SMALL-TIME EXPANSIONS FOR LOCAL JUMP-DIFFUSIONS … · SMALL-TIME EXPANSIONS FOR LOCAL JUMP-DIFFUSIONS MODELS WITH INFINITE JUMP ACTIVITY JOSE E. FIGUEROA-L OPEZ AND CHENG OUYANG

SMALL-TIME EXPANSIONS FOR LOCAL JUMP-DIFFUSIONS MODELS WITHINFINITE JUMP ACTIVITY

JOSE E. FIGUEROA-LOPEZ AND CHENG OUYANG

Abstract. We consider a Markov process X(x)t t≥0 with initial condition X

(x)t = x, which is the

solution of a stochastic differential equation driven by a Levy process Z and an independent Wienerprocess W . Under some regularity conditions, including non-degeneracy of the diffusive and jumpcomponents of the process as well as smoothness of the Levy density of Z, we obtain a small-timesecond-order polynomial expansion in t for the tail distribution and the transition density of theprocess X. The method of proof combines a recent approach for regularizing the tail probability

P(X(x)t ≥ x + y) with classical results based on Malliavin calculus for purely-jump processes, which

have to be extended here to deal with the mixture model X. As an application, the leading term forout-of-the-money option prices in short maturity under a local jump-diffusion model is also derived.

1. Introduction

It is well-known by practitioners that the market implied volatility skewness is more pronouncedas the expiration time approaches. Such a phenomenon indicates that a jump risk should be includedinto classical purely-continuous financial models (e.g. local volatility models and stochastic volatilitymodels) to reproduce more accurately the implied volatility skews observed in short-term optionprices (see [8], Chapter 1 for further motivation of jumps in asset price modeling). Moreover, furtherstudies have shown that accurate modeling of the option market and asset prices requires a mixture ofa continuous diffusive component and a jump component. For instance, based on statistical methods,several empirical studies have rejected statistically the null hypothesis of either a purely-continuousor a purely-jump model (see, e.g., [1], [2], [3], [26]). Similarly, by contrasting the observed behaviorof at-the-money (ATM) and out-of-the-money (OTM) call option prices near expiration with theiranalog theoretical behaviors with and without jumps, [7] and [20] argued that both continuous andjump component are necessary to explain the implied volatility behavior of S&P500 index options.

The study of small-time asymptotics of option prices and implied volatilities has grown significantlyduring the last decade, as it provides a convenient tool for testing various pricing models and cali-brating parameters in each model. For recent accounts of the subject in the case of purely-continuousmodes, we refer the reader to [15] for local volatility models, [9] for the Heston model, [13] for ageneral uncorrelated local-stochastic volatility model, and [5] for general stochastic volatility models.Broadly, for purely-continuous models, OTM option prices are asymptotically equivalent to O(e−c/t),where t is the time-to-maturity. In this case, the corresponding implied volatility converges to a non-negative constant which is characterized by the Riemannian distance from the current stock price tothe strike price under the Riemannian metric induced by the diffusion coefficient for the model. For

1991 Mathematics Subject Classification. 60J75, 60G51, 60F10.First author supported in part by NSF Grant DMS-0906919.

1

2 JOSE E. FIGUEROA-LOPEZ AND CHENG OUYANG

Levy process, [10], [28] and [31] show that OTM option prices are generally asymptotically equiv-alent to the time-to-maturity t. This fact in turn implies that the corresponding implied volatilityexplodes as t → 0 in such models. The explosion rate was proved to be asymptotically equivalentto (t log(1/t))−1 in [31], while the second order and higher order asymptotics where obtained in [10]and [14], respectively. A similar result was also obtained for a stochastic volatility model with inde-pendent Levy jumps in [12], while [21] studied the first order short-time asymptotics for OTM andATM option prices under a relatively general class of Ito semimartingales with jumps.

In spite of the ample literature on the asymptotic behavior of the transition densities and optionprices for either purely-continuous or purely-jump models, results on local jump-diffusion models arescarce. One difficulty to deal with these models arises from the more complex interplay of the jumpand continuous components. The current article is thus an important contribution in this direction,by analyzing the higher-order asymptotic behavior of the transition distributions and densities in localjump-diffusion models. In addition to the asymptotics of vanilla option prices, short-time asymptoticsfor the transition densities and distributions of Markov processes have also proved to be useful inother financial applications such as non-parametric estimation methods of the model under high-frequency sampling data and Monte-Carlo based pricing of path-dependent options such as lookbackor Asian options. In all these applications, a certain discretization of the continuous-time objectunder study is needed and, in that case, short-time asymptotics are important in the study of therate of convergence of the discrete-time proxies to their continuous-time analogs.

The small-time behavior for the transition densities of Markov processes X(x)t t≥0 with determin-

istic initial condition X(x)0 = x has been studied for a long time in the literature, especially for either

a purely-continuous or a purely-discontinuous process. Starting from the problem of existence, thereare several sets of sufficient conditions for the existence of the transition densities, y → pt(y;x),largely based on the machinery of Malliavin calculus, originally developed for continuous diffusions(see the monograph [22]) and, then, extended to Markov process with jumps (see the monograph[6]). This approach can also yield estimates of the transition density pt(y;x) in small time t. Forpurely-jump Markov processes, the key assumption is that the Levy measure of the process admitsa smooth Levy density. The pioneer of this approach was Leandre [17], who obtained the first-ordersmall-time asymptotic behavior of the transition density for fully supported Levy densities. Thisresult was extended in [16] to the case where the point y cannot be reached with only one jump fromx but rather with finitely many jumps, while [24] developed a new method for proving existence ofthe transition densities which allows Levy measures with a possibly singular component.

The main result in [17] states that

limt→0

1

tpt(y;x) = g(y;x),

where g(y;x) is the Levy density of the process X to be defined below (see (1.6)). Leandre’s approachconsists of separating the small jumps (say, those with sizes smaller than an ε > 0) and the large jumpsof the underlying Levy process, and condition on the number of large jumps by time t. Malliavin’scalculus is then applied to control the density given that there is no large jump. For ε > 0 smallenough, the term when there is only one large jump is proved to be equivalent, up to a remainder oforder o(t), to the resulting term resulting as if there were no small-jump component at all. Finally,the terms when there is more than one large jump are shown to be of order O(t2).

Higher order expansions of the transition density of Markov processes with jumps have been con-sidered quite recently and only for processes with finite jump activity (see, e.g., [32]) or for Levy

SMALL-TIME EXPANSIONS FOR JUMP-DIFFUSIONS 3

processes with possibly infinite jump-activity. We focus on the literature of the latter case due toits close connection to the present work. The first attempt in obtaining higher order expansions forLevy processes was given in [29] using Leandre’s approach. Concretely, the following expansion forthe transition densities pt(y)t≥0 of a Levy process Ztt≥0 was proposed there:

(1.1) pt(y) :=d

dyP(Zt ≤ y) =

N−1∑n=1

an(y)tn

n!+O(tN), (y 6= 0, N ∈ N).

As in [17], the idea was to justify that each higher order term (say, the term corresponding to klarge jumps) can be replaced, up to a remainder of order O(tN), by the resulting density as if therewere no small-jump component. However, this method of proof fails for k ≥ 2 and the expressionsproposed there were in general incorrect for any higher order term (in fact, only valid for compoundPoisson processes). This issue was subsequently resolved in [11] using a new approach, where theexpansion (1.1) was obtained under the assumption that the Levy density of the Levy process Ztt≥0

is sufficiently smooth and the following technical condition

(1.2) sup0<s<t0

sup|y|>δ|p(k)s (y)| <∞,

for any k ∈ N and δ > 0, and some t0 := t0(δ, k) > 0. There are two key ideas in [11]. Firstly,instead of working directly with the transition densities, the following analog expansions for the tailprobabilities were considered:

(1.3) P(Zt ≥ y) =N−1∑n=1

An(y)tn

n!+ tNRt(y), (y > 0, N ∈ N),

where sup0<t≤t0 |Rt(y)| <∞, for some t0 > 0. Secondly, by considering a smooth thresholding of thelarge jumps (so that the density of large jumps is smooth) and conditioning on the size of the firstjump, it was possible to regularized the discontinuous functional 1Zt≥x and, subsequently, proceedto use an iterated Dynkin’s formula to expand the resulting smooth moment functions E(f(Zt)) as apower series in t. (1.1) was then obtained by differentiation of (1.3), after justifying that the functionsAn(y) and the remainder Rt(y) were differentiable in y. Recently, [12] formally proved that (1.1) canbe derived from (1.3) using only the conditions of Leandre [17], without the technical condition (1.2).

The results described in the previous paragraph open the door to the study of higher order ex-pansions for the transition densities of more general Markov models with infinite jump-activity. Wetake the analysis one step further and consider a jump-diffusion model with non-degenerate diffusionand jump components. Our analysis can also be applied to purely-discontinuous processes as in [17],but we prefer to take a “mixture model” due to its relevance in financial applications where, as ex-plained above, both continuous and jump components are necessary. More concretely, we considerthe following Stochastic Differential Equations (SDE):

(1.4) Xt(x) = x+

∫ t

0

b(Xu(x))du+

∫ t

0

σ (Xu(x)) dWu +∑

u∈(0,t]: ∆Zu 6=0

γ (Xu−(x),∆Zu) ,

where b, σ : R→ R, and γ : R×R→ R are some suitable deterministic functions, Wtt≥0 is a Wienerprocess, and Ztt≥0 is an independent purely-jump Levy process of bounded variation. The lattercondition on Z is not essential for our analysis and it is only imposed for the easiness of notation.


Under some regularization conditions on b, σ and γ, as well as the Levy measure ν of Z, we showthe following second order expansion (as t→ 0) for the tail distribution of Xt(x)t≥0:

(1.5) P(Xt(x) ≥ x+ y) = tA1(x; y) +t2

2A2(x; y) +O(t3), for x ∈ R, y > 0.

The key assumptions on the SDE’s coefficients are some non-degeneracy conditions on σ and γ thatensure the existence of the transition density and that allow us to use Malliavin calculus. As in [17],the key assumption on the Levy measure ν of Z is that this admits a density h : R\0 → R+ thatis bounded and sufficiently smooth outside any neighborhood of the origin. In that case, the leadingterm A1(x; y) depends only on the jump component of the process as follows

A1(x; y) = ν(ζ : γ(x, ζ) ≥ y) =

∫ζ:γ(x,ζ)≥y

h(ζ)dζ.

The second order term A2(x; y) admits a slightly more complicated (but explicit) representation. Asa consequence of this “correction” term A2(x; y), one can explicitly characterize the effects of the driftb and the diffusion σ of the process in the likelihood of a “large” positive move (say, a move of sizemore than y) during a short time period t. For instance, in the case that γ does not dependent onthe current state x (i.e. γ(x, ζ) = γ(ζ) for all x), we found that a positive spot drift value of b(x) willincrease the probability of a “large” positive move by b(x)g(y)t2(1 + o(1)), where g(y) is the so-calledLevy density of the process X defined by g(y) := − d

dyν(ζ : γ(ζ) ≥ y). Similarly, since typically

g′(y) < 0 when y > 0, the effect of a non-zero spot volatility σ(x) is to increase the probability of a“large” positive move by 2−1σ2(x)|g′(y)|t2(1 + o(1)).

