0
�2(s)
2
�2(⌫,'s)ds.
71
By the previous computation, we know that IT (', ⌫) = sup�2ST
F�(', ⌫)for every (', ⌫) 2 CT ⇥MT . For every � 2 ST , F�(⌫,') is continuous on⌫ due to Lemma 3.7.1. �en IT (', ⌫) is lower semi-continuous on ⌫ sinceit is the pointwise supremum of continuous functions. Hence the secondinequality is established. �e claim is proved. ⇤
3.6 �e lower bound for the local LDP�is section studies the lower bound of the local LDP. To this end, it is real-ized that only �nite rate functions need to be investigated. �e rate function˜IT (⌫) is �nite for every ⌫ 2 MT since 0 6 ˜IT (⌫) 6 �T
Pdi=1Qii. We fur-
ther observe that the rate function IT (', ⌫) is �nite for every (', ⌫) 2 HT ⇥MT if infi,x �2(i, x) > 0. Hence we consider the case of infi,x �2(i, x) > 0
�rst. Let (', ⌫) 2 HT ⇥MT . We de�ne
(3.14) ¯N ✏t :=
1p✏
Z t
0
'0s � ˆb('s, ⌫)
�('s, ⌫)dBs.
�en its stochastic exponential is
E (
¯N ✏)t = exp
✓¯N ✏t �
1
2
h ¯N ✏it◆,whereh ¯N ✏it =
1
✏
Z t
0
"'0s � ˆb('s, ⌫)
�('s, ⌫)
#2ds.
For simplicity, we denote
hs :='0s � ˆb('s, ⌫)
�('s, ⌫)
throughout this section. Recall from (3.11) that, for a given u(·, ·) 2 U,
E (
˜N ✏)t =
u(t,X✏t )
u(0, X✏0)
exp
�Z t
0
@@su(s,X
✏s) + (Q✏u)(s,X✏
s)
u(s,X✏s)
ds
!.
In order to perform a change of measure in Proposition 3.6.3, we showthat {E (
˜N ✏)tE (
¯N ✏)t}t2[0,T ] is a true martingale. A�er that, we will derive
a lower bound for [E (
˜N ✏)T ]
�1.
72
Lemma 3.6.1 For every (', ⌫) 2 HT ⇥MT , u 2 U, {E (
˜N ✏)tE (
¯N ✏)t}t2[0,T ]
is a martingale if infi,x �2(i, x) > 0.
Proof We have shown in last section that E (
˜N ✏)t is a martingale. Since
' 2 HT and recalling that we assumed infi,x �2(i, x) > 0, it follows thath ¯N ✏iT =
1✏
R T0 h2sds < 1. �en Novikov’s condition implies that E (
¯N ✏)t
is also a martingale. Since X✏t is independent of the Brownian motion Bt,
E (
˜N ✏)t is also independent of E (
¯N ✏)t. So,
E[E (
˜N ✏)TE (
¯N ✏)T ] = E[E (
˜N ✏)T ]E[E (
¯N ✏)T ] = E[E (
˜N ✏)0]E[E (
¯N ✏)0]
= E[E (
˜N ✏)0E (
¯N ✏)0].
By the same reasoning as in the proof of Lemma 3.5.1, E (
˜N ✏)tE (
¯N ✏)t is
a supermartingale. Hence, it is a martingale by Liptser and Shiryaev [41,Lemma 6.4]. ⇤Lemma 3.6.2 For every ⌫ 2 MT , every u 2 U and every �, � > 0, there existpositive constants Cu, C 0
u,K 0u andKQ,u not depending on ✏ or � such that
[E (
˜N ✏)T ]
�1 > K 0u exp
✓�Cu�d� �d� C 0
uTd�1
✏KQ,u�d
+
1
✏
Z T
0
dX
i=1
Qu(s, i)
u(s, i)K⌫(s, i)ds
!
on the set {⇢T (M ✏,') + dT (⌫✏, ⌫) 6 �}.
Proof According to the computation in Lemma 3.5.3,
[E (
˜N ✏)T ]
�1=
u(0, X✏0)
u(T,X✏T )
exp
dX
i=1
Z T
0
@@su(s, i)
u(s, i)[1{X✏
s
=i} �K⌫(s, i)]ds
+
dX
i=1
Z T
0
@@su(s, i)
u(s, i)K⌫(s, i)ds
!⇥
exp
1
✏
dX
i=1
Z T
0
Qu(s, i)
u(s, i)[1{X✏
s
=i} �K⌫(s, i)]ds
+
1
✏
dX
i=1
Z T
0
Qu(s, i)
u(s, i)K⌫(s, i)ds
!.
73
De�ningK 0u := mini,j u(0, j)/u(T, i), we have that
[E (
˜N ✏)T ]
�1 > K 0u exp
�
dX
i=1
�����
Z T
0
@@su(s, i)
u(s, i)[1{X✏
s
=i} �K⌫(s, i)]ds
�����
�dX
i=1
�����
Z T
0
@@su(s, i)
u(s, i)K⌫(s, i)ds
�����
!⇥
exp
�1
✏
dX
i=1
����Z T
0
Qu(s, i)
u(s, i)[1{X✏
s
=i} �K⌫(s, i)]ds
����
+
1
✏
dX
i=1
Z T
0
Qu(s, i)
u(s, i)K⌫(s, i)ds
!.
On the set {⇢T (M ✏,') + dT (⌫✏, ⌫) 6 �}, Lemma 3.7.1 implies�����
Z T
0
@@su(s, i)
u(s, i)[1{X✏
s
=i} �K⌫(s, i)]ds
����� 6 Cu� + �, 8i 2 S, 8� > 0,
����Z T
0
Qu(s, i)
u(s, i)[1{X✏
s
=i} �K⌫(s, i)]ds
���� 6 KQ,u�, 8i 2 S.
Also, �����
Z T
0
@@su(s, i)
u(s, i)K⌫(s, i)ds
����� 6 C 0uT.
Hence,
[E (
˜N ✏)T ]
�1 > K 0u exp
✓�Cu�d� �d� C 0
uTd�1
✏KQ,u�d
+
1
✏
Z T
0
dX
i=1
Qu(s, i)
u(s, i)K⌫(s, i)ds
!.
We have thus derived the desired lower bound. ⇤
We proceed to prove the lower bound of the local LDP under the condi-tion infi,x �2(i, x) > 0.
74
Proposition 3.6.3 For every (', ⌫) 2 HT ⇥MT , if infi,x �2(i, x) > 0,
lim inf
�!0lim inf
✏!0✏ logP(⇢T (M ✏,') + dT (⌫
✏, ⌫) 6 �) > �LT (', ⌫).
Proof For any ⌫ 2 MT , there is a sequence ⌫⌘ 2 M++T such that ⌫⌘ ! ⌫ by
Lemma 3.4.1. Actually, the convergence happens in the way thatK⌘⌫ (·, i) !
K⌫(·, i) a.e.. �en by Lemma 3.4.3, the rate function LT (', ⌫) satis�es thecontinuity property required in Lemma 1.2.6. Hence we only need to provethe lower bound on the dense subset HT ⇥M++
T . Recall that for every ⌫ 2M++
T , Lemma 3.4.2 implies that there is an optimizer u⇤(·, ·) of
inf
u2U
"dX
i=1
(Qu)(i)
u(i)K⌫(s, i)
#
such that u⇤ 2 U and
E u⇤t =
u⇤(t,X✏t )
u⇤(0, X✏0)
exp
�Z t
0
@@su
⇤(s,X✏
s) + (Q✏u⇤)(s,X✏s)
u⇤(s,X✏s)
ds
!.
By Lemma 3.6.1, we knowE[E u⇤T E (
¯N ✏)T ] = 1.On (⌦,FT ), we de�ne a new
probability measure Pu⇤ through dPu⇤= E u⇤
T E (
¯N ✏)TdP. Since E u⇤
T E (
¯N ✏)T
is strictly positive, Pu⇤ is equivalent to P and dP =
⇥E u⇤T E (
¯N ✏)T
⇤�1dPu⇤ .
So that we can translate the probability of our interest under the originalmeasure P into the mean of a certain random quantity under the alternativemeasure Pu⇤ :
P(⇢T (M ✏,') + dT (⌫✏, ⌫) 6 �)(3.15)
=
Z
{⇢T
(M✏,')+dT
(⌫✏,⌫)6�}
hE u⇤T E (
¯N ✏)T
i�1dPu⇤ .
ByGirsanov’s theorem, ˜Bt := Bt� 1p✏
R t0 hsds is aPu⇤-Brownianmotion
on (⌦, (Ft)t6T ). We substitute the above equation in (3.14), and obtain
¯N ✏T � 1
2
h ¯N ✏iT =
1p✏
Z T
0hsd ˜Bs +
1
2✏
Z T
0h2sds.
75
It thus follows that [E u⇤T E (
¯N ✏)T ]
�1 is equal to
u⇤(0, X✏0)
u⇤(t,X✏t )
exp
Z t
0
@@su
⇤(s,X✏
s) + (Q✏u⇤)(s,X✏s)
u⇤(s,X✏s)
ds� 1p✏
Z T
0hsd ˜Bs
� 1
2✏
Z T
0h2sds
◆.
Now let L be a positive constant. We de�ne
⇥
✏:=
⇢⇢T (M
✏,') + dT (⌫✏, ⌫) 6 �,
����Z T
0hsd ˜Bs
���� 6 L
�.
�en (3.15) implies
P(⇢T (M ✏,') + dT (⌫✏, ⌫) 6 �) >
Z
⇥✏
hE u⇤T E (
¯N ✏)T
i�1dPu⇤ .
By Lemma 3.6.2, we obtain the following non-random lower bound of[E u⇤
T E (
¯N ✏)T ]
�1, valid on the set ⇥✏:
K 0u⇤ exp
�Cu⇤�d� C 0
u⇤Td�1
✏KQ,u⇤�d�
˜IT (⌫)
✏� IT (', ⌫)
✏� Lp
✏
!.
As a consequence, we have the following lower bound of the probabilityP(⇢T (M ✏,') + dT (⌫✏, ⌫) 6 �):
K 0u⇤ exp
�Cu⇤�d� C 0
u⇤Td�1
✏KQ,u⇤�d�
˜IT (⌫)
✏� IT (', ⌫)
✏� Lp
✏
!
⇥Pu⇤(⇥
✏).
�is, in turn, leads to the following lower bound on the corresponding expo-nential decay rate:
✏ logP(⇢T (M ✏,') + dT (⌫✏, ⌫) 6 �)(3.16)
> ✏ logK 0u⇤ � ✏(Cu⇤�d+ C 0
u⇤Td)�KQ,u⇤�d
�˜IT (⌫)� IT (', ⌫)�p✏L+ ✏ logPu⇤
(⇥
✏).
76
�en a su�cient condition for desired result to hold is lim✏!0 Pu⇤(⇥
✏) >
0. It is evident that
Pu⇤(⇥
✏) > 1� Pu⇤
✓����Z T
0hsd ˜Bs
���� > L
◆(3.17)
�Pu⇤(dT (⌫
✏, ⌫) > �)� Pu⇤(⇢T (M
✏,') > �).
We proceed by consecutively proving that the three probabilities in the right-hand side of (3.17) vanishes as ✏ ! 0. We start by analyzing the �rst proba-bility. By Chebyshev’s inequality,
˜Pu⇤
✓����Z T
0hsd ˜Bs
���� > L
◆6
˜Eu⇤
���R T0 hsd ˜Bs
���2
L2=
R T0 h2sds
L2.
SinceR T0 h2sds < 1, we can make this upper bound arbitrarily small by
picking L su�ciently large.Next we consider the second probability in the right-hand side of (3.17).
We notice that the part
exp
✓1p✏
Z T
0hsd ˜Bs +
1
2✏
Z T
0h2sds
◆
in the change of measure procedure is not related to the Markov chain . �enby Proposition 11.2.3 in Bielecki and Rutkowski [3] (See also Corollary 2.1.2in van Handel [20]), a Markov chain Xt with transition intensity matrix Qunder P becomes a Markov chain under Pu⇤ with transition intensity matrixQ(u⇤)(t) where
Q(u⇤)(t)ij = Qiju⇤(t, j)
u⇤(t, i)for i 6= j; Q(u⇤)(t)ii = �
X
j 6=i
Qiju⇤(t, j)
u⇤(t, i).
Hence, Q(u⇤)(t)/✏ is the transition intensity matrix of X✏t under Pu⇤ . By
Lemma 3.7.2, for every t 2 [0, T ], K⌫(t) = (K⌫(t, 1), . . . ,K⌫(t, d)) is theunique solution of
µ(t)Q(u⇤)(t) = 0,dX
i=1
µ(t, i) = 1, µ(t, i) > 0.
77
Also, all entries of the matrixQ(u⇤)(t) are smooth on [0, T ] by Lemma 3.4.2.�en by Corollary 5.8 in Yin and Zhang [61],
Pu⇤
sup
t6T,i2S
����Z t
01{X✏
s
=i}ds�Z t
0K⌫(i, s)ds
���� > ✏1/4!
6 K exp
⇢� CT
✏1/4(T + 1)
3/2
�,
where CT is a strictly positive constant. For any � > 0, the following (obvi-ous) inequality holds for every ✏ such that ✏ 2 (0, �4)
Pu⇤
sup
t6T,i2S
����Z t
01{X✏
s
=i}ds�Z t
0K⌫(i, s)ds
���� > �
!
6 Pu⇤
sup
t6T,i2S
����Z t
01{X✏
s
=i}ds�Z t
0K⌫(i, s)ds
���� > ✏1/4!.
Hence we have
Pu⇤
sup
t6T,i2S
����Z t
01{X✏
s
=i}ds�Z t
0K⌫(i, s)ds
���� > �
!! 0 as ✏! 0.
�at is, Pu⇤(dT (⌫✏, ⌫) > �) ! 0, as ✏! 0.
Now we proceed by showing the third probability in the right-hand sideof (3.17) vanishes as ✏! 0. We substitute ˜Bt for Bt in (3.1), yielding
M ✏t =
Z t
0b(X✏
s,M✏s) + hs�(X
✏s,M
✏s)ds+
p✏
Z t
0�(X✏
s,M✏s)d
˜Bs.
By se�ing ˜M ✏t := M ✏
t � 't, we obtain
˜M ✏t =
p✏
Z t
0�(X✏
s,M✏s)d
˜Bs +
Z t
0[b(X✏
s,M✏s)� b(X✏
s,'s)]ds
+
Z t
0[b(X✏
s,'s)� ˆb(⌫,'s)]ds+
Z t
0hs[�(X
✏s,M
✏s)� �(X✏
s,'s)]ds
+
Z t
0hs[�(X
✏s,'s)� �(⌫,'s)]ds.
78
Using the Lipschitz continuity featuring in (A1), we �nd that both
sup
t6T
����Z t
0[b(X✏
s,M✏s)� b(X✏
s,'s)]ds
���� 6Z T
0K ˜M ✏⇤
s ds,
and
sup
t6T
����Z t
0hs[�(X
✏s,M
✏s)� �(X✏
s,'s)]ds
���� 6Z T
0|hs|K ˜M ✏⇤
s ds.
Recalling that we denote throughout this chapter running maximum pro-cesses by adding an asterisk (‘⇤’), it is now immediate that
˜M ✏⇤T 6 I1⇤T + I2⇤T + I3⇤T +
Z T
0K(1 + |hs|) ˜M ✏⇤
s ds,
where
I1t :=
p✏
Z t
0�(X✏
s,M✏s)d
˜Bs, I2t :=
Z t
0[b(X✏
s,'s)� ˆb(⌫,'s)]ds,
I3t :=
Z t
0hs[�(X
✏s,'s)� �(⌫,'s)]ds.
�en Gronwall’s inequality implies
(3.18) ˜M ✏⇤T 6 [I1⇤T + I2⇤T + I3⇤T ] exp
✓Z T
0K(1 + |hs|)ds
◆.
�e next step is to study the impact of I1⇤T , I2⇤T , and I3⇤T separately. Forany � > 0, it is an immediate consequence of Chebyshev’s inequality thatPu⇤
(I1⇤T > �) 6 ��3˜Eu⇤
[(I1⇤T )
3]. We notice the close similarity between
I1t and C✏t in the proof of Proposition 3.3.3. �e quantity Eu⇤
[(I1⇤T )
3] can
be dealt with using essentially the same procedure that was used to boundE[(C✏⇤
T )
1/✏]: we derive an inequality similar to (3.7), i.e.,
Eu⇤[(I1⇤T )
3] 6 27
8
Eu⇤
3✏
Z T
0|I1s |�2(X✏
s,M✏s)ds
�.
79
We thus obtain
Eu⇤[(I1⇤T )
3] 6 81✏kT
8
exp
✓81✏kT
8
◆,
where k is a positive constant. As a consequence, lim✏!0 Pu⇤(I1⇤T > �) = 0.
�e claim lim✏!0 Pu⇤(I2⇤T > �) = 0 can be established as follows. As a
�rst step we observe that since
sup
t6T
����Z t
0[b(X✏
s,'s)� ˆb(⌫,'s)]ds
����
= sup
t6T
�����
dX
i=1
Z t
0b(i,'s)[1{X✏
s
=i} �K⌫(s, i)]ds
�����
6 d sup
t6T,i2S
����Z t
0b(i,'s)[1{X✏
s
=i} �K⌫(s, i)]ds
���� ,
the following upper bound applies:
Pu⇤(I2⇤T > �)
= Pu⇤
✓sup
t6T
����Z t
0[b(X✏
s,'s)� ˆb(⌫,'s)]ds
���� > �
◆
6 Pu⇤
sup
t6T,i2S
����Z t
0b(i,'s)[1{X✏
s
=i} �K⌫(s, i)]ds
���� > �/d
!
Since b(i, x) is Lipschitz continuous in x and 't is absolutely continuous,b(i,'t) is bounded on [0, T ]. �en by Corollary 5.8 in Yin and Zhang [61]again, Pu⇤
(I2⇤T > �) ! 0, as ✏! 0.Similar to the above computation, we can obtain that
Pu⇤(I3⇤T > �)
6 Pu⇤
sup
t6T,i2S
����Z t
0hs�(i,'s)[1{X✏
s
=i} �K⌫(s, i)]ds
���� > �/d
!.