Once the asymptotic expansion for tail distribution is obtained, we take the analysis one step furtherand obtain a second order expansion for the transition density function pt(y;x) by taking derivativeof the tail distribution with respect to y, as long as proper regularity conditions are assumed. Asexpected, the leading term of the transition density pt(y;x) is of the form tg(y;x), where g(y;x) isthe so-called Levy density of the process Xt(x)t≥0 defined by

(1.6) g(y;x) := − d

dyν(ζ : γ(x, ζ) ≥ y).

One of the main difficulties here comes from controlling the first term (which corresponds to the case ofno “large” jumps). Hence, we generalize the result in [17] to the case where there is a non-degenaratediffusion component. Again, Malliavin calculus is proved to be the key tool for this task.

As an application of the above asymptotic formulas, we derive the leading term of the small-timeexpansion for option prices of out-of-the-money European call options with strike price K and expiryT . Specifically, let Stt≥0 be the stock price process and denote Xt = logSt for each t ≥ 0. Weassume that P is the option pricing measure and that under this measure the process Xtt≥0 is ofthe form in (1.4). Then, under certain regularity conditions, we prove that

(1.7) limt→0

1

tE(St −K)+ =

∫ ∞−∞

(S0e

γ(x,ζ) −K)

+h(ζ)dζ.

A special case of the above result is when γ(x, ζ) = ζ. The model reduces to an exponential Levymodel. In this case, the above first order asymptotics becomes

limt→0

1

tE(St −K)+ =

∫ ∞−∞

(S0e

ζ −K)

+h(ζ)dζ,


which recovers the well-known first-order asymptotic behavior for exponential Levy model (see, e.g.,[28] and [31]). It is important to note that (1.7) is derived from the asymptotics for the tail distri-butions (1.5) rather than from those for the transition density, given that the former requires lessstringent conditions on the coefficients of (1.4). An important related paper is [19], where (1.7) wasobtained for a wide class of multi-factor Levy Markov models under certain technical conditions (seeTheorem 2.1 therein), including the requirement that limt→0 E(St −K)+/t exists in the “out-of-the-money region” and an integrability condition such as

∫(|γ(x, ζ)|2 ∧ 1)eγ(x,ζ)+ε|γ(x,ζ)|h(ζ)dζ < ∞, for

some ε > 0.

The paper is organized as follows. In Section 2, we introduced the model and the assumptionsneeded for our results. The probabilistic tools, such as the iterated Dynkin’s formula as well as tailestimates for semimartingales with bounded jumps, are presented in Section 3. The main results of thepaper are then stated in Sections 4 and 5, where the second order expansion for the tail distributionsand the transition densities are obtained, respectively. The application of the expansion for the taildistribution to option pricing in local jump-diffusion financial models is presented in Section 6. Theproofs of our main results as well as some preliminaries of Malliavin calculus on Wiener-Poisson spacesare given in several appendices.

2. The Setup and Assumptions

In this paper, let Ztt≥0 be a driftless pure-jump Levy process of bounded variation and letWtt≥0 be a Wiener process independent of Z, both of which are defined on a complete probabilityspace (Ω,F ,P) equipped with the natural filtration (Ft)t≥0, generated by W and Z and augmentedby all the null sets in F so that it satisfies the usual conditions (see, e.g., [27]). We consider thefollowing local jump-diffusion model

(2.1) Xt(x) = x+

∫ t

0

b(Xu(x))du+

∫ t

0

σ (Xu(x)) dWu +∑

0<u≤t

γ (Xu−(x),∆Zu) .

where b, σ : R → R and γ : R × R → R are suitable deterministic functions so that (2.1) admits aunique solution (see, e.g., [23]). It is convenient to write Z in terms of its Levy-Ito representation:

Zt =∑

0<u≤t

∆Zu =

∫ t

0

∫R0

ζM(du, dζ).

Here, R0 = R\0 and M is the jump measure of the process Z, which is a Poisson random measureon R+ × R0. Furthermore, we make the following standing assumptions about Z:

(C1) The Levy measure ν of Z has a C∞(R\0) strictly positive density h such that,for every ε > 0 and n ≥ 0,

(2.2) (i)

∫(1 ∧ |ζ|)h(ζ)dζ <∞, and (ii) sup

|ζ|>ε|h(n)(ζ)| <∞.

Remark 2.1. The condition of bounded variation on Z is not essential for our results and it isimposed only for the easiness of notation. We could have taken a general Levy process Z and replacethe last term in (2.1) with a finite-variation pure-jump process plus a compensated Poisson integral


as follows:

Xt(x) = x+

∫ t

0

b(Xu(x))du+

∫ t

0

σ (Xu(x)) dWu

+

∫ t

0

∫|ζ|>1

γ (Xu−(x), ζ)M(du, dζ) +

∫ t

0

∫|ζ|≤1

γ (Xu−(x), ζ) M(du, dζ),

where as usual M(du, dζ) is the compensated Poisson integral (M(du, dζ)− ν(dζ)du).

Throughout the paper, the generator γ satisfies the following assumption:

(C2) For any x, ζ ∈ R, γ(x, 0) = 0, γ(·, ·) ∈ C∞, and

(2.3) supx

∣∣∣∣∂iγ(x, ζ)

∂xi

∣∣∣∣ ≤ C(ζ2 ∧ 1),

for i = 0, 1, 2 and C <∞. In particular, supx,ζ

∣∣∣∂iγ(x,ζ)∂xi

∣∣∣ <∞, for i = 0, 1, 2.

We also assume the following non-degeneracy and boundedness conditions, which were also imposedin [17]:

(C3-a) The functions b(x) and v(x) := σ2(x)/2 belong to C∞b (R) and there exists aconstant δ > 0 such that σ(x) ≥ δ for all x.

(C3-b) For the same constant δ above, we have∣∣∣∣1 +∂γ(x, ζ)

∂x

∣∣∣∣ ≥ δ for all x, ζ ∈ R.

As it is usually the case with Levy processes, we shall decompose Z into a compound Poissonprocess and a process with bounded jumps. More specifically, let φε ∈ C∞(R) be a truncationfunction such that 1|ζ|≥ε ≤ φε(ζ) ≤ 1|ζ|≥ε/2 and let Zt(ε)t≥0 and Z ′t(ε)t≥0 be independent driftlessLevy processes of bounded variation with Levy measures φε(ζ)h(ζ) and (1−φε(ζ))h(ζ), respectively.Without loss of generality, we assume hereafter that

(2.4) Zt = Zt(ε) + Z ′t(ε).

Note that Zt(ε) is a compound Poisson process with intensity of jumps λε :=∫φε(ζ)h(ζ)dζ and

jumps with probability density function

(2.5) hε(ζ) :=φε(ζ)h(ζ)

λε.

Throughout the paper, J := Jε will represent a random variable with density hε(ζ). The followingassumptions will also be needed in the sequel:

(C4-a) For any u ∈ R and ε > 0 small-enough (independent of u), the random variableγ(u, Jε) admits a density Γ(ζ;u) := Γε(ζ;u) in C∞b (R0×R), the class of function withcontinuous partial derivatives of arbitrary order such that, for every n,m ≥ 0,

supu,ζ

∣∣∣∣∂n+mΓ(ζ;u)

∂ζn∂um

∣∣∣∣ <∞.


(C4-b) In addition to (C4-a), we also have that, for any z ∈ R, δ′ > 0, and m ≥ 0,there exists a δ > 0 such that∫ ∞

δ′sup

u∈(z−δ,z+δ)

∣∣∣∣∂mΓ(ζ;u)

∂um

∣∣∣∣ dζ <∞.Remark 2.2. Note that

Γε(ζ;u) = − ∂

∂ζ

∫1γ(u;y)≥ζhε(y)dy = − 1

λε

∂

∂ζ

∫1γ(u;y)≥ζφε(y)h(y)dy.

In particular, if for each u ∈ R, the function y → γ(u, y) is strictly increasing, then

Γε(ζ;u) = hε(γ−1(u; ζ)

)(∂γ∂ζ

(u, γ−1(u; ζ))

)−1

,

where γ−1(u; ζ) is the inverse of the function y → γ(u, y) (i.e. γ(u; γ−1(u; ζ)) = ζ). As a consequence,since γ−1(u; ζ) 6= 0 for any ζ 6= 0, there exists an ε0(u) > 0 small enough such that

λεΓε(ζ;u) = h(γ−1(u; ζ)

)(∂γ∂ζ

(u; γ−1(u; ζ))

)−1

=: a1(u; ζ),

for any ε < ε0. Note also that for Γε(ζ;u) to be bounded, it suffices that (2.2-ii) with n = 0 and

infu,ζ

∣∣∣∂γ(u;ζ)∂ζ

∣∣∣ > 0 hold true. Similarly, ∂Γε(ζ;u)/∂ζ can be written as

h′ε(γ−1(u; ζ)

)(∂γ∂ζ

(u, γ−1(u; ζ))

)−2

−hε(γ−1(u; ζ)

)∂2γ

∂ζ2(u, γ−1(u; ζ))

(∂γ

∂ζ(u, γ−1(u; ζ))

)−3

,

which can be bounded uniformly if, in addition, (2.2-ii) with n = 1 and (2.3) hold true. One cansimilarly deal with higher order derivatives of Γε. This observation justifies the usage of a truncationfunction φε to regularize the Levy density h around the origin.

We next define an important class of processes. For a collection of times s1, . . . , sn in R+, letXt(ε, s1, . . . , sn, x)t≥0 be the solution of the SDE:

Xt(ε, s1, . . . , sn, x) := x+

∫ t

0

b(Xu(ε, s1, . . . , sn, x))du+

∫ t

0

σ (Xu(ε, s1, . . . , sn, x)) dWu

+∑

0<u≤t

γ(Xu−(ε, s1, . . . , sn, x),∆Z ′u(ε)) +∑i:si≤t

γ(Xs−i(ε, s1, . . . , sn, x), Jεi ),

where Jεi i≥1 represents a random sample from the distribution φε(ζ)h(ζ)dζ/λε, independent of Z ′.Similarly, we define Xt(ε, ∅, x)t≥0 as the solution of the SDE:

(2.6) Xt(ε, ∅, x) := x+

∫ t

0

b(Xu(ε, ∅, x))du+

∫ t

0

σ (Xu(ε, ∅, x)) dWu+∑

0<u≤t

γ (Xu−(ε, ∅, x),∆Z ′u(ε)) .

3. Probabilistic tools

Throughout this section, Cnb (I) (resp. Cn

b ) denotes the class of functions having continuous andbounded derivatives of order 0 ≤ k ≤ n in an open interval I ⊂ R (resp. in R). Also, ‖g‖∞ =supy |g(y)|.