We know that hs�(i,'s) is square-integrable for every i 2 S. �en by themethod of molli�cation in�eorem C.6 in Evans [14], there exists a sequence
80
of smooth functions h⌘(i, s) such that h⌘(i, s) ! hs�(i,'s) as ⌘ ! 0 inL2
[0, T ], which is the space of square-integrable functions on [0, T ]. By theCauchy-Schwarz inequality,
����Z t
0[hs�(i,'s)� h⌘(i, s)][1{X✏
s
=i} �K⌫(s, i)]ds
����
6p2t
✓Z t
0[hs�(i,'s)� h⌘(i, s)]2ds
◆1/2
.
�en,
sup
t6T,i2S
����Z t
0hs�(i,'s)[1{X✏
s
=i} �K⌫(s, i)]ds
����
6 sup
t6T,i2S
����Z t
0h⌘(i, s)[1{X✏
s
=i} �K⌫(s, i)]ds
����
+sup
i2S
p2T
✓Z T
0[hs�(i,'s)� h⌘(i, s)]2ds
◆1/2
.
We let H(⌘) := supi2Sp2T
⇣R T0 [hs�(i,'s)� h⌘(i, s)]2ds
⌘1/2. It is clear
that H(⌘) ! 0 as ⌘ ! 0. Hence,
Pu⇤(I3⇤T > �)
6 Pu⇤
sup
t6T,i2S
����Z t
0h⌘(i, s)[1{X✏
s
=i} �K⌫(s, i)]ds
����+H(⌘) > �/d
!.
For any � > 0, we can choose all ⌘ > 0 small enough such thatH(⌘) < �/2d.It yields
Pu⇤(I3⇤T > �) 6 Pu⇤
sup
t6T,i2S
����Z t
0h⌘(i, s)[1{X✏
s
=i} �K⌫(s, i)]ds
���� > �/2d
!.
Since h⌘(i, s) is bounded on [0, T ], the probability in the right-hand of aboveinequality vanishes as ✏! 0 for any small enough ⌘ by Corollary 5.8 in Yinand Zhang [61]. Hence we conclude that lim✏!0 Pu⇤
(I3⇤T > �) = 0.We have thus shown that Pu⇤
(⇥
✏) remains bounded away from 0 as ✏!
0. Upon combining all the above, the proof of the lemma is now complete. ⇤
81
So far we have focused on the case infi,x �2(i, x) > 0. To complete theanalysis, we next consider the situation that this condition is li�ed, in whichcase �('s, ⌫) can be singular. Our proof uses arguments used in the methodpresented by Liptser [37, Lemma A.6]. Given � > 0, we study the SDE
(3.19) M ✏,�t =
Z t
0b(X✏
s,M✏,�s )ds+
p✏
Z t
0�(X✏
s,M✏,�s )dBs +
p✏�Wt,
where M ✏,�0 ⌘ 0 and Wt is another standard P-Brownian motion, indepen-
dent ofBt andX✏t . We provide an auxiliary lemma which is to be used when
proving the lower bound; informally, it states that M ✏,� and M ✏ are ‘super-exponentially close’.
Lemma 3.6.4 For every T > 0 and ⌘ > 0,
(3.20) lim
�!0lim sup
✏!0✏ logP (⇢T (M
✏,� ,M ✏) > ⌘) = �1.
Proof We de�ne A✏,�t := M ✏,�
t �M ✏t , and
↵✏t :=
b(X✏t ,M
✏,�t )� b(X✏
t ,M✏t )
M ✏,�t �M ✏
t
, �✏t :=�(X✏
t ,M✏,�t )� �(X✏
t ,M✏t )
M ✏,�t �M ✏
t
.
By (A.1), i.e., the Lipschitz condition, we conclude
(3.21) |↵✏t | 6 K, |�✏t | 6 K, 8t 2 [0, T ].
As an immediate consequence of (3.1) and (3.19), we have
dA✏,�t = ↵✏
tA✏,�t dt+
p✏�✏tA
✏,�t dBt +
p✏�dWt.
We de�ne
E✏t := exp
✓Z t
0
h↵✏s �
✏
2
(�✏s)2ids+
p✏
Z t
0�✏sdBs
◆.
We apply Ito’s formula to (E✏t )
�1, so as to obtain
d(E✏t )
�1= (E✏
t )�1
�✏(�✏t )
2dt� ↵✏
tdt�p✏�✏tdBt
�.
82
Since Wt is independent of Bt, we have
dhA✏,�t , (E✏
t )�1i = h�
p✏�✏t (E
✏t )
�1dBt,
p✏�✏tA
✏,�t dBt +
p✏�dWti
= �✏(�✏t )2(E✏t )
�1A✏,�t dt.
By applying the integration-by-parts formula,
dA✏,�t (E✏
t )�1
= A✏,�t d(E✏
t )�1
+ (E✏t )
�1dA✏,�
t + dhA✏,�t , (E✏
t )�1i
=
p✏�(E✏
t )�1
dWt,
and henceA✏,�
t =
p✏ �E✏
t
Z t
0(E✏
s)�1
dWs.
We de�ne the set �N := {1/N 6 inft6T E✏t 6 supt6T E✏
t 6 N}, forN 2 N. Observe that it holds that ⇢T (M ✏,� ,M ✏
) = (A✏,�)
⇤T , and therefore
P(⇢T (M ✏,� ,M ✏) > ⌘) 6 P((A✏,�
)
⇤T > ⌘,�N ) + P(⌦ \ �N )
6 2max {P((A✏,�)
⇤T > ⌘,�N ),P(⌦ \ �N )} .
Wenow consider each of the probabilitiesP((A✏,�)
⇤T > ⌘,�N ) andP(⌦\�N )
separately. On the set �N ,
(A✏,�)
⇤T 6
p✏�E✏⇤
T sup
t6T
����Z t
0(E✏
s)�1
dWs
���� 6p✏�N sup
t6T
����Z t
0(E✏
s)�1
dWs
���� .
Since ↵✏t and �✏t are bounded as ✏ ! 0, it follows that
R t0 [↵
✏s � ✏
2(�✏s)
2]ds
is bounded as well, and therefore we omit it for brevity when analyzing E✏t .
Based on the above, the stated holds if we can prove that (A) for all N 2 N,covering the contribution of P((A✏,�
)
⇤T > ⌘,�N ),
(3.22)
lim
�!0lim sup
✏!0✏ logP
✓p✏�N sup
t6T
����Z t
0(E✏
s)�1
dWs
���� > ⌘,�N
◆= �1,
and (B), covering the contribution of P(⌦ \ �N ),
(3.23) lim
N!1lim sup
✏!0✏ logP
✓p✏ supt6T
����Z t
0�✏sdBs
���� > logN
◆= �1.
83
Let us �rst consider contribution (A). To this end, de�ne
⌧ := T ^ inf
⇢t 6 T :
����Z t
0(E✏
s)�1
dWs
���� >⌘p✏�N
�.
�en (3.22) is equivalent to, for all N 2 N,(3.24)
lim
�!0lim sup
✏!0✏ logP
✓p✏�N
Z ⌧
0(E✏
s)�1
dWs > ⌘ (or 6 �⌘),�N
◆= �1.
ForN 2 N and ⌘ > 0, we de�ne the process ˜E✏t and its stochastic exponential
E (
˜E✏)t:
˜E✏t :=
⌘p✏�N3T
Z t
0(E✏
s)�1
dWs, E (
˜E✏)t = exp
✓˜E✏t �
1
2
h ˜E✏it◆.
Since E (
˜E✏)t is a supermartingale, we have
Eh1{p✏�N
R⌧
0 (E✏
s
)�1dWs
> ⌘,�N
}E (
˜E✏)⌧
i6 1
On the set {p✏�N
R ⌧0 (E
✏s)
�1dWs > ⌘,�N}, we have
E (
˜E✏)⌧ > exp
⌘p
✏�N3T
⌘p✏�N
� 1
2
✓⌘p
✏�N3T
◆2
N2T
!
= exp
✓⌘2
2✏�2N4T
◆,
and consequently
exp
✓⌘2
2✏�2N4T
◆P✓p
✏�N
Z ⌧
0(E✏
s)�1
dWs > ⌘,�N
◆6 1.
We conclude that the part corresponding to “> ⌘” in (3.24) is valid, but it isimmediately veri�ed that the part corresponding to “6 �⌘” in (3.24) can beaddressed in the same way.
84
We now turn to contribution (B). �e validity of (3.23) can be proved ina similar way by de�ning the stopping time
⌧ 0 := inf
⇢t 6 T :
����Z t
0�✏sdBs
���� >logNp
✏
�
and the process ˜�✏t and its stochastic exponential E (
˜�✏)t:
˜�✏t :=logNp✏K2T
Z t
0�✏sdBs, E (
˜�✏)t = exp
✓˜�✏t �
1
2
h˜�✏it◆,
whereK is the constant in (3.21). ⇤
�e following result establishes the lower bound of the local LDP.
Proposition 3.6.5 For every (', ⌫) 2 CT ⇥M,
lim inf
�!0lim inf
✏!0✏ logP(⇢T (M ✏,') + dT (⌫
✏, ⌫) 6 �) > �LT (', ⌫).
Proof As mentioned in the beginning of this section, only the case (', ⌫) 2HT ⇥MT such that
Z T
0
['0t � ˆb(⌫,'t)]
2
�2(⌫,'t)dt < 1
needs to be considered. If infi,x �2(i, x) > 0, then the result is valid due toProposition 3.6.3. If infi,x �2(i, x) = 0, then we consider M ✏,�
t as de�ned in(3.19). �e idea is that we decompose the probability
P✓⇢T (M
✏,� ,') + dT (⌫✏, ⌫) 6 �
2
◆
into the sum of
(3.25) P✓⇢T (M
✏,� ,') + dT (⌫✏, ⌫) 6 �
2
, ⇢T (M✏,') + dT (⌫
✏, ⌫) 6 �
◆
and
(3.26) P✓⇢T (M
✏,� ,') + dT (⌫✏, ⌫) 6 �
2
, ⇢T (M✏,') + dT (⌫
✏, ⌫) > �
◆.
85
Obviously, (3.25) is majorized by P(⇢T (M ✏,') + dT (⌫✏, ⌫) 6 �). Using thetriangle inequality, we �nd that ⇢T (M ✏,') 6 ⇢T (M ✏,M ✏,�
)+⇢T (M ✏,� ,').So that (3.26) is majorized by P
�⇢T (M ✏,M ✏,�
) > �2
�. Hence,
P✓⇢T (M
✏,� ,') + dT (⌫✏, ⌫) 6 �
2
◆
is not greater than
2max
P(⇢T (M ✏,') + dT (⌫
✏, ⌫) 6 �),P✓⇢T (M
✏,M ✏,�) >
�
2
◆�.
By Lemma 3.6.4,
lim
�!0lim sup
✏!0✏ logP
✓⇢T (M
✏,� ,M ✏) >
�
2
◆= �1,
and, as a result,
lim inf
�!0lim
�!0lim inf
✏!0✏ logP
✓⇢T (M
✏,� ,') + dT (⌫✏, ⌫) 6 �
2
◆
6 lim inf
�!0lim inf
✏!0✏ logP(⇢T (M ✏,') + dT (⌫
✏, ⌫) 6 �).
Next we compute the term on the le�-hand side of the above inequality. SinceM ✏,� meets the conditions in Lemma 3.6.3, (M ✏,� , ⌫✏) satis�es the inequality(3.16). �en for every � > 0, we obtain
lim inf
✏!0✏ logP
✓⇢T (M
✏,� ,') + dT (⌫✏, ⌫) 6 �
2
◆
> �KQ,u⇤�
2
d� ˜IT (⌫)�1
2
Z T
0
['0s � ˆb('s, ⌫)]2
�2('s, ⌫) + �2ds.
By the monotone convergence theorem (recall the convention 0/0 = 0),
1
2
Z T
0
['0s � ˆb('s, ⌫)]2
�2('s, ⌫) + �2ds ! 1
2
Z T
0
['0s � ˆb('s, ⌫)]2
�2('s, ⌫)ds = IT (', ⌫), as � ! 0
86
which implies that
lim inf
�!0lim
�!0lim inf
✏!0✏ logP
✓⇢T (M
✏,� ,') + dT (⌫✏, ⌫) 6 �
2
◆
> �˜IT (⌫)� IT (', ⌫).
We have proven the claim. ⇤
3.7 AppendixLemma 3.7.1 Let f(t, i) be a continuous function on [0, T ] for every i 2 S.Let µ, ⌫ 2 M such that dT (µ, ⌫) 6 �. For any � > 0 and [t1, t2] ⇢ [0, T ],there exists a constant C > 0 such that
(3.27) sup
i2S
����Z t2
t1
f(s, i)[Kµ(s, i)�K⌫(s, i)]ds
���� 6 C� + �.
Proof We �rst look at functions f(t, i) that are of bounded variation. Byintegration by parts, we have
Z t2
t1
f(s, i)[Kµ(s, i)�K⌫(s, i)]ds = [µ(s, i)� ⌫(s, i)]f(s, i)|t2t1
�Z t2
t1
[µ(s, i)� ⌫(s, i)]df(s, i).
�en����Z t2
t1
f(s, i)[Kµ(s, i)�K⌫(s, i)]ds
���� 6 |µ(t2, i)� ⌫(t2, i)||f(t2, i)|
+|µ(t1, i)� ⌫(t1, i)||f(t1, i)|+Z t2
t1
|µ(s, i)� ⌫(s, i)||df(s, i)|
6 C1� + C2� + TVf [t1, t2]�,
where TVf [t1, t2] denotes the total variation of f on [t1, t2] and C1, C2 aretwo positive constants. Since S has �nite elements, we can �nd a constant Csuch that the claim holds. If f(t, i) is only continuous, it can be uniformly
87
approximated by a continuously di�erentiable function (see [47]), that is, forany � > 0, there exists a continuously di�erentiable function f�
(t, i) suchthat
sup
t2[t1,t2],i2S|f(t, i)� f�
(t, i)| < �/2 (t2 � t1).
�en
sup
i2S
����Z t2
t1
f(s, i)[Kµ(s, i)�K⌫(s, i)]ds
����
6 sup
i2S
����Z t2
t1
[f(s, i)� f�(s, i)] [Kµ(s, i)�K⌫(s, i)]ds
����
+sup
i2S
����Z t2
t1
f�(s, i)[Kµ(s, i)�K⌫(s, i)]ds
����
6 � + C�.
�is �nishes our proof. ⇤
For any u 2 U, let Q(u)(t) be the transition matrix resulting from themeasure change induced by the stochastic exponential E (
˜N ✏). It is known,
see Proposition 11.2.3 in Bielecki and Rutkowski [3], that
Q(u)(t)ij = Qiju(t, j)
u(t, i)if i 6= j; Q(u)(t)ii = �
X
j 6=i
Qiju(t, j)
u(t, i).
For a �xed t, we suppress this t, so as to make the notation more compact. Inmatrix notation, (where we throughout write diag(u) to denote the diagonalmatrix with entries ui�ij) we have
(3.28) Q(u) = diag(u)�1Qdiag(u)� diag(u)�1diag(Qu).
Lemma 3.7.2 Let ⌫ be a d-dimensional vector such thatPd
i=1 ⌫(i) = 1 and⌫(i) > 0. Let u⇤ 2 U be an optimizer of
inf
u2U
X
i
(Qu)iui
⌫i = inf
u2U⌫Tdiag(u)�1Qu.
�en ⌫ is the unique invariant vector of the transition matrix Q(u⇤).
88
Proof To �nd aminimizing u⇤ = u(⌫) (where it observed that minimizers arenot necessarily unique) for the above problem, we �rst note that all u⇤i > 0.Hence the minimizer solves the system of �rst order conditions. Di�erenti-ation with respect to uk yields (ek denoting the k-th basis vector)
�⌫Tdiag(u)�1ekeTk diag(u)
�1Qu+ ⌫Tdiag(u)�1Qek = 0.
In vector notation these equations can be conveniently summarized as
(3.29) ⌫Tdiag(u)�1��diag(u)�1
diag(Qu) +Q�= 0.
From (3.28) we deduce by commutation of diagonal matrices the relation
Q(u) = diag(u)�1(Q� diag(u)�1
diag(Qu))diag(u),
and hence
diag(u)Q(u)diag(u)�1= Q� diag(u)�1
diag(Qu).
We can therefore rewrite (3.29) as
⌫TQ(u)diag(u)�1= 0.
It follows that ⌫TQ(u⇤) = 0. Since ⌫(i) > 0, ⌫ is the unique invariant vectorof Q(u⇤). ⇤
89
90
Chapter 4
Limit theorems for re�ectedOrnstein-Uhlenbeckprocesses
4.1 Introduction�e se�ing considered in this chapter is again a complete stochastic basis(⌦,F , {Ft}t2R+ ,P). Given a ROU process (de�ned in Section 1.3) withsmall noise
dY ✏t = (↵� �Y ✏
t )dt+p✏�dBt + dL✏
t, Y ✏0 = x > 0,
whereR[0,T ] 1{Zt
>0}dLt = 0 for any T > 0. We are interested in large-deviations asymptotics of the type
(4.1) lim
✏!0✏ logP(Y ✏
T > b | Y ✏0 = x),
for x > 0, T > 0, and b > E(Y ✏T | Y ✏
0 = x) (so that the event underconsideration is rare).