3.1. Uniform tail probability estimates. The following general result will be important in thesequel.

Proposition 3.1. Let M be a Poisson random measure on R+×R0 and M be its compensated randommeasure. Let Y := Y (x) be the solution of the SDE

Yt = x+

∫ t

0

b(Ys)ds+

∫ t

0

σ(Ys)dWs +

∫ t

0

∫γ(Ys−, ζ)M(ds, dζ).

Assume that b(x) and σ are uniformly bounded and that γ(x, ζ) are regular enough so that the secondand third integral on the right hand side of the above equation are martingales. Furthermore, supposethe jumps of Ytt≥0 are bounded by some constant S. Then there exists a constant C(S) dependingon S, such that, for all 0 ≤ t ≤ 1,

P

sup0≤s≤t

|Y (x)s − x| ≥ 2pS

≤ C(S)tp.

Proof. Let Mt be either ∫ t

0

σ(Ys)dWs, or

∫ t

0

∫γ(Ys−, z)M(ds, dz).

It is clear that Mt is a martingale with its jumps bounded by S. Moreover, the quadratic variation〈M,M〉 satisfies

〈M,M〉t ≤ kt

for some constant k. By [18] (or (1.32) in [17]) we have

P

sup0≤s≤t

|Ms| ≥ C

≤ 2 exp

[−λC +

λ2

2S2kt(1 + exp[λS])

].(3.1)

Now take C = 2pS and λ = | log t|/2S. The rest of the proof is then clear.

As a direct corollary of the above proposition, we have the following tail probability estimate forthe small-jump component Xt(ε, ∅, x)t≥ of X introduced in (2.6).

Lemma 3.2. Fix any η > 0 and a positive integer N . Then, under the Conditions (C2-C3) ofSection 2, there exists an ε > 0 and C := C(N, η) <∞ such that

(1) For all t < 1,

(3.2) sup0<ε′<ε,x∈R

P(|Xt(ε′, ∅, x)− x| ≥ η) < CtN .

(2) For all t < 1,

supε′<ε,x∈R

∫ ∞η

Pe|Xt(ε′,∅,x)−x| ≥ sds = supε′<ε,x∈R

∫ ∞η

P|Xt(ε′, ∅, x)− x| ≥ log sds < CtN .

Proof. The first statement is a special case of Proposition 3.1, which can be applied in light of theboundedness conditions (C3) as well as the condition (C2) that ensure that the components∫ t

0

σ (Xu(ε′, ∅, x)) dWu,

∫ t

0

∫γ (Xu−(ε′, ∅, x), ζ) M ′(du, dζ),


are martingales. Here, M ′ is the compensated Poisson measure of Z ′(ε′) (see Remark 3.4 below forthe notation). To prove the second statement, we keep the notation of the proof of Proposition 3.1.By (3.1) there exists a constant C > 0 such that∫ ∞

η

P|Xt(ε, ∅, x)− x| ≥ log sds ≤ C

∫ ∞η

exp

[−λ log s+

λ2

2ε2kt(1 + exp[λε])

]ds

=Cη

(λ− 1)ηλexp

[λ2

2ε2kt(1 + exp[λε])

].

Now it suffices to take λ = | log t|/2ε and ε = log η/2N .

3.2. Iterated Dynkin’s formula. The following well-known formula will be important in the sequel.The case n = 0 can be proved using Ito’s formula (see Section 1.5 in [23]), while the general case isproved by induction.

Lemma 3.3. Let Y := Y (x) be a solution of the SDE

dYt = b(Yt)dt+ σ(Yt)dWt +

∫|ζ|≤1

γ(Yt− , ζ)M(dt, dζ) + +

∫|ζ|>1

γ(Yt− , ζ)M(dt, dζ),(3.3)

Y0 = x,(3.4)

and let L be its infinitesimal generator, defined by

(Lf)(y) :=σ2(y)

2f ′′(y) + b(y)f ′(y) +

∫ (f(y + γ(y, ζ))− f(y)− γ(y, ζ)f ′(y)1|ζ|≤1

)ν(dζ),(3.5)

for any f ∈ C2b . Then, we have

(3.6) Ef(Yt) = f(x) +n∑k=1

tk

k!Lkf(x) +

tn+1

n!

∫ 1

0

(1− α)nELn+1f(Yαt)

dα,

valid for any n ≥ 0 and f ∈ C2b such that Lkf ∈ C2

b , for each k = 1, . . . , n.

Remark 3.4. Below, Y is taken to be the process (2.6), which is of the form (3.3-3.4) with underlyingdriving Levy process Z ′t(ε)t≥0 (the small-jump component of Z defined in (2.4)). In particular,denoting M ′ the jump measure of Z ′(ε), we have

Xt(ε, ∅, x) = x+

∫ t

0

b (Xu(ε, ∅, x)) du+

∫ t

0

σ (Xu(ε, ∅, x)) dWu +

∫ t

0

∫γ (Xu−(ε, ∅, x), ζ)M ′(du, dζ).

Since (1 − φε(ζ))ν(dζ) is the Levy measure of Z ′(ε), we can write Xt(ε, ∅, x) as in (3.3-3.4) withγ(y, ζ) = γ(y, ζ), σ(y) = σ(y), and

b(y) = b(y) +

∫|ζ|≤1

γ(y, ζ)(1− φε(ζ))ν(dζ).

After substitution of these coefficient functions in (3.5) and some simplification, we get the followinginfinitesimal generator for Xt(ε, ∅, x)t≥0:

Lεf(y) :=σ2(y)

2f ′′(y) + b(y)f ′(y) +

∫(f(y + γ(y, ζ))− f(y)) hε(ζ)dζ,(3.7)

where hε(ζ) := (1− φε(ζ))h(ζ).


The following result gives conditions for the validity of the second-order Dynkin’s formula (3.6)when Y is the small-jump component of Z (its proof is reported in Appendix D).

Lemma 3.5. Supposed that γ, b, and σ satisfy the Conditions (C2-C3) of Section 2 and let g be abounded functions that is C4

b in a neighborhood of x. Then, the expansion

(3.8) Ef (Xt(ε, ∅, x)) = f(x) + tLεf(x) +t2

2Rt(ε, x),

holds true with Rt(ε, x) such that

(3.9) sup0<t<1, ε′<ε

|Rt(ε′, x)| ≤ K

for a constant K < ∞ depending only on∫

(1 ∧ ζ2)h(ζ)dζ, ‖f (k)1(x−δ,x+δ)‖∞ for some small-enough

δ > 0, ‖b(k)‖∞, and ‖v(k)‖∞ (k = 0, 1, 2). Moreover, if f ∈ C4b , then

(3.10) sup0<t<1, ε′<ε,x∈R

|Rt(ε′, x)| ≤ K,

for some K <∞.

The following small-time expansion will also be needed (see the Appendix D for a proof).

Lemma 3.6. Let s ∈ (0, t) and f ∈ C2b . Then,

Ef (Xt(ε, s, x)) = G0(x) + sRs(x) + (t− s)ERt−s(Xs(ε, ∅, x)),

where G0(x) := E [f (x+ γ(x, J))], supu>0,z∈R |Ru(z)| <∞, and supu>0,z∈R |Ru(z)| <∞.

4. Second order expansion for the tail distributions

In this section, we consider the small-time behavior of the tail distribution of Xt(x)t≥0:

(4.1) Ft(x, y) := P(Xt(x) ≥ x+ y), (y > 0).

The key idea is to use the decomposition (2.4) by conditioning on the number of “large” jumpsoccurring before time t. Concretely, denoting N ε

t t≥0 and λε :=∫φε(ζ)h(ζ)dζ the jump counting

process and the jump intensity of the compound Poisson process Zt(ε)t≥0, respectively, we have

(4.2) P(Xt(x) ≥ x+ y) = e−λεt∞∑n=0

P(Xt(x) ≥ x+ y|N εt = n)

(λεt)n

n!.

The first term in (4.2) (when n = 0) can be written as

P(Xt(x) ≥ x+ y|N εt = 0) = P(Xt(ε, ∅, x) ≥ x+ y).

In light of (3.2), this term can be made O(tN) for an arbitrarily large N ≥ 1, by taking ε small enough.In order to deal with the other terms in (4.2), we use the iterated Dynkin’s formula introduced inSection 3.2.

The following is the main result of this section (see Appendix B for the proof). Below, J denotesa random variable with density (2.5), J1 and J2 denote independent copies of J , and Γ(ζ;x) denotesthe density of γ(x, J).


Theorem 4.1. Let x ∈ R and y > 0. Then, under the Conditions (C1-C3) of Section 2, we have

(4.3) Ft(x, y) := P(Xt(x) ≥ x+ y) = tA1(x; y) +t2

2A2(x; y) +O(t3),

as t→ 0, where A1(x; y) and A2(x; y) admit the following representations (for ε > 0 small-enough):

A1(x; y) := λεH0,0(x; y) =

∫γ(x,ζ)≥y

h(ζ)dζ,

A2(x; y) := λε [H0,1(x; y) +H1,0(x; y)] + λ2ε [H2,0(x; y)− 2H0,0(x; y)] ,

with

λε =

∫φε(ζ)h(ζ)dζ, H0,0(x; y) = P [γ(x, J) ≥ y] =

∫ ∞y

Γ(ζ;x)dζ,

H0,1(x; y) = b(x)

(∫ ∞y

∂Γ(ζ;x)

∂xdζ + Γ(y;x)

)+σ2(x)

2

(∫ ∞y

∂2Γ(ζ;x)

∂x2dζ + 2

∂Γ(y;x)

∂x− ∂Γ(y;x)

∂y

)+ H0,1(x; y),

H0,1(x; y) =

∫(P [γ(x+ γ(x, ζ), J) ≥ y − γ(x, ζ)]− P [γ(x, J) ≥ y]) hε(ζ)dζ,(4.4)

H1,0(x; y) = b(x)Γ(y;x)− σ2(x)

2

∂Γ(y;x)

∂y+ H1,0(x; y),

H1,0(x; y) =

∫(P [γ(x, J) ≥ y − γ(x, ζ)]− P [γ(x, J) ≥ y]) hε(ζ)dζ,

H2,0(x; y) = P (γ(x+ γ(x, J2), J1) ≥ y − γ(x, J2)) .

Remark 4.2. Note that if supp(ν) ∩ ζ : γ(x, ζ) ≥ y = ∅ (so that it is not possible to reach thelevel y from x with only one jump), then A1(x; y) = 0 and P(Xt(x) ≥ x + y) = O(t2) as t → 0. If,in addition, it is possible to reach the level y from x with two jumps, then H2,0(x; y) 6= 0 and, hence,A2(x; y) 6= 0, implying that P(Xt(x) ≥ x + y) decreases at the order of t2. These observations areconsistent with the results in [16] and [25].

Example 4.3. In the case that the generator γ(x, ζ) does not depend on x, we get the followingexpansion for P(Xt(x) ≥ x+ y):

λεP(γ(J) ≥ y) t+ λ2ε (P(γ(J1) + γ(J2) ≥ y)− 2P(γ(J) ≥ y))

t2

2

+ λε

(b(x)Γ(y)− σ2(x)

2Γ′(y) +

∫(P [γ(J) ≥ y − γ(ζ)]− P [γ(J) ≥ y]) hε(ζ)dζ

)t2 +O(t3).