�e strategy for the computation of (4.1) is discussed brie�y in Overviewat the beginning of this thesis. Here we give a detailed explanation of it. Inour strategy, a �rst step is to study the above decay rate for the non-re�ected
91
OU process. �is decay rate is computed as the solution of a certain vari-ational problem, relying on standard calculus-of-variations: it minimizes arate function over all paths f such that f(0) = x and f(T ) > b. �e op-timizing path f? has the informal interpretation of ‘most likely path’ (or:‘minimal cost path’): given the rare event under study happens, with over-whelming probability it does so through a path ‘close to’ f?. In fact, f? doesnot hit the level 0 between 0 and T . For ROU processes, one can computethe cost of staying at the boundary 0, and performing all calculations it turnsout that the most likely path for ROU processes stays away from 0, and coin-cides with the most likely path for OU.�e computations for OU process arepresented in Section 4.2. �e results are in line with what could be computedfrom the explicitly known distribution of X✏
T conditional on X✏0 = x, but
provide us in addition with the most likely path.In Section 4.3, we recall the LDP for certain re�ected di�usion processes
in Doss and Priouret [10]. �e integrand of the rate function at the re�ectingboundary is di�erent from the one in the interior. We derive explicit expres-sions of the rate functions. Section 4.4 then focuses on the computation ofthe decay rate for ROU process. Above we described the intuitively appeal-ing approach we followed, but it should be emphasized that at the technicallevel there are some non-trivial steps to be taken. We use the explicit for-mula of the rate function derived in Section 4.3. But, calculus-of-variationtechniques cannot be used immediately to identify the most likely path. �isis due to the fact that we need to minimize over all non-negative continu-ous paths. However, the non-negativity of the optimizing path for the OUprocess facilitates the computation of the decay rates for the ROU process.In Section 4.5, we compute the decay rate for the DROU process with smallnoise by the same strategy as the one for ROU processes.
�e second part of this chapter focuses on obtaining properties of the lossprocessUt and the idle processLt of DROU processes (de�ned in Section 1.3),
dZt = (↵� �Zt)dt+ �dBt + dLt � dUt, Z0 = x 2 [0, d],
whereR[0,T ] 1{Zt
>0}dLt = 0 andR[0,T ] 1{Zt
<d}dUt = 0 for any T > 0.We assume that the upper boundary d is larger than ↵/� to guarantee thathi�ing d does not happen too frequently, which is a reasonable assumptionfor most of applications.
92
In Section 4.6 we use Ito’s formula to identify a central limit theorem(CLT) for Ut: we �nd explicit expressions for qU and ⌘U such that (Ut �qU t)/
pt converges to a Normal random variable with mean 0 and variance
⌘2U . A similar result is established for Lt. Finally, we re�ne these results tofunctional central limit theorems by the application of Shiryaev’s conver-gence of triplets.
4.2 Transient asymptotics for OU processes
�e primary goal of this section is to compute the decay rate (4.1) with Y ✏
replaced by X✏ which is the solution to
dX✏t = (↵� �X✏
t )dt+p✏�dBt, X0 = x > 0.
In other words, we now consider the OU case (that is, no re�ection). Be-fore we a�ack this problem, we �rst identify the OU process’ average be-havior. To this end, we �rst describe the ‘zeroth-order approximation’ ofone-dimensional di�usion processes. �e SDE (more general than the onede�ning OU) we consider here is
dJ ✏t = b(J ✏
t )dt+p✏�(J ✏
t )dBt, J ✏0 = x,
and the corresponding ODE is
dx(t) = b(x(t))dt, x(0) = x.
�eorem 4.2.1 (Freidlin and Wentzell [16,�eorem 2.1.2]) Suppose that b(·)and �(·) are Lipschitz continuous and increase no faster than linearly, i.e.,
[b(x)� b(y)]2 + [�(x)� �(y)]2 6 K2|x� y|2,
b2(x) + �2(x) 6 K2(1 + |x|2),
where K is a constant. �en for all t > 0 and ✏ > 0 we have
E|J ✏t � x(t)|2 6 ✏a(t),
93
where a(t) is a monotone increasing function, which is expressed in terms of |x|and K . Moreover, for all t > 0 and � > 0
lim
✏!0P✓
sup
06s6t|J ✏
s � x(s)| > �
◆= 0.
In the speci�c case of OU processes, it is checked that the limiting processx(t) is given by
x(t) = ↵� �x(t), x(0) = x,
which has the solution
x(t) =↵
�+
✓x� ↵
�
◆e��t.
Note that x(t) = EX✏t . Popularly, as ✏ # 0, with high probability X✏
t iscontained in any �-neighborhood of x(t) on the interval [0, T ]. Assumingthat b > x(T ), it is now seen that the probability of our interest, of whichwe wish to identify the decay rate, relates to a rare event.
We now recall the Freidlin-Wentzell theorem from Dembo and Zeitouni[7,�eorem 5.6.7], which is the cornerstone behind the results of this section.It states that X✏ satis�es the sample-path LDP with the rate function
Ix(f) :=
((2�2)�1
R T0 (f 0
(t)� ↵+ �f(t))2dt if f 2 Hx,
1 if f /2 Hx,
where Hx := {f 2 C[0,T ] : f(t) = x +
R t0 �(s)ds,� is square integrable}.
�e LDP states that for any closed set F and open set G in (C[0,T ], k · k1),
lim sup
✏!0✏ logP(X✏
t 2 F ) 6 � inf
f2FIx(f),
lim inf
✏!0✏ logP(X✏
t 2 G) > � inf
f2GIx(f).
�ese upper and lower bounds obviously match for Ix-continuity sets S, thatis, sets S such that inff2clS Ix(f) = inff2intS Ix(f).
We now return to the decay rate under consideration. Let us �rst in-troduce some notation, following standard conventions in Markov process
94
theory. We write Px(E) for the probability of an event E in terms of theprocessX✏ if this process starts in x. We will mainly work with a �xed timehorizon T > 0 and write X⇧ for {Xt, t 2 [0, T ]}. Our �rst step is to ex-press the probability under study in terms of probabilities featuring in thesample-path LDP. Observe that we can write Px(X✏
T > b) = Px(X✏⇧ 2 S),
with
S :=
[
a>b
Sa, Sa :=
�f 2 C[0,T ] : f(0) = x, f(T ) = a
.
Below we �rst solve a calculus-of-variation problem to �nd inff2Sa
Ix(f)explicitly. Secondly, we prove that S is an Ix-continuity set. A combinationof these �ndings gives us an expression for the decay rate.
Proposition 4.2.2 Let a > b > x(T ). �en
inf
f2Sa
Ix(f) =[a� x(T )]2
[1� e�2�T](�2/�)
=
[a� (
↵� + (x� ↵
� )e��T
)]
2
[1� e�2�T](�2/�)
.
�e optimizing path is given by
f?(t) = (C � ↵
�)e�t + (x� C)e��t
+
↵
�,
where
C :=
a� ↵� +
↵� e
�T � xe��T
e�T � e��T.
Moreover, f?(t) > 0 on t 2 [0,1) when the starting point x > 0; f?
(t) 2[0, d] on t 2 [0, T ] when the starting point x 2 [0, d], a 2 [0, d] and ↵/� < d.
Proof We can see
inf
f2Sa
Ix(f) = inf
⇢1
2�2
Z T
0(f 0
(t) + �f(t)� ↵)2dt, f 2 Hx \ Sa
�.
According to Euler’s necessary condition in Shwartz andWeiss [49,�eoremC.13], the initial condition and the boundary condition, we have that theoptimizing path satis�es
f 00(t)� �2f(t) + ↵� = 0, f(0) = x, f(T ) = a.
95
�e general solution of the ODE (unique up to the choice of the two con-stants) reads
f(t) = C1e�t+ C2e
��t+
↵
�.
It is now checked that the stated expression follows, by imposing the initialcondition and the boundary condition. Hence,
inf
f2Sa
Ix(f) =(2C1�)2
2�2
Z T
0e2�tdt =
[a� (
↵� + (x� ↵
� )e��T
)]
2
[1� e�2�T](�2/�)
.
We proceed with proving that f?(t) > 0 and f?
(t) 2 [0, d] on t 2 [0,1)
under the two stipulated assumptions. First we note that x(t) = xe��t+
(1� e��t)↵/�, a convex combination of x and ↵/�. Since both of these are
nonnegative by assumption, so is x(t). For f?(t)we have the following alter-
native expressions with q(t) := sinh(�t)/sinh(�T ), as a direct computationshows:
f?(t) = x(t) + (a� x(T ))q(t)
= q(t)a+ (e��t � q(t)e��T)x+
�1� e��t � q(t)(1� e��T
)
�↵�.
It follows from the �rst equality that f?(t) > x(t), because a > x(T ), and
hence f?(t) is nonnegative. Moreover, the second equality shows that f?
(t)is a convex combination of a, x and ↵/�, see below. Since all three of theseare assumed to be less than d, the same holds true for f?
(t). Finally we showthat we indeed have the claimed convex combination, by showing that allcoe�cients are nonnegative and sum to one. �e la�er is obvious, as well asq(t) 2 [0, 1]. Furthermore e��t � q(t)e��T > (1� q(t))e��T > 0. To provethat the third coe�cient is nonnegative we use the basic equality
sinh(x) =(1 + ex)(1� e�x
)
2
.
�en observe that
1� e��t � q(t)(1� e��T) = 1� e��t � (1 + e�t)(1� e��t
)
(1 + e�T )(1� e��T)
(1� e��T)
= (1� e��t)
✓1� 1 + e�t
1 + e�T
◆> 0.
96
�is completes the proof. ⇤
Proposition 4.2.3 S is an Ix-continuity set.
Proof Consider the topological space (C[0,T ], ⌧), where the topology ⌧ is in-duced by the metric d(f, g). We next consider ¯Sx = {f 2 C[0,T ] : f(0) = x}with the subspace topology ⌧S
x
= {U \ ¯Sx : U 2 ⌧}. �e set S is a closedsubset in ¯Sx since the coordinate mapping f 7! f(T ) is ⌧ -continuous. Bythe same property and the fact that the coordinate mapping is ⌧ -open, the⌧S
x
-interior of S is intS = {f 2 C[0,T ] : f(0) = x, f(T ) > b}. We thushave
inf
f2clSIx(f) = inf
f2SIx(f) = inf
a>binf
f2Sa
Ix(f),
inf
f2intSIx(f) = inf
a>binf
f2Sa
Ix(f).
Using Proposition 4.2.2 and the fact that a > b > x(T ),
inf
a>binf
f2Sa
Ix(f) = inf
a>binf
f2Sa
Ix(f) =[b� (
↵� + (x� ↵
� )e��T
)]
2
[1� e�2�T](�2/�)
.
Consequently, S is an Ix-continuity set. ⇤
Now the decay rate under consideration can be determined.
Proposition 4.2.4 Let b > x(T ). �en
lim
✏!0✏ logPx(X
✏T > b) = �
[b� (
↵� + (x� ↵
� )e��T
)]
2
[1� e�2�T](�2/�)
.
Moreover, the minimal cost path is as given in Proposition 4.2.2 (with a replacedby b).
Proof Apply ‘Freidlin-Wentzell’ to the Ix-continuity set S:
lim
✏!0✏ logPx(X
✏T > b) = lim
✏!0✏ logPx(X
✏⇧ 2 S)
= � inf
f2SIx(f) = � inf
a>binf
f2Sa
Ix(f).
By the computations in the proof of Proposition 4.2.3, we obtain the desiredresult. �e minimal cost path is directly obtained from Proposition 4.2.2. ⇤
97
Wementioned in the introduction to this chapter that there is an alterna-tive method to compute the decay rate under study. It follows relatively di-rectly from the fact thatX✏
T (withX✏0 = x) is normally distributed withmean
µT = x(T ) = ↵� (1� e��T
) + xe��T and variance �2T (✏) =✏�2
2� (1� e�2�T),
in conjunction with the standard inequality in Shwartz and Weiss [49, p. 19]
1
y + y�1e�
12y
2 6Z 1
ye�
12 t
2dt 6 1
ye�
12y
2.
We have followed our sample-path approach, though, for two reasons: (i) theresulting most likely path is interesting in itself, as it gives insight into thebehavior of the system conditional on the rare event, but, more importantly,(ii) it is useful when studying the counterpart of the decay rate for ROU(rather than OU), which we pursue in Section 4.4. We also note that
lim
T!1lim
✏!0✏ logPx(X
✏T > b) = �
(b� ↵� )
2
�2/�.
It is known that the steady-state distribution ofX✏t withX✏
0 = x is normallydistributed with mean ↵/� and variance ✏ �2/(2�). We conclude that thisshows that the result is invariant under changing the orders of taking limits(T ! 1 and ✏! 0).
4.3 LDPs for re�ected di�usion processesWe recall a sample-path LDP for a certain re�ected di�usion process withconstant di�usion coe�cient. �e se�ing is less general than (1.3), but isrelevant to the ROU process. Let D� 2 R be an open interval, @D and Ddenote its boundary and closure. Let ⌫(x) denote the function giving theinward normal at x 2 @D, i.e. ⌫(x) = 1 if x is a �nite le� endpoint ofD and⌫(x) = �1 if x is a �nite right endpoint of D. Let (H✏, ⇠✏) be the uniquesolution to
dH✏t = b(H✏
t )dt+p✏�dBt + d⇠✏t , H✏
0 = x 2 D,(4.2)|⇠✏|t =
R t0 1@D(H
✏s)d|⇠✏|s, ⇠✏t =
R t0 ⌫(Hs)d|⇠✏|s.
98
We assume b(·) is uniformly Liptschitz continuous and � is a nonzero con-stant. We denote by H+
x the nonnegative functions in Hx and by !t a func-tion from [0, T ] to R.
�eorem 4.3.1 (Doss and Priouret [10,�eorem 4.2]) If b(·) is uniformly Lip-schitz continuous and bounded, and � is a nonzero constant, then H✏ satis�esthe LDP in C[0,T ](D) with the rate function
I(h) = inf
!>0
1
2�2
Z T
0(h0t � b(ht)� ⌫(ht)!t1@D(ht))
2dt.
if h 2 H+x and1 else.
For re�ected di�usions with a single re�ecting boundary at 0, we canidentify the optimal !?
t and have the following explicit expression of the ratefunction. As usual, we de�ne x+ = max{x, 0} and x� = �min{x, 0}.
Proposition 4.3.2 Let D = [0,1). When b(0) > 0, H✏ satis�es the LDP inC[0,T ](R+) with the rate function
I+(h) =1
2�2
Z T
0
�h0t � b(ht)
�2dt
if h 2 H+x and 1 else. When b(0) < 0, H✏ satis�es the LDP in C[0,T ](R+)
with the rate function
I+(h) =1
2�2
Z T
0
�h0t � b(ht)
�2dt� 1
2�2b(0)2
Z T
01{0}(ht) dt.
if h 2 H+x and1 else. In short, for h 2 H+
x and b(0) 2 R we have
I+(h) =1
2�2
Z T
0
�h0t � b(ht)
�2dt� 1
2�2(b(0)�)2
Z T
01{0}(ht) dt.
Proof In this case, D = R+, @D = {0} and ⌫(0) = 1. �e rate functionbecomes
I+(h) = inf
{!t
>0}
1
2�2
Z T
0(h
0t � b(ht)� !t1{0}(ht))
2dt.
99
Weminimize for each t separately under the integral. If h0t� b(ht) < 0, then!t = 0 is optimal. If h0t � b(ht) > 0 and ht > 0, then 1{0}(ht)!t ⌘ 0, whichmeans that any value of ! is optimal. If h0t � b(ht) > 0 and ht = 0, then!t = h0t � b(ht) is optimal. Hence !?
t = (h0t � b(ht))+ is the optimizer. Itgives the following explicit expression:
I+(h) =1
2�2
Z T
0
�h0t � b(ht)� 1{0}(ht)(h
0t � b(ht))
+�2
dt
if h 2 H+x and 1 else. For any h 2 C[0,T ](R+) which is di�erentiable a.e.,
note that h0t = 0 if ht = 0. �en we have
I+(h) =
1
2�2
Z T
0
�h0t � b(ht)� 1{0}(ht)b(0)
��2dt
=
1
2�2
Z T
0
�h0t � b(ht)
�2dt+
1
2�2
Z T
01{0}(ht)
�b(0)�
�2dt
+
1
�2
Z T
01{0}(ht)b(0)
�b(0) dt.
When b(0) > 0, the last two terms are zero, and for b(0) < 0 they sum to
� 1
2�2b(0)2
Z T
01{0}(ht) dt.
�is completes our proof. ⇤
We can still derive an explicit expression for the optimal !?t in the case
of two re�ecting boundaries, and hence have the following simpli�ed ratefunction.
Proposition 4.3.3 Given D = [0, d], the rate function I(h) in �eorem 4.3.1can be rewri�en as follows:
I++(h) =
1
2�2
Z T
0
�h0t � b(ht)� 1{0}(ht)b(0)
�+ 1{d}(ht)b(d)
+�2
dt
if h 2 H++x := {f 2 Hx : 0 6 f 6 d} and1 else.
100
Proof Since @D = {0, d}, the rate function becomes
I++(h) = inf
!>0
1
2�2
Z T
0
�h0t � b(ht)� [1{0}(ht)� 1{d}(ht)]!t
�2dt.
Let !? denote the optimizer of the above problem. We discuss the value of!?t in two cases.
• Case 1: h0t � b(ht) < 0. If ht 2 (0, d), then !?t can be any value; if
ht = 0, then !?t = 0; if ht = d, then !?
t = �(h0t � b(ht)).
• Case 2: h0t � b(ht) > 0. If ht 2 (0, d), then !?t can be any value; if
ht = d, then !?t = 0; if ht = 0, then !?
t = h0t � b(ht).
As a consequence, I++(h) has the following explicit expression:
1
2�2
Z T
0
�h0t � b(ht)� 1{0}(ht)(h
0t � b(ht))
++ 1{d}(ht)(h
0t � b(ht))
��2dt
if h 2 H++x and 1 else. Also, h 2 [0, d] and h is di�erentiable a.e. imply
that 8t 2 (0, T ), h0t = 0 when ht = 0 or ht = d. So I++(h) can be further
simpli�ed to
1
2�2
Z T
0
�h0t � b(ht)� 1{0}(ht)(�b(0))+ + 1{d}(ht)(�b(d))�
�2dt
if h 2 H++x and 1 else. ⇤
4.4 Transient asymptotics for ROU processes�is section determines the decay rate (4.1) for ROU processes. �eorem 4.3.1requires b(·) to be bounded which is a condition that ROU processes do notsatisfy. But the mapping in (4.2) from Brownian motion Bt to re�ected dif-fusion process H✏
t is continuous since � is a constant and b(·) is uniformlycontinuous. One can directly apply the contraction principle of �eorem1.2.7 to the rate function of Brownian motion. Details can be found in theproof of�eorem 3.1 on page 10 in Doss and Priouret [10]. As a result, Propo-sition 4.3.2 is valid for ROU processes.