The leading term in the above expression is dominated by the jump component of the process and ithas a natural interpretation: if within a very short time interval there is a “large” positive move (say,a move by more than y), this move must be due to a jump. It is until the second term, when thediffusion and drift terms of the process appear. Denoting

g(y) := − d

dyν(ζ : γ(ζ) ≥ y) = − d

dy

∫γ(ζ)≥y

h(ζ)dζ,


the so-called Levy density of the process Xtt≥0, the effect of a positive spot drift b(x) > 0 is toincrease the probability of a “large” positive move by b(x)g(y)t2(1 + o(1)). Similarly, since typicallyg′(y) < 0 when y > 0, the effect of a non-zero spot volatility σ(x) is to increase the probability of a

“large” positive move by σ2(x)2|g′(y)|t2(1 + o(1)).

5. Expansion for the transition densities

Our goal here is to obtain a second-order small-time approximation for the transition densitiespt(·;x)t≥0 of Xt(x)t≥0. As it was done in the previous section, the idea is to work with theexpansion (4.2) by first showing that each term there is differentiable with respect y, and thendetermining their rates of convergence to 0 as t → 0. One of the main difficulties of this approachcomes from controlling the term corresponding to no “large” jumps. As in the case of purely diffusionprocesses, Malliavin calculus is proved to be the key tool for this task. This analysis is presented inthe following subsection before our main result is presented in Section 5.2.

5.1. Density estimates for SDE with bounded jumps. In this part, we analyze the term cor-responding to N ε

t = 0:

P(Xt(x) ≥ x+ y|N εt = 0) = P(Xt(ε, ∅, x) ≥ x+ y).

We will prove that, for any fix positive integer N and η > 0, there exists an ε0 > 0 small enough anda constant C <∞ such that the density pt(·; ε, ∅, x) of Xt(ε, ∅, x) satisfies

(5.1) sup|y−x|>δ,ε<ε0

pt(y; ε, ∅, x) < CtN ,

for all 0 < t ≤ 1.

The estimate (5.1) holds true for a larger class of SDE with bounded jumps. Concretely, throughoutthis section, we consider the more general equation

Xxt = x+

∫ t

0

b(Xxs−)ds+

∫ t

0

σ(Xxs−)dWs +

∫ t

0

∫γ(Xx

s−, ζ)M(ds, dζ),(5.2)

where, as before, M(ds, dζ) is a Poisson random measure on R+ × R\0 with mean measureµ(ds, dζ) = ds × ν(dζ) and M = M − µ is its compensated measure. As before, we assume that νadmits a smooth density h with respect to the Lebesgue measure on R\0, but moreover, we nowassume h is supported in a ball B(0, r) for some r > 0. Note that the solution (Xt(ε, ∅, x))t of (2.6)perfectly fits within this framework.

The following assumptions on the coefficient functions of (5.2) are used in the sequel. Note that,when Xx

t is taken to be Xt(ε, ∅, x), these assumptions follow from the assumptions in Section 2.

Assumption 5.1. The coefficients b(x), σ(x) and γ(x, ζ) are 4 times differentiable. There exists aconstant c > 0 and a function κ ∈

⋂2≤p<∞ Lp(ν) such that:

(1) |∂nxnb(x)| , |∂nxnσ(x)| ,∣∣∣ 1κ(ζ)

∂nxnγ(x, ζ)∣∣∣ ≤ c, for 0 ≤ n ≤ 5.

(2)∣∣∣∂n+kxnζk

γ(x, ζ)∣∣∣ ≤ c, for 1 ≤ n+ k ≤ 5 and k ≥ 1.


Malliavin calculus is the main tool to analyze the existence and smoothness of density for Xxt . For

the sake of completeness, we are presenting some necessary preliminaries of Malliavin calculus onWiener-Poisson spaces in Appendix A following the approach in [6]. As described therein, there aredifferent ways to define a Malliavin operator for such spaces. For our purposes, it suffices to considerthe Malliavin operator corresponding to ρ = 0 (see Section A.2 for the details). The intuitiveexplanation of ρ = 0 is that when making perturbation of the sample path on the Wiener-Poissonspace, we only perturb the Brownian path.

Let us start by noting that Assumption 5.1 ensures that Xxt is a C2–diffeomorphism and also has

a continuous density (see, e.g., [6]). Furthermore, recalling that H∞ is the extended domain of theMalliavin operator, one has Xt ∈ H∞ and

(5.3) Γ(Xt, Xt) :=Ut = ∇xXxt

∫ t

0

(∇xXxs )−1σ2(Xx

s )(∇xXxs )−1ds

∇xX

xt .

Remark 5.2. Under the Condition (C3-b) of Section 2, one can show that ∇xXxt is almost surely

invertible. Indeed, one can show that ( cf. [6])

d (∇xXxt ) = 1 + ∂xb(X

xt−)∇xX

xt−dt+ ∂xσ(Xx

t−)∇xXxt−dWt

+∂xγ(Xxt−, ζ)∇xX

xt−M(dt, dζ).

Moreover, ∇xXxt admits an inverse Yt = (∇xX

xt )−1 which satisfies a similar equation

dYt = 1 + Yt−Dtdt+ Yt−EtdWt + Yt−FtM(dt, dζ).

Here Dt, Et and Ft are determined by b(x), σ(x), γ(x, ζ) and Xxt . As a consequence, together with our

assumption on b, σ and γ, one have

E sup0≤t≤1

(∇xXxt )p , E sup

0≤t≤1(∇xX

xt )−p <∞

for all p > 1.

The main result for this section is Theorem 5.7. For this purpose, we state the following integrationby parts formula (which is also the main ingredient for existence and smoothness of the density ofXxt ), which is a special case of Lemma 4-14 together with the discussion of Chapter IV in [6].

Proposition 5.3. For any f ∈ C∞c (R), there exists a random variable Jt ∈ Lp for all p ∈ N, suchthat

EDxf(Xxt ) = EJtU−2

t f(Xxt ).

The following existence and regularity result for the density of a finite measure is well known (see,e.g., Theorem 5.3 in [30]).

Lemma 5.4. Let m be a finite measure supported in an open set O ⊂ Rn. Take any p > n. Supposethat there exists g = (g1, ..., gn) ∈ Lp(m) such that∫

Rn

∂jfdm =

∫Rn

fgjdm, f ∈ C∞c (O), j = 1, ..., n.

Then m has a bounded density function q ∈ Cb(O) satisfying

‖q‖∞ ≤ C‖g‖nLp(m)m(O)1−n/p.

Here the constant C depends on n and p.


The following lemma is where we use assumption (C3-a). It is the first ingredient in provingTheorem 5.7. Its proof is deferred to Appendix D.

Lemma 5.5. Recall Ut = Γ(Xt, Xt). Under the Condition (C3-a) of Section 2, we have

EU−pt ≤ Ct−p.

The final ingredient toward our main result of this section is the proposition below, which is justa restatement of Proposition 3.1.

Proposition 5.6. Let Xt be the solution to equation (5.2). Recall that since the Levy density h iscompactly supported and γ(x, z) is uniformly bounded, the jumps of Xt is bounded by some constantS. There exists a constant C(S) depending on S, such that, for all 0 ≤ t ≤ 1,

P

sup0≤s≤t

|Xxs − x| ≥ 2pS

≤ C(S)tp.

Finally, we can state and prove our main result of this section.

Theorem 5.7. Assume the Condition (C3) of Section 2 is satisfied. Let Xxt t≥0 be the solution to

equation (5.2) and denote by pt(y;x) the density of Xxt . Fix any η > 0. Then for any N > 0, there

exists r(η,N) > 0 such that if µ is supported in B(0, r) with r ≤ r(η,N), we have for all 0 ≤ t ≤ 1

supy:|x−y|≥η

pt(y;x) ≤ C(η,N)tN .

Proof. For a fix t ≥ 0, define a finite measure mηt on R by

mηt (A) = P

(Xxt ∈ A ∩ Bc(x, η)

), A ⊂ R,

where Bc(x, r) denotes the complement of closure of B(x, r). Thus, to prove our result it sufficesto prove that mη

t admits a density that has the desired bound. Next, for any smooth function fcompactly supported in Bc(x, η), we have∫

R(∂xf)(y)mη

t (dy) = E∂xf(Xxt ) = EJtU−2

t f(Xxt )

=

∫RE[JtU

−2t

∣∣Xxt = y

]f(y)mη

t (dy),

by Proposition 5.3. The rest of the proof follows from Lemmas 5.4 and 5.5 and Proposition 5.6.

5.2. Expansion for the transition density. We are ready to state our main result of this section,namely, the second order expansion for the transition densities of the process Xt(x)t≥0 in terms ofthe density Γ(·;x) of γ(x, Jε). The proof is deferred to Appendix C.

Theorem 5.8. Let x ∈ R and y > 0. Then, under the Conditions (C1-C3) of Section 2, we have

(5.4) pt(y;x) := −∂P(Xt(x) ≥ x+ y)

∂y= ta1(x; y) +

t2

2a2(x; y) +O(t3),


as t→ 0, where a1(x; y) and a2(x; y) admit the following representations (for ε > 0 small-enough):

a1(x; y) := −∂A1(x; y)

∂y= λεΓ(y;x),

a2(x; y) := −∂A1(x; y)

∂y= −λε [h0,1(x; y) + h1,0(x; y)]−λ2

ε [h2,0(x; y) + 2Γ(y;x)] ,

with

h0,1(x; y) =σ2(x)

2

(−∂

2Γ(y;x)

∂x2+ 2

∂2Γ(y;x)

∂y∂x− ∂2Γ(y;x)

∂y2

)+ h0,1(x; y),

h0,1(x; y) =

∫Γ(y;x)− Γ(y − γ(x, ζ);x+ γ(x, ζ) hε(ζ)dζ,

h1,0(x; y) = −σ2(x)

2

∂2Γ(y;x)

∂y2+ h1,0(x; y),(5.5)

h1,0(x; y) =

∫Γ(y;x)− Γ(y − γ(x, ζ);x) hε(ζ)dζ,

h2,0(x; y) = −E Γ (ζ − γ(x, J);x+ γ(x, J)) .

6. The first order term of the option price expansion

In this section, we use our previous results to derive the leading term of the small-time expansionfor option prices of out-of-the-money (OTM) European call options with strike price K and expiryT . This can be achieved by either the asymptotics of the tail distributions or the transition density.Given that the former requires less stringent conditions on the coefficients of the SDE, we choose toemploy our result for tail distribution.

Throughout this section, let Stt≥0 be the stock price process and let Xt = logSt for each t ≥ 0.We assume that P is the option pricing measure and that under this measure the process Xtt≥0 isof the form in (2.1). As usual, without loss of generality we assume that the risk-free interest rate ris 0. In particular, in order for St = expXt to be a Q-(local) martingale, we fix

b(x) := −1

2σ2(x)−

∫ (eγ(x,z) − 1

)h(z)dz

We also assume that σ and γ are such that the Conditions (C1-C4) of Section 2 are satisfied.

By the Markov property of the system, it will suffice to compute a small-time expansion for

vt = E(St −K)+ = E(eXt −K

)+.

In particular, using the well-known formula

EU1U>K = KPV > logK+

∫ ∞K

PV > log sds,

where U is a positive random variable and V := logU , we can write

E(eXt −K)+ =

∫ ∞K

PSt > sds = S0

∫ ∞KS0

PXt − x > log sds,


where x = X0 = logS0. Recall that

(6.1) P(Xt − x ≥ y) = e−λεt∞∑n=0

P(Xt − x ≥ y|N εt = n)

(λεt)n

n!,

where λε :=∫φε(ζ)h(ζ)dζ is the jump intensity of N ε

t t≥0. We proceed as in Section 4. First, notethat

vt = S0

∫ ∞KS0

PXt − x > log sds = S0e−λεt(I1 + I2 + I3),(6.2)

where

I1 =

∫ ∞KS0

PXt − x ≥ log s|N εt = 0ds =

∫ ∞KS0

PXt(ε, ∅, x)− x ≥ log sds,

I2 = λεt

∫ ∞KS0

PXt − x ≥ log s|N εt = 1ds,

I3 = λ2εt

2

∞∑n=2

(λεt)n−2

n!

∫ ∞KS0

PXt − x ≥ log s|N εt = nds.

It is clear that I1/t→ 0 as t→ 0 by Lemma 3.2. We show that the same is true for I3, which is thecontent of the following lemma. Its proof is given in Appendix D.

Lemma 6.1. With the above notation, we have

supn∈N,t∈[0,1]

1

n!

∫ ∞0

P(|Xt − x| ≥ log y|N εt = n)dy <∞.

As a consequence, I3/t→ 0 as t→ 0.

Note that the above lemma actually implies that Ee|Xt−x| < ∞ for all t ∈ [0, 1). We are ready tostate the main result of this section.

Theorem 6.2. Let vt = E(St−K)+ be the price of a European call option with strike K. Under theConditions (C1-C4) of Section 2, we have

limt→0

1

tvt =

∫ ∞−∞

(S0e

γ(x,ζ) −K)

+h(ζ)dζ.

Proof. We use the notation introduced in (6.2). Following a similar argument as in the proof ofLemma 6.1, one can show that

(6.3)

∫ ∞KS0

supt∈[0,1]

PXt − x ≥ log s|N εt = 1ds <∞.

Also, it is clear that I1/t converges to zero when t approaches to zero by Lemma 3.2. Using thelatter fact, (6.3), Lemma 6.1, (6.2), and dominated convergence theorem, we have

limt→0

vtt

= limt→0

S0I2

t= λεS0 lim

t→0

∫ ∞KS0

PXt−x ≥ log s|N εt = 1ds = λεS0

∫ ∞KS0

limt→0

PXt−x ≥ log s|N εt = 1ds.


Next, using Theorem 4.1, it follows that

limt→0

vtt

= S0

∫ ∞KS0

A1(x, log s)ds = S0

∫ ∞KS0

∫γ(x,ζ)≥log s

h(ζ)dζds =

∫ ∞−∞

(S0e

γ(x,ζ) −K)

+h(ζ)dζ,

where we had used Fubini’s theorem for the last equality above.

Remark 6.3. As a special case of our result, let γ(x, ζ) = ζ. The model reduces to an exponentialLevy model. The above first order asymptotics becomes to

limt→0

1

tvt =

∫ ∞−∞

(S0e

ζ −K)

+h(ζ)dζ.

This recovers the well-known first-order asymptotic behavior for exponential Levy model (see, e.g.,[28] and [31]).

Appendix A. Preliminaries of Malliavin calculus on Wiener-Poisson spaces

In this section, we follow [6] and introduce briefly the Malliavin calculus on Wiener-Poisson spaceto our need. All the details can be found in [6]. We first introduce, in an abstract manner, thenotion of Malliavin operator (infinitesimal generator of the Ornstein-Uhlenbeck semigroup). Fix aprobability space (Ω,F ,P). Let C2

p(Rn) be the space of all C2 functions on Rn which, together withall their partial derivatives up to the second order, have at most polynomial growth.

Definition A.1. A Malliavin operator (L,R) is a linear operator L on a domain R ⊂⋂p<∞ Lp,

taking values in⋂p<∞ Lp, and such that the following conditions hold true:

(1) If Φ ∈ Rn and F ∈ C2p(Rn), then F (Φ) ∈ R.

(2) F is the σ-field generated by all functions in R.(3) L is symmetric in L2, i.e. E(ΦLΨ) = E(ΨLΦ) for all Φ,Ψ ∈ R.(4) Define

Γ(Φ,Ψ) = L(ΦΨ)− ΦLΨ−ΨLΦ, for all Φ,Ψ ∈ R.Then Γ is nonnegative on R×R.

(5) For Φ = (Φ1, ...,Φn) ∈ Rn and F ∈ C2p(Rn), one have

L(F Φ) =n∑i=1

∂F

∂xi(Φ)LΦi +

1

2

n∑i,j=1

∂2F

∂xi∂xj(Φ)Γ(Φi,Φj).

A.1. Malliavin operator on Wiener-space. We consider the Wiener space of continuous paths:

ΩW =(C([0, 1],R), (FW

t )0≤t≤1,PW)

where:

(1) C([0, 1],R) is the space of continuous functions [0, 1]→ R;(2) (Wt)t≥0 is the coordinate process defined by Wt(f) = f (t), f ∈ C([0, 1],R);

(3) PW is the Wiener measure;(4) (FW

t )0≤t≤1 is the (PW -completed) natural filtration of (Wt)0≤t≤1.


Now, (ΩW ,FW ,PW ) is the canonical 1-dimensional Wiener space. Set

S =

Φ = F (Wt1 , ...,Wtn), F ∈ C2p(Rn), 0 ≤ ti ≤ 1

.

For Φ given as above, define

LWΦ = −1

2

n∑i=1

∂F

∂xi(Wt1 , ...,Wtn)Wti +

1

2

n∑i,j=1

∂2F

∂xi∂xj(Wt1 , ...,Wtn)(ti ∧ tj).

It is well-known that (LW ,SW ) is a Malliavin operator. Furthermore, for Φ = F (Wt1 , ...,Wtn) andΨ = H(Ws1 , ...,Wsm)

ΓW (Φ,Ψ) =n∑i=1

m∑j=1

∂F

∂xi(Wt1 , ...,Wtn)

∂H

∂xj(Ws1 , ...,Wsm)(ti ∧ sj).

A.2. Malliavin operator on Poisson space. In this section, we consider the canonical probabil-ity space (ΩP ,F P ,PP ) of a prefixed Poisson random measure M(dt, dζ) on [0, 1] × (R\0). Letµ(dt, dζ) = dt× ν(dζ) be the intensity measure of M(dt, dζ) and denote by M = M − µ its compen-sated Poisson random measure. Throughout our discussion, we also assume that ν admits a smoothdensity h with respect to the Lebesgue measure on R, i.e. ν(dζ) = h(ζ)dζ.

We denote by Cb,2c ([0, 1]×R\0) the set of all compactly supported, Borel functions f on [0, 1]×

R\0 that are of C2 on R\0 with f, ∂ζf, ∂2ζ2f uniformly bounded. For such f , we write

M(f) =

∫f(t, ζ)M(dt, dζ).

Let ρ : R\0 → [0,∞) be an auxiliary function which is of class C1b . Set

SP = Φ = F (M(f1), ...,M(fk));F ∈ C2p(Rk), fi ∈ Cb,2

c ([0, 1]× R\0).

Given any Φ ∈ SP as above, define

LPΦ =1

2

k∑i=1

∂F

∂xi(M(f1), ...,M(fk))M

(ρ4ζfi + ∂zρ∂ζfi + ρ

∂ζh

h∂ζfi

)

+1

2

k∑i,j=1

∂2F

∂xi∂xj(M(f1), ...,M(fk))M(ρ∂ζfi∂ζfj).

In the above, 4ζ stands for the Laplacian on R\0.

Proposition A.2. (LP ,SP ) defines a Malliavin operator. Moreover forΦ(M(f1), ...,M(fk)) and Ψ = H(M(h1), ...,M(hl)), we have

ΓP (Φ,Ψ) =k∑i=1

l∑j=1

∂F

∂xi(M(f1), ...,M(fk))

∂H

∂xj(M(h1), ...,M(hl))M(ρDζfiDζhj).

Proof. See Proposition 9-3 in [6].

Remark A.3. As we can see from the above definition, the Malliavin operator for Poisson space isnot uniquely defined. Indeed, each different choice of ρ gives a different operator. More remarks onthis point are given at the end of this section below.


A.3. Malliavin operator on Wiener-Poisson space. Keep notations as in previous sections. Wedefine a probability space (Ω,F ,P) by

(Ω,F ,P) = (ΩW ,FW ,PW )⊗ (ΩP ,F P ,PP ).

Clearly (Ω,F ,P) is the canonical Wiener-Poisson space. Then define (L,S) to be the direct productof (LW ,SW ) and (LP ,SP ). More precisely,

(1) S = Φ = F (Φ1, ...,Φn; Ψ1, ...,Ψm); for all Φi ∈ SW ,Ψj ∈ SP , F ∈ C2p.

(2) For Φ as given in (1), and ω = (ωW , ωP ) ∈ Ω,

LΦ(ω) = LWΦ(·, ωP ) + LPΦ(ωW , ·).

It can be shown that (L,S) is a Malliavin operator on (Ω,F ,P). A few properties of (L,S) arelisted below.

Proposition A.4. Let Φ ∈ SW and Ψ ∈ SP , we have

(1) LΦ = LWΦ, and Γ(Φ,Φ) = ΓW (Φ,Φ).(2) LΨ = LPΨ, and Γ(Ψ,Ψ) = ΓP (Ψ,Ψ).(3) Γ(Φ,Ψ) = 0.

Finally, we extend the L to a larger domain that is suitable to work on. For this purpose, set

‖Φ‖Hp = ‖Φ‖Lp + ‖LΦ‖Lp + ‖Γ(Φ,Φ)1/2‖Lp

and denote by Hp the closure of S under ‖ · ‖Hp . Also set

H∞ =⋂

2≤p<∞

Hp.

Proposition A.5. The operator (L,H∞) is a Malliavin operator.

Proof. This is the content of Section 8-b in [6].

Remark A.6. As before, a different choice of the auxiliary function ρ in the definition for LP resultsin a different Malliavin operator L on the Wiener-Poisson space. Hence, the extended domain H∞may also be different.

Remark A.7. Consider the following stochastic differential equation with regular enough coefficients

Xxt = x+

∫ t

0

b(Xxs−)ds+

∫ t

0

σ(Xxs−)dWs +

∫ t

0

∫γ(Xx

s−, ζ)M(ds, dζ).