101
Above we observed (i) that the most likely path for OU was non-negative(Proposition 4.2.2), (ii) the rate functions I and I+ for OU and ROU pro-cesses are the same as long as their arguments are nonnegative paths on[0, T ] (Proposition 4.3.2). �is suggests that the decay rates for OU and ROUprocesses (and the corresponding most likely paths) coincide.
�e idea is now that we �nd the decay rate (4.1) for ROU by using thesample-path results that we derived for OU processes. Recall that the zeroth-order approximation of OU is x(t) = ↵/� + (x � ↵/�)e��t. It is checkedthat x(t) > 0 when the starting point x > 0. So we still assume b > x(T )in the decay rate for ROU process. We de�ne S+
:= {f 2 C[0,T ](R+) :
f(0) = x, f(T ) > b}, corresponding to the rare event {Y ✏⇧ 2 S+}, so as
to compute the decay rate (4.1); the set S+a is de�ned as {f 2 C[0,T ](R+) :
f(0) = x, f(T ) = a}. Below we keep the notation Px for probabilities ofevents in terms of Y ✏ when this process starts in x.
�eorem 4.4.1 Let b > x(T ). �en, similar to the result of Proposition 4.2.4,
lim
✏!0✏ logPx(Y
✏T > b) = �
[b� (
↵� + (x� ↵
� )e��T
)]
2
[1� e�2�T](�2/�)
.
Moreover, the minimal cost path is as given in Proposition 4.2.2 (with a replacedby b).
Proof Since b(0) = ↵ > 0, by Proposition 4.3.2, Y ✏ satis�es the sample pathLDP in C[0,T ](R+) with the rate function
I+x (h) =1
2�2
Z T
0
�h0t � ↵+ �ht)
�2dt
if h 2 H+x and1 else. By an argument that is similar to the one used in the
proof of Proposition 4.2.3, S+ is an I+x -continuity set. �en,
lim
✏!0✏ logPx(Y
✏T > b) = lim
✏!0✏ logPx(Y
✏⇧ 2 S+
) = � inf
h2S+I+x (h)
= � inf
a>binf
h2S+a
I+x (h).
102
We have
inf
h2S+a
I+x (h) = inf
⇢1
2�2
Z T
0(h0t + �ht � ↵)2dt, h 2 Hx \ S+
a
�
> inf
⇢1
2�2
Z T
0(h0t + �ht � ↵)2dt, h 2 Hx \ Sa
�
= inf
f2Sa
Ix(f).
�e optimizer f? of inff2Sa
Ix(f) is always positive, for any starting pointx > 0, due to Proposition 4.2.2. �at is, f? 2 S+
a . So we conclude thatinfh2S+
a
I+x (h) = inff2Sa
Ix(f). �en the results follows immediately fromProposition 4.2.2 and a > b > x(T ),
lim
✏!0✏ logPx(Y
✏T > b) = � inf
a>b
[a� (
↵� + (x� ↵
� )e��T
)]
2
[1� e�2�T](�2/�)
= �[b� (
↵� + (x� ↵
� )e��T
)]
2
[1� e�2�T](�2/�)
.
�is proves the claim. ⇤
4.5 Transient asymptotics for DROU processes
�is section computes the decay rate (4.1) for DROU processes. �e case ofDROU processes corresponds to choose the set D = [0, d] in �eorem 4.3.1.For DROU processes, b(·) is bounded, and as a consequence it ful�lls all re-quirements in�eorem 4.3.1 and its rate function can be obtained by Propo-sition 4.3.3 directly. Now, by a similar argument as employed in the last sec-tion, we prove that the decay rates (4.1) for OU and DROU processes (and thecorresponding most likely paths) coincide. Recall from the introduction thatwe have assumed ↵/� < d for DROU processes. Under this assumption, thezeroth-order approximation x(t) belongs to (0, b) when the starting point xis in [0, d]. We consider crossing levels d > a > b > x(T ). De�ne
S++:= {f 2 C[0,T ]([0, d]) : f(0) = x, f(T ) > b},
103
so that our rare event corresponds to Z✏⇧ 2 S++; the set S++
a is de�ned as{f 2 C[0,T ]([0, d]) : f(0) = x, f(T ) = a}. Finally, we arrive at the mainresult for DROU processes.
�eorem 4.5.1 Let d > b > x(T ). �en
lim
✏!0✏ logPx(Z
✏T > b) = �
[b� (
↵� + (x� ↵
� )e��T
)]
2
[1� e�2�T]�2/�
.
Moreover, the minimal cost path is the one given in Proposition 4.2.4.
Proof Since b(0) = ↵ > 0 and b(d) = ↵ � �d < 0, by Proposition 4.3.3, Z✏
satis�es the sample-path LDP in C[0,T ]([0, d]) with the rate function
I++x (h) =
1
2�2
Z T
0
�h0t � ↵+ �ht
�2dt
if h 2 H++x and 1 else. Due to Proposition 4.2.2, it holds that f?
=
arg inff2Sa
Ix(f) is in [0, d] on t 2 [0, T ], when the starting point x 2[0, d], a 2 [0, d] and ↵/� < d, i.e., f? 2 S++
a . As a consequence, f?=
arg inf�2S++a
I++x (�), and infh2S++
a
I++x (h) = inff2S
a
Ix(f). �e rest ofproof is similar to that of�eorem 4.4.1. ⇤
4.6 CLTs and FCLTs for the loss and idle processesIn this section we derive a central limit theorem for the loss process Ut of theDROU process Zt satisfying
dZt = (↵� �Zt)dt+ �dBt + dLt � dUt, Z0 = x 2 [0, d].
We do so relying on martingale techniques. A similar procedure can be fol-lowed for the idle process Lt. To prepare for the main result of this section,we start with some preliminary calculations. Let h be a twice continuouslydi�erentiable function on R, and Z be the DROU process de�ned earlier. ByIto’s formula, we have:
dh(Zt) =�(↵� �Zt)h
0(Zt) +
�2
2
h00(Zt)�dt+ �h0(Zt)dBt
+ h0(Zt)dLt � h0(Zt)dUt.
104
Based on the key properties of L and U , this reduces to
(4.3) dh(Zt) = (Lh)(Zt)dt+ �h0(Zt)dBt + h0(0)dLt � h0(d)dUt,
where the operator L is de�ned through
L := (↵� �x)d
dx+
�2
2
d
2
dx2.
�e following lemma is the key to the proof of Proposition 4.6.2 below. Itgives the solution to a linear ODE whose inhomogeneous term is consideredas a variable as well, under initial and boundary conditions.
Lemma 4.6.1 Consider the ODE with real variable right-hand side q 2 R
(Lh) = q, 0 6 x 6 d,
under the mixed initial/boundary conditions h(0) = 0, h0(0) = 0, and h0(d) =1. It has the unique solution (h, q) 2 C2
(R)⇥ R given by
q = qU :=
�2
2
W (d)R d0 W (v)dv
, h(x) =2qU�2
Z x
0
Z u
0
W (v)
W (u)dv du,
where
W (v) := exp
✓2↵v
�2� �v2
�2
◆.
Proof By applying reduction of order, the ODE can be wri�en a system of�rst-order ODEs:
h0(x) = f(x), f 0(x) +
2↵� 2�x
�2f(x) =
2q
�2.
�e integrating factor of the second �rst-order ��� isW (x). Hence,
f(x) =C1
W (x)+
2q
�2
Z x
0
W (u)
W (x)du.
�en the general solution is
h(x) = C2+
Z x
0f(u)du = C2+C1
Z x
0
1
W (u)du+
2q
�2
Z x
0
Z u
0
W (v)
W (u)dv du.
105
�en the initial conditions h(0) = 0, h0(0) = 0 uniquely determine the val-ues of C1, C2, while h0(d) = 1 uniquely determines qU . Hence, we obtainedthe desired unique solution. ⇤
We now present some additional observations. Let X be distributed ac-cording to the stationary distribution of the �� process, i.e. X has aN(
↵� ,
�2
2� )
distribution, and denote by pX its continuous density. We see that up to amultiplicative constant, the integrating factorW in the above proof is equalto pX . It then follows that
qU =
�2
2
pX(d)
P(0 X d).
We will use the stationary distribution ⇡Z of the DROU process Z . Let, witha light abuse of notation, Z also denote a random variable with that distri-bution and let pZ denote its density, FZ its cumulative distribution function.�e density pZ of ⇡Z is obtained in Ward and Glynn [58, Proposition 1] (seealso Linetsky [35, Equation 31]) as, with N(m, s2) denoting a Normal ran-dom variable with mean m and variance s2,
pZ(z) =
d
dzP✓N
✓↵
�,�2
2�
◆ z
���� 0 6 N
✓↵
�,�2
2�
◆6 d
◆
=
r2�
�2
'
✓(z � ↵
� )
q2��2
◆
�
✓(d� ↵
� )
q2��2
◆� �
✓(�↵
� )
q2��2
◆ ,
where ' and � are the density function and cumulative density function ofa standard Normal random variable. In short, for 0 z d we have
pZ(z) =pX(z)
P(0 X d),
and hence qU =
�2
2 pZ(d). For values of u, v in [0, d] one also has
W (v)
W (u)=
pX(v)
pX(u)=
pZ(v)
pZ(u).