To make use of the power of Malliavin calculus to analyze the above SDE, one needs to impose furtherconditions on ρ so that Xx

t is in the domain H∞. Those further conditions are generally determinedby the support of the intensity measure ν and are need to ensure that some approximation procedurecan work out. For more details of the role that ρ plays, see Section 9-1 and Theorem 10-3 in [6]. Itis important to note that, among various choices of ρ, one can always choose ρ = 0, as we imposethe non-degeneracy Assumption (C3-a). The intuitive explanation of ρ = 0 is that when makingperturbation of the sample path on the Wiener-Poisson space, we only perturb the Brownian path.


Appendix B. Proof of the tail distribution expansion

The proof of Theorem 4.1 is decomposed into two parts described in the following two subsections.For future reference, we try to keep track of the reminder terms when applying Dynkin’s formula(3.6).

B.1. The leading term. In order to determine the leading term of (4.1), we analyze the secondterm in (4.2) corresponding to n = 1 (only one “large” jump). By conditioning on the time of thejump (necessarily uniformly distributed on [0, t]), we can write

P(Xt(x) ≥ x+ y|N εt = 1) =

1

t

∫ t

0

P(Xt(ε, s, x) ≥ x+ y)ds.(B.1)

Conditioning on Fs− (similar to the procedure in (D.3) below), we can write

P(Xt(ε, s, x) ≥ x+ y) = E (Gt−s (Xs(ε, ∅, x))) ,(B.2)

where

Gt(z) := Gt(z;x, y) := P [Xt(ε, ∅, z) + γ(z, J) ≥ x+ y] .(B.3)

Conditioning the last expression on Xt(ε, ∅, z), we have

(B.4) Gt(z) = EH(Xt(ε, ∅, z)),with

H(w) := H(w;x, y, z) := P [γ(z, J) ≥ x+ y − w] .

Note that ∂wH(w;x, y, z) = Γ(x+ y − w; z) and

(B.5) ∂kwH(w;x, y, z) = (−1)k−1∂k−1Γ(x+ y − w; z)

∂ζk−1.

Here and throughout the proof, Γ(ζ;u) is the density of γ(z, J), whose existence is ensured fromthe Condition (C4) in the introduction, and ∂kζΓ = ∂kΓ/∂ζk and ∂kuΓ = ∂kΓ/∂uk denote its partialderivatives. Next, in light of (C4-a), we can apply Dynkin’s formula and get

(B.6) Gt(z) = EH(Xt(ε, ∅, z)) = H0(z;x, y) +H1(z;x, y)t+t2

2R1t (z;x, y),

where

H0(z;x, y) := H(z;x, y, z) = P [γ(z, J) ≥ x+ y − z] ,

H1(z;x, y) := LεH(z;x, y, z) = b(z)Γ(x+ y − z; z)− σ2(z)

2∂ζΓ(x+ y − z; z)

+

∫(H(z + γ(z, ζ);x, y, z)−H(z;x, y, z))hε(ζ)dζ

= b(z)Γ(x+ y − z; z)− σ2(z)

2∂ζΓ(x+ y − z; z) + H1,0(z;x+ y − z),

R1t (z;x, y) =

∫ 1

0

(1− α)EL2εH(Xαt(ε, ∅, z);x, y, z)dα,(B.7)

where H1,0 is given as in (4.4). Next, using (3.10), (C4-a), and (B.5),

(B.8) supx,y,z|R1

t (z;x, y)| ≤ C <∞,


for a constant C depending on∫

(1 ∧ ζ2)hε(ζ)dζ, ‖b(k)‖∞, ‖v(k)‖∞, and supu,ζ |∂kζΓ(ζ;u)|, for k =0, . . . , 4. Plugging (B.6) in (B.2), we get

P(Xt(ε, s, x) ≥ x+ y) = EH0(Xs(ε, ∅, x);x, y) + (t− s)EH1(Xs(ε, ∅, x);x, y)(B.9)

+(t− s)2

2ER1

t−s(Xs(ε, ∅, x);x, y),

Finally, we apply Dynkin’s formula to EH0(Xs(ε, ∅, x);x, y). To this end, we show that H0(z;x, y) isC4b in a neighborhood of x. Note that

1

δ(H0(z + δ;x, y)−H0(z;x, y)) =

∫ ∞x+y−z−δ

∫ 1

0

∂Γ(ζ; z + δβ)

∂udβdζ − 1

δ

∫ x+y−z−δ

x+y−zΓ(ζ; z)dζ,

where, as before, Γ(ζ;u) denotes the density of γ(u, J). Due to (C4-b), the limit of the previousexpression as δ → 0 exists and we get

(B.10)∂H0(z;x, y)

∂z=

∫ ∞x+y−z

∂Γ(ζ; z)

∂udζ + Γ(x+ y − z; z),

which is locally bounded for values of z in a neighborhood of x because of (C4). Similarly, we canshow that

(B.11) ∂2zH0(z;x, y) =

∫ ∞x+y−z

∂2uΓ(ζ; z)dζ + 2∂uΓ(x+ y − z; z)− ∂ζΓ(x+ y − z; z),

which is also locally bounded for values of z in a neighborhood of x because of (C4). We can similarlyproceed to show that ∂kzH0(z;x, y) exist and is bounded for k = 3, 4. Using Lemma 3.5,

(B.12) EH0(Xs(ε, ∅, x);x, y) = H0,0(x; y) + sH0,1(x; y) +s2

2R2s(x; y),

where

H0,0(x; y) := H0(x;x, y) = P [γ(x, J) ≥ y] ,

H0,1(x; y) := LεH0(x;x, y) = b(x)∂H0(z;x, y)

∂z

∣∣∣∣z=x

+σ2(x)

2

∂2H0(z;x, y)

∂z2

∣∣∣∣z=x

+

∫(H0(x+ γ(x, ζ);x, y)−H0(x;x, y))hε(ζ)dζ

= b(x)∂H0(z;x, y)

∂z

∣∣∣∣z=x

+σ2(x)

2

∂2H0(z;x, y)

∂z2

∣∣∣∣z=x

+ H0,1(x; y),(B.13)

R2s(x; y) :=

∫ 1

0

(1− α)EL2εH0,δ(Xαs(ε, ∅, x);x, y)dα + EH0,δ(Xs(ε, ∅, x);x, y),

where H0,1(x; y) is defined as in (4.4) and H0,δ(z) = H0(z)ψδ(z), and H0,δ(z) = H0(z)ψδ(z) with ψδdefined as in the proof of Lemma 3.5. Note the R2

s(x; y) = O(1), as s→ 0. Plugging (B.12) in (B.9),

P(Xt(ε, s, x) ≥ x+ y) = P [γ(x, J) ≥ y] +O(t),

which can then be plugged in (B.1) to get

P(Xt(x) ≥ x+ y|N εt = 1) = P [γ(x, J) ≥ y] +O(t).


Finally, (4.2) can be written as

P(Xt(x) ≥ x+ y) = e−λεttλεP [γ(x, J) ≥ y] +O(t2) = t

∫1γ(x,ζ)≥yh(ζ)dζ +O(t2),

where the second equality above is valid for ε > 0 small enough.

B.2. Second order term. In addition to (B.12), we also need to consider the leading terms in theterm EH1(Xs(ε, ∅, x);x, y) of (B.9) and the term P(Xt(x) ≥ x+ y|N ε

t = 2) of (4.2). Let us first showthat z → H1(z;x, y) is C2

b . Let

K(ζ;x, y, z) := P [γ(z, J) ≥ x+ y − z − γ(z, ζ)]− P [γ(z, J) ≥ x+ y − z] ,

and recall that

H1(z;x, y) = b(z)Γ(x+ y − z; z)− σ2(z)

2∂ζΓ(x+ y − z; z) +

∫K(ζ;x, y, z)hε(ζ)dζ.

Obviously, the first two terms on the right-hand side of the previous expression are C2b in light of

(C3) and (C4-a). Hence, for the derivative ∂H1(z;x,y)∂z

to exist, it suffices to show that ∂K(ζ;x,y,z)∂z

existsand that

supz,x,y

∣∣∣∣∂K(ζ;x, y, z)

∂z

∣∣∣∣ < C(ζ2 ∧ 1),

for any |ζ| < ε. Note that

∂K(ζ;x, y, z)

∂z= Γ(x+ y − z − γ(z, ζ); z)− Γ(x+ y − z; z)︸︷︷︸

T1

+ Γ(x+ y − z − γ(z, ζ); z)∂γ(z, ζ)

∂z︸︷︷︸T2

+

∫ x+y−z

x+y−z−γ(z,ζ)

∂Γ(ζ ′; z)

∂udζ ′︸︷︷︸

T3

.

The following bounds are evident from (C2) and (C4-a), for ε > 0 small enough,

supx,y,z|T1| ≤ sup

u,ζ′

∣∣∣∣∂Γ(ζ ′;u)

∂ζ

∣∣∣∣ supz|γ(z, ζ)| ≤ C(1 ∧ ζ2),


u,ζ′|Γ(ζ ′;u)| sup

z

∣∣∣∣∂γ(z, ζ)

∂z

∣∣∣∣ ≤ C(1 ∧ ζ2),


u,ζ′

∣∣∣∣Γ(ζ ′;u)

∂u

∣∣∣∣ supz|γ(z, ζ)| ≤ C(1 ∧ ζ2).

We can similarly prove that ∂2H1(z;x,y)∂z2

exists and is bounded. Hence, we can apply Dynkin’s formulato get

(B.14) EH1(Xs(ε, ∅, x);x, y) = H1,0(x, y) + sR3s(x; y),


where

H1,0(x; y) := H1(x;x, y) = b(x)Γ(y;x)− σ2(x)

2∂ζΓ(y;x)

+

∫(P [γ(x, J) ≥ y − γ(x, ζ)]− P [γ(x, J) ≥ y]) hε(ζ)dζ

R3s(x; y) :=

∫ 1

0

ELεH1(Xαs(ε, ∅, x);x, y)dα = O(1), as s→ 0.

In order to handle P(Xt(x) ≥ x+ y|N εt = 2), we again condition on the times of the jumps, which

are necessarily distributed as the order statistics of two independent uniform [0, t] random variables.Concretely,

P(Xt(x) ≥ x+ y|N εt = 2) =

2

t2

∫ t

0

∫ t

s1

P(Xt(ε, s1, s2, x) ≥ x+ y)ds2ds1.(B.15)

As before, P(Xt(ε, s1, s2, x) ≥ x+ y) is expressed in the following way:

E(P[Xt(ε, s1, s2, x) ≥ x+ y| Fs−2

])= E

(P [Xt−s2(ε, ∅, z) + γ(z, J) ≥ x+ y]

∣∣z=Xs2 (ε,s1,x)

)= E (Gt−s2 (Xs2(ε, s1, x))) ,

where, as in (B.4),

Gt(z) = P [Xt(ε, ∅, z) + γ(z, J) ≥ x+ y] = EH(Xt(ε, ∅, z)).(B.16)

Using Dynkin’s formula,Gt(z) = H0(z;x, y) + tR4

t (z;x, y)

with

H0(z;x, y) := H(z;x, y, z) = P(γ(z, J) ≥ x+ y − z),

R4t (z;x, y) :=

∫ 1

0

ELεH(Xαt(ε, ∅, z);x, y, z)dα = O(t), as t→ 0.

Hence, we have

P(Xt(ε, s1, s2, x) ≥ x+ y) = E (H0 (Xs2(ε, s1, x);x, y))(B.17)

+ (t− s2)ER4t−s2(Xs2(ε, s1, x);x, y).

Next, applying Lemma 3.6 (see also equations (D.5), (D.6), and (D.7) below), we have

E (H0 (Xs2(ε, s1, x);x, y)) = H2,0(x; y) + s1R5s1

(x; y)(B.18)

+ (s2 − s1)ER6s2−s1(Xs1(ε, ∅, x);x, y),

where

H2,0(x; y) := E [H0 (x+ γ(x, J);x, y)] = P (γ(x+ γ(x, J2), J1) ≥ y − γ(x, J2)) ,

R5s(x; y) :=

∫ 1

0

ELεH2,0(Xαs(ε, ∅, x); y)dα, R6s(z;x, y) :=

∫ 1

0

ELεH0(Xαs(ε, ∅, z);x, y, z)dα

H0(w;x, y, z) := EH0(w + γ(z, J);x, y),

with J1, J2 above denoting two independent copies of J . Therefore,

(B.19) P (Xt(x) ≥ x+ y|N εt = 2) = H2,0(x; y) +O(t).


In light of (B.1), (B.9), (B.9), (B.12), (B.14), and (B.19), we have the following second orderdecomposition of the tail distribution P(Xt(x) ≥ x+ y):

P(Xt(x) ≥ x+ y) = e−λεtλεtH0,0(x; y) + e−λεtλεt

2

2(H0,1(x; y) +H1,0(x; y))

+ e−λεt(λεt)

2

2H2,0(x; y) +O(t3)

= λεtH0,0(x; y) +t2

2

λε [H0,1(x; y) +H1,0(x; y)]

+ λ2ε [H2,0(x; y)− 2H0,0(x; y)]

+O(t3).

This prove the result of Theorem 4.1 after plugging (B.10-B.11) in (B.13).

Appendix C. Proof of the expansion for the transition densities

The following result will allow us to control the higher order terms (see Appendix D for a proof):

Lemma C.1. Let

(C.1) Rt(x, y) := e−λεt∞∑n=3

P(Xt(x) ≥ x+ y|N εt = n)

(λεt)n

n!.

Then, under the conditions of Theorem 5.8, there exists a constant B = B(ε) <∞ such that, for anyt ≤ 1,

supx,y

∣∣∂yRt(x, y)∣∣ ≤ Bt3.

Proof of Theorem 5.8. Let us consider the terms corresponding to one and two “large” jumps in (4.2).From (B.1), (B.9), (B.12), and (B.14), it follows that

P(Xt(x) ≥ x+ y|N εt = 1) = H0,0(x; y) +

t

2[H0,1(x; y) +H1,0(x; y)](C.2)

+1

t

∫ t

0

s2

2R2s(x; y) + (t− s)sR3

s(x; y) +(t− s)2

2ER1

t−s (Xs(ε, ∅, x);x, y)

ds.

Similarly, from (B.15), (B.17), and (B.18), we have

P(Xt(x) ≥ x+ y|N εt = 2) = H2,0(x; y)

+2

t2

∫ t

0

∫ t

s1

s1R5

s1(x; y) + (s2 − s1)ER6

s2−s1 (Xs1(ε, ∅, x);x, y)(C.3)

+ (t− s2)ER4t−s2 (Xs2(ε, s1, x);x, y)

ds2ds1.

Equations (C.2-C.3) show that in order for the derivatives

a1(x; y) :=∂

∂yP(Xt(x) ≥ x+ y|N ε

t = 1), a2(x; y) :=∂

∂yP(Xt(x) ≥ x+ y|N ε

t = 2)


to exist, it suffices that the partial derivatives of the functions Hi,j(x; y) with respect to y exist andalso that the partial derivatives of the functions Ri

t with respect to y exist and are uniformly boundedin y, w. Furthermore, under the later boundedness property, we will then obtain that

a1(x; y) =∂H0,0(x; y)

∂y+t

2

[∂H0,1(x; y)

∂y+∂H1,0(x; y)

∂y

]+O(t2), (t→ 0),(C.4)

a2(x; y) =∂H2,0(x; y)

∂y+O(t), (t→ 0).(C.5)

(1) Differentiability of Hi,j(x; y):

In light of our condition (C4-a), it is easy to check that ∂yH0,0(x; y) = −Γ(y;x), ∂yH0,1(x; y) =h0,1(x; y), and ∂yH1,0(x; y) = h1,0(x; y), as given in (5.5). For ∂yH2,0(x; y), note that

∂

∂yH2,0(x; y) =

∂

∂y

∫P (γ(x+ γ(x, ζ2), J1) ≥ y − γ(x, ζ2))hε(ζ2)dζ2

=

∫∂

∂y

∫ ∞y−γ(x,ζ2)

Γ (ζ1;x+ γ(x, ζ2)) dζ1hε(ζ2)dζ2

= −∫

Γ (y − γ(x, ζ2);x+ γ(x, ζ2))hε(ζ2)dζ2 = h2,0(x; y),

where the second equality above follows from condition (C4-a). Before proving the existence andboundedness of ∂yRi

t(w;x, y), note that (C.4) suffices to conclude (5.4) in light of (4.2), Theorem 5.7,and Lemma C.1.

(2) Boundedness of ∂yRi(w;x, y):

The existence and boundedness of the partial derivatives of the remainder terms follow from (C2),(C3-a), and (C4). For instance, in view of (B.7), ∂yR1

t (z;x, y) exists and is uniformly bounded inz and y, if ∂yL

2εH(w;x, y, z) exists and is uniformly bounded in w, y, z. Here, H(w;x, y, z) is defined

as in (B.4). Throughout a lengthy but straightforward computation, one can express L2εH(w;x, y, z)

as sums of terms of the following form:

A(w)∂rwH(w;x, y, z),

∫ ∫ 1

0

∂rwH(w + γ(w; ζ)β;x, y, z)dβ∂iwγ(w; ζ)γ(w; ζ)hε(ζ)dζ,∫ ∫ 1

0

A(w + γ(w; ζ)β)∂rwH(w + γ(w; ζ)β;x, y, z)dβ∂iwγ(w; ζ)hε(ζ)dζ,∫ ∫ 1

0

∫ ∫ 1

0

∂rwH(w + γ(w; ζ)β + γ(w + γ(w; ζ)β; ζ)β;x, y, z)dβ

× γ(w + γ(w; ζ)β; ζ))hε(ζ)dζdβγ(w; ζ)hε(ζ)dζ,∫ ∫ 1

0

∫∂rwH(w + γ(w; ζ)β + γ(w + γ(w; ζ)β; ζ);x, y, z)

× ∂wγ(w + γ(w; ζ)β; ζ))hε(ζ)dζdβγ(w; ζ)hε(ζ)dζ,

where i, j, r ∈ N and A(w) is the product of derivatives of the functions b or v. In light of (B.5) and(C4-a), all the above terms are differentiable with respect to y and their resulting derivatives areuniformly bounded in w, y, z. The same will then hold for L2

εH(w;x, y, z). One can similarly dealwith all other reminders terms in (C.2-C.3).


Appendix D. Proof of other technical lemmas

Proof of Lemma 3.5. We start by proving the second statement in the lemma via Lemma 3.3. Forthis end, we need to show that Lεf belongs to C2

b . This is easy to check if f ∈ C4b . Indeed, let

Bf(y) = b(y)f ′(y), Df(y) =σ2(y)

2f ′′(y), If(y) =

∫(f(y + γ(y, ζ))− f(y)) hε(ζ)dζ,

so that Lε = B +D + I. Note that

|∂y (f(y + γ(y, ζ))− f(y)) | = |f ′(y + γ(y, ζ))(1 + ∂yγ(y, ζ))− g′(y)|≤ ‖f ′′‖∞|γ(y, ζ)|+ ‖f ′‖∞|∂yγ(y, ζ)|,

which can be bounded by C(‖f ′′‖∞ + ‖f ′‖∞)(1 ∧ ζ2) in light of (C2). Here, ∂y denotes the partialderivative with respect to y. Given that

∫(1 ∧ ζ2)hε(ζ)dζ < ∞, we can interchange differentiation

and integration, and get

(If)′(y) =

∫(f ′(y + γ(y, ζ))(1 + ∂yγ(y, ζ))− f ′(y)) hε(ζ)dζ

=

∫ ∫ 1

0

f ′′(y + γ(y, ζ)β)dβγ(y, ζ))hε(ζ)dζ +

∫f ′(y + γ(y, ζ))∂yγ(y, ζ))hε(ζ)dζ.

In particular, supy |(If)′(y)| ≤ C(‖f ′′‖∞ + ‖f ′‖∞)∫

(1 ∧ ζ2)hε(ζ)dζ := K and

supy|(Lεf)′(y)| ≤ ‖b′‖∞‖f ′‖∞ + ‖b‖∞‖f ′′‖∞ + ‖v′‖∞‖f ′′‖∞ + ‖v‖∞‖f (3)‖∞ +K.

We can similarly show that supy |(Lεf)′′(y)| <∞ if f ∈ C4b . We also have

L2εf(y) = b(y)(Lεf)′(y) +

σ2(y)

2(Lεf)′′(y) +

∫[Lεf(y + γ(y, ζ))− Lεf(y)] hε(ζ)dζ

= b(y)(Lεf)′(y) +σ2(y)

2(Lεf)′′(y) +

∫ ∫ 1

0

(Lεf)′(y + γ(y, ζ)β)dβγ(y, ζ)hε(ζ)dζ.

Therefore,

(D.1) ‖L2εf‖∞ ≤ ‖b‖∞‖(Lεf)′‖∞ + ‖v‖∞‖(Lεf)′′‖∞ + ‖(Lεf)′‖∞

∫(1 ∧ ζ2)hε(ζ)dζ <∞,

and (3.10) will follow from the form of the reminder in (3.5). This concludes the proof of the secondstatement of the lemma.

Let us now consider a bounded function f that is C4b in only a neighborhood of x. Let ψδ be

a C∞ function such that 1(x−δ/2,x+δ/2)(y) ≤ ψδ(y) ≤ 1(x−δ,x+δ)(y) and let fδ(y) = f(y)ψδ(y) andfδ(y) = f(y)(1− ψδ(y)). Then, since f is bounded,∣∣Efδ (Xt(ε, ∅, x))

∣∣ ≤ ‖f‖∞P (|Xt(ε, ∅, x)− x| ≥ δ/2) ,

which can be made O(t2) due to Lemma 3.2. Now, since fδ is C4b in R, we can apply the second order

Dynkin’s expansion derived above to get

Efδ (Xt(ε, ∅, x)) = fδ(x) + tLεfδ(x) +t2

2

∫ 1

0

(1− α)EL2εfδ(Xαt(ε, ∅, x))

dα.