106
Hence the function h in Lemma 4.6.1 has for x 2 [0, d] the alternative ex-pressions
h(x) = pZ(d)
Z x
0
P(0 X u)
pX(u)du = pZ(d)
Z x
0
P(0 Z u)
pZ(u)du
= pZ(d)
Z x
0
FZ(u)
pZ(u)du.
For the derivative h0(x) we have for x 2 [0, d]
h0(x) =pZ(d)
pZ(x)FZ(x).
Proposition 4.6.2 �e loss process U satis�es the central limit theorem, with⌘2U de�ned in (4.5),
Ut � qU tpt
) N (0, ⌘2U ), as t ! 1.
Proof We insert the unique solution h(x) of Lemma 4.6.1 into (4.3). Sinceh0(0) = 0, h0(d) = 1, and (Lh)(Zt) = qU , we have the following integralexpression:
(4.4) Ut � qU t+ h(Zt)� h(Z0) = �
Z t
0h0(Zs)dBs.
We then observe thatMt := Ut�qU t+h(Zt)�h(Z0) is a zero-mean squareintegrable martingale. As usual, hMi denotes the quadratic variation processof M . By the ergodic theorem Gihman and Skorokhod [17, p. 134],
(4.5) t�1hMit = t�1�2Z t
0h0(Zs)
2ds
P! �2Z d
0h0(x)2⇡Z(dx) =: ⌘2U ,
where⇡Z is the stationary distribution ofZt. �enwe obtain fromLemma 4.6.1
⌘2U =
4q2U�2
Z d
0
✓Z x
0
W (v)
W (x)dv
◆2
pZ(x) dx.
107
By the martingale central limit theorem in Komorowski et al. [33, �eorem2.1], t�
12Mt ) N (0, ⌘2U ) as t ! 1. Since Z 2 [0, d] and h is continuous,
h(Z) is bounded. So,h(Zt)� h(Z0)p
t! 0
a.s. as t ! 1, which implies the claim. ⇤
�e loss process at 0 can be treated analogously. De�ne
(4.6) qL :=
�2
2
1
R d0 W (v)dv
,
(4.7) ⌘2L := �2Z d
0
✓� 1
W (x)+
2qL�2
Z x
0
W (v)
W (x)dx
◆2
⇡(x) dx.
Proposition 4.6.3 �e loss process L satis�es the central limit theorem
Lt � qLtpt
) N (0, ⌘2L), as t ! 1.
Let us add some additional observations. �e considerations just aboveProposition 4.6.2 enable us to write an alternative expression for ⌘2U as well.Using these, we obtain from (4.5)
⌘2U = �2pZ(d)2Z d
0
FZ(x)2
pZ(x)dx.
Analogously, one derives that the function h in Lemma 4.6.1 satisfying thealternative boundary conditions h0(0) = 1, h0(d) = 0 is, for x 2 [0, d], givenby
h0(x) =pZ(0)
pZ(x)P(Z � x) =:
pZ(0)
pZ(x)¯F (x),
and for qL and ⌘2L in Proposition 4.6.3
qL =
�2
2
pZ(0)
⌘2L = �2pZ(0)2Z d
0
¯FZ(x)2
pZ(x)dx.
108
�e duality between the expressions for qU , ⌘2U and qL, ⌘2L can be com-pletely explained by looking at the OU process X 0
:= d � X , its doublyre�ected version and the associated increasing processes U 0 and L0. One canshow that Z 0, denoting a random variable having the stationary distributionof this doubly re�ected process, has the propertyP(Z 0 � x) = P(Z d�x),for all x 2 [0, d], so that pZ0
(0) = pZ(d). Similarly, the processes U 0 and L0
can be interpreted as U and L swapped.We proceed to re�ne the assertions of Propositions 4.6.2 and 4.6.3 to weak
convergence results.
Proposition 4.6.4 Let ⌘2U and qU be de�ned in (4.5) and Lemma 4.6.1 respec-tively. As n ! 1, the scaled loss process
Unt :=
Unt � qUnt
⌘Upn
converges weakly with respect to CR+ to Bt,
where Bt is a standard Brownian motion. Likewise, the scaled loss processLnt :=
Lnt
�qL
nt⌘L
pn
, with ⌘2L and qL as in (4.6), converges weakly with respectto CR+ to a standard Brownian motion.
Proof It su�ces to give the proof forUnt . By speeding up the time and scaling
down the space, the expression (4.4) in the proof of Proposition 4.6.2 becomes
Unt � qUntpn
=
h(Z0)� h(Znt)pn
+
�pn
Z nt
0h0(Zs)dBs.
By the ergodic theorem in Gihman and Skorohod [17, p. 134], for arbitraryt 2 R+,
h �pn
Z n·
0h0(Zs)dBsit =
�2t
nt
Z nt
0h0(Zs)
2ds
P! ⌘2U t, as n ! 1.
Since Znt 2 [0, d] and h is continuous, h(Znt) is bounded. So,
sup
t>0
h(Z0)� h(Znt)pn
! 0 a.s. as n ! 1.
�en the claim is proved by applying�eorem 1.1.1, which is the functionalcentral limit theorem for semimartingales, to Un
t . ⇤
109
110
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Summary
Limit theorems forMarkov-modulated and re�ecteddi�usion processes
In this thesis, asymptotic properties of two variants of one-dimensional di�u-sion processes, which are Markov-modulated and re�ected Ornstein-Uhlen-beck processes, are studied. Besides the random term of the Brownian mo-tion, the Markov-modulated di�usion process evolves in an extra randomenvironment, namely the �nite-state Markov chain. �e re�ected Ornstein-Uhlenbeck process behaves as an Ornstein-Uhlenbeck process which has in-stantaneous re�ection at boundaries. �ey are widely used in modeling dueto abovementioned their distinctive features and great analytical tractability.
We obtain four limit theorems from the perspectives of weak conver-gence and large deviations. Firstly, we prove weak convergence of a se-quence of Markov-modulated di�usion processes with rapid switching toan ordinary di�usion process by verifying its tightness property. Secondly,a sample-path large deviations principle for the coupling of the Markov-modulated di�usion process with small noise and the occupation measureof the rapid switching Markov chain is obtained. �e large deviations princi-ples for each individual term are derived by the contraction principle. �oseresults reveal interesting behavior of Markov-modulated di�usion processwhen the modulating Markov chain switches fast. �irdly, transient asymp-totics of large deviations type are acquired for re�ected and doubly re�ectedOrnstein-Uhlenbeck processes. Fourthly, we prove central limit theoremsand functional central limit theorems for the centered and scaled loss andidle processes of doubly re�ected Ornstein-Uhlenbeck processes.
117
118
Samenvatting
Limietstellingen voorMarkovgemoduleerde en gere-�ecteerde di�usieprocessen
In dit proefschri� bestuderen we asymptotische eigenschappen van tweevarianten op di�usieprocessen, Markovgemoduleerde di�usieprocessen engere�ecteerde Ornstein-Uhlenbeckprocessen. Een Markovgemoduleerd dif-fusieproces wordt niet alleen als een gewone di�usie aangedreven door eenBrownse beweging, maar beweegt zich tevens in een stochastische omge-ving, namelijk die van een Markovketen met eindig veel toestanden. Hetgere�ecteerde Ornstein-Uhlenbeckproces gedraagt zich als een gewoon Orn-stein-Uhlenbeckproces in het inwendige van zijn toestandsruimte, maar metinstantane re�ectie op de rand ervan. Zulke processen worden veel in mo-dellen gebruikt waar hun kenmerkende eigenschappen van belang zijn, envanwege hun analytische hanteerbaarheid.
Voor deze processen verkrijgenwe vier limietstellingen op het gebied vanzwakke convergentie en grote afwijkingen. Als eerste bewijzen we zwakkeconvergentie van een rij Markovgemoduleerde di�usieprocessen naar eengewone di�usie als de sprongintensiteiten van de Markovketen onbeperktopgevoerd worden, waarin veri�catie van tightness (uniforme beperktheid)centraal staat. Als tweede verkrijgen we een limietstelling voor kansen opgrote afwijkingen op het niveau van de paden van een Markovgemoduleerdedi�usie met verdwijnende ruisterm, tezamen met grote afwijkingen voor depaden van de verblijfstijden van de Markovketen in zijn toestanden waar-van de sprongintensiteiten onbegrensd stijgen. Resultaten voor kansen opgrote afwijkingen van elk van de componenten afzonderlijk worden afge-
119
leid via het contractieprincipe. Deze resultaten geven een interessant in-zicht in het gedrag van een Markovgemoduleerde di�usie als de Markovke-ten zeer snel door de toestanden springt. In de derde plaats verkrijgen weasymptotische resultaten voor het overgangsgedrag van gere�ecteerde endubbelgere�ecteerde Ornstein-Uhlenbeckprocessen. In de vierde plaats be-wijzen we gewone en functionele centrale limietstellingen voor de gecen-treerde en geschaalde verliesprocessen van dubbelgere�ecteerde Ornstein-Uhlenbeckprocessen.
120