Putting together the two previous facts,

Ef (Xt(ε, ∅, x)) = Efδ (Xt(ε, ∅, x)) + Efδ (Xt(ε, ∅, x)) = fδ(x) + tLεfδ(x) +t2

2Rt(ε, δ, x).

Using that γ(x, 0) = 0 and the continuity of γ(x, ·), we can check that for ε > 0 small enough,Lεfδ(x) = Lεf(x). Now, in light of (D.1) and Lemma 3.2, the reminder

(D.2) Rt(ε, x) :=

∫ 1

0

(1− α)EL2εfδ(Xαt(ε, ∅, x))

dα + 2t−2Efδ (Xt(ε, ∅, x)) ,

can be uniformly bounded assup

0<t<1,ε′<ε|Rt(ε

′, x)| ≤ K,

for a constant K <∞ depending only on ‖f (k)1(x−δ,x+δ)‖∞, ‖b(k)‖∞, ‖v(k)‖∞, and∫

(1∧ζ2)h(ζ)dζ.

Proof of Lemma 3.6. By the Markov property, we have

Ef (Xt(ε, s, x)) = E (E [f (Xt(ε, s, x))| Fs− ])

= E(E [f (Xt−s(ε, ∅, z) + γ(z, J))]|z=Xs(ε,∅,x)

)= E (Ft−s (Xs(ε, ∅, x))) ,(D.3)

where J is a random variable independent of X(ε, ∅, z) with density hε and

Ft(z) := E [f (Xt(ε, ∅, z) + γ(z, J))] .

By conditioning on Xt(ε, ∅, z),Ft(z) = EG(Xt(ε, ∅, z)),

where G(w) := G(w; z) := E [f (w + γ(z, J))] . By formal differentiation and induction, we can seethat G ∈ C2

b in w for each fixed z and, furthermore,

∂kwG(w; z) =1

λε

∫f (k)(w + γ(z, ζ))h(ζ)φε(ζ)dζ, sup

w,z

∣∣∂kwG(w; z)∣∣ ≤ ‖f (k)‖∞ <∞, k = 1, 2.

Applying Dynkin’s formula (3.6), we have

(D.4) Ft(z) = G0(z) + tRt(z),

where

G0(z) := G(z; z) = E [f (z + γ(z, J))] , Rt(z) :=

∫ 1

0

ELεG(Xαt(ε, ∅, z); z)dα.(D.5)

Writing

LεG(w; z) = b(y)∂wG(w; z) + v(y)∂2wG(w; z) +

∫ ∫ 1

0

∂wG(w + γ(w, ζ)β; z)dβγ(w, ζ)hε(ζ)dζ

and using the conditions (C2-C3), we see that, for some C <∞,

supt>0,z|Rt(z)| ≤ ‖v‖∞ sup

w,v|∂2wG(w; v)|+ C sup

w,v|∂wG(w; v)|[‖b‖∞ +

∫(ζ2 ∧ 1)hε(ζ)dζ].

Plugging (D.4) in (D.3)

Ef (Xt(ε, s, x)) = EG0(Xs(ε, ∅, x)) + (t− s)ERt−s(Xs(ε, ∅, x)).


Applying Dynkin’s formula again,

Ef (Xt(ε, s, x)) = G0(x) + sRs(x) + (t− s)ERt−s(Xs(ε, ∅, x)),(D.6)

where

(D.7) Rs(x) =

∫ 1

0

ELεG0(Xαs(ε, ∅, x))dα.

Since

supz|∂zf(z + γ(z, J))| ≤ ‖f ′‖∞(1 + sup

z,ζ|∂zγ(z, ζ)|) <∞,

supz|∂2zf(z + γ(z, J))|| ≤ ‖f ′′‖∞(1 + sup

z,ζ|∂zγ(z, ζ)|)2 + ‖f ′‖∞ sup

z,ζ|∂2zγ(z, ζ)| <∞,

we have

G′0(z) = E [f ′ (z + γ(z, J)) (1 + ∂zγ(z, J))]

G′′0(z) = E[f ′′ (z + γ(z, J)) (1 + ∂zγ(z, J))2 + f ′ (z + γ(z, J)) ∂2

zγ(z, J)].

and, thus, supz |G(k)0 (z)| <∞, k = 1, 2. Again, using that

LεG0(z) = b(y)G′0(z) + v(z)G′′0(z) +

∫ ∫ 1

0

G′0(z + βγ(z, ζ))dβγ(z, ζ)hε(ζ)dζ,

and conditions (C2-C3), we get

supz|LεG0(z)| := ‖v‖∞‖G′′0‖∞ + C‖G′0‖∞[‖b‖∞ +

∫(ζ2 ∧ 1)hε(ζ)dζ] <∞,

and also, supx,s>0 |Rs(x)| ≤ supz |LεG0(z)| <∞.

Proof of Lemma 5.5. The proof is a direct consequence of assumption (C3-a) and Remark 5.2. Moreprecisely,

EU−pt = E(∇xX

xt )−2p(∫ t

0(∇xXx

s )−2σ(Xxs )2ds

)p ≤ 1

tpE

(∇xXxt )−2p

δ2p inf0≤s≤t(∇xXxs )−2p

=1

tpδ−2pE

((∇xX

xt )−2p sup

0≤s≤1(∇xX

xs )2p

).

The proof is completed.

Proof of Lemma C.1. By conditioning on the times of the jumps, which are necessarily distributedas the order statistics of n independent uniform [0, t] random variables, we have

P(Xt(x) ≥ x+ y|N εt = n) =

n!

tn

∫∆

P(Xt(ε, s1, . . . , sn, x) ≥ x+ y)dsn . . . ds1,

where ∆ := (s1, . . . , sn) : 0 < s1 < s2 < · · · < sn < t. Hence,

P(Xt(ε, s1, . . . , sn, x) ≥ x+ y) = E[P(Xt(ε, s1, . . . , sn, x) ≥ x+ y

∣∣Fs−n )]= E

[P [Xt−sn(ε, ∅, z) + γ(z, J) ≥ x+ y]

∣∣z=Xsn (ε,s1,...,sn−1,x)

]= E [Gt−sn (Xsn(ε, s1, . . . , sn−1, x);x, y)] ,


where

Gt(z;x, y) = P (Xt(ε, ∅, z) + γ(z, J) ≥ x+ y) .

In terms of the densities pt(·; z) := pt(·; ε, ∅, z) and Γ(·; z) of Xt(ε, ∅, z) and γ(z, J), respectively, weobviously have that

∂yGt(z;x, y) = −Γ ∗ pt(x+ y; z) :=

∫ ∞−∞

pt(ζ; z)Γ(x+ y − ζ; z)dζ,

supz,x,y|∂yGt(z;x, y)| ≤ sup

x,z|Γ(x; z)| sup

z

∫ ∞−∞

pt(ζ; z)dζ = supx,z|Γ(x; z)| <∞.

Since this bound is uniform in x, y, z, we can interchange partial differentiation and expectationto deduce the following representation for the transition density of Xt(ε, s1, . . . , sn, x), denotedhereafter pt(·; ε, s1, . . . , sn, x),

pt(x+ y; ε, s1, . . . , sn, x) = −∂yP (Xt(ε, s1, . . . , sn, x) ≥ x+ y)

= −E [∂yGt−sn (Xsn(ε, s1, . . . , sn−1, x);x, y)]

= E[Γ ∗ pt(x+ y; z)

∣∣z=Xsn (ε,s1,...,sn−1,x)

],

which can be bounded uniformly as follows:

supx,y,t,z,s1,...,sn

pt(x+ y; ε, s1, . . . , sn, x) ≤ supx,z|Γ(x; z)| := B <∞.

Using this property, we can easily check that

supx,y,t|∂yP(Xt(x) ≥ x+ y|N ε

t = n)| ≤ B,

and, hence,

supx,y

∣∣∂yRt(x, y)∣∣ = sup

x,ye−λεt

∞∑n=3

|∂yP(Xt(x) ≥ x+ y|N εt = n)| (λεt)

n

n!

≤ Be−λεt∞∑n=3

(λεt)n

n!≤ Bλ3

εt3.

The proof is then complete.

Proof of Lemma 6.1. By conditioning on the times of the jumps, which are necessarily distributed asthe order statistics of n independent uniform [0, t] random variables, we have

P( |Xt − x| ≥ log y|N εt = n) =

n!

tn

∫∆

P(|Xt(ε, s1, . . . , sn, x)− x| ≥ log y)dsn . . . ds1,

where ∆ := (s1, . . . , sn) : 0 < s1 < s2 < · · · < sn < t. Hence, we only need to bound

supn∈N,t∈[0,1]

1

n!

∫ ∞0

P(|Xt(ε, s1, ..., sn, x)− x| ≥ log y)dy

uniformly. By conditioning again,

P(|Xt(ε, s1, . . . , sn, x)− x| ≥ log y) = E[P(|Xt(ε, s1, . . . , sn, x)− x| ≥ log y

∣∣Fs−n )]≤ E

[P [|Xt−sn(ε, ∅, z)− x|+ |γ(z, J)| ≥ log y]

∣∣z=Xsn (ε,s1,...,sn−1,x)

].


Let M > 0 be a constant such that supx,ζ γ(x, ζ) < M (which existence is ensured from conditionC2). Fix any positive constant A and t ≤ 1, we have

Ee|Xt(ε,s1,...,sn,x)−x| =

∫ ∞0

P|Xt(ε, s1, . . . , sn, x)− x| > log ydy

=

∫ A

0


+

∫ ∞A


≤ A+

∫ ∞0

E[P |Xt−sn(ε, ∅, z)− x|+ |γ(z, J)| ≥ log y

∣∣z=Xsn (ε,s1,...,sn−1,x)

]dy

≤ A+ 2e−λ1M+ 12λ21ε

2k(1+expλ1ε)1

Aα1

α

(Eeλ1|Xsn (ε,s1,...,sn−1,x)−x|) .

Above, we used (3.1) for the last inequality with λ = λ1 = 1 + α, where α > 0 is to be chosen later.Now we iterate the above procedure by taking λi = (1 + α)i, i = 1, 2, ..., n, at each step, and chooseλn = (1 + α)n = e. We conclude that there exists a large enough constant C independent of n and tsuch that ∫ ∞

0

P|Xt(ε, s1, . . . , sn, x)− x| > log ydy ≤ Cn

(1

α

)n.

In what follows, we only need to show Cn (1/α)n /n! → 0 as n → ∞. Recall that α = e1/n − 1. Wehave

log

[Cn

(1

α

)n]∼ n

(C + log

1

n

)as n→∞.

On the other hand, we know log n! ∼ n2/2 as n→∞. The proof is then complete.

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Department of Statistics, Purdue University, West Lafayette, IN 47907, USA

E-mail address: [email protected]

Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago,Chicago, IL 60607, USA

E-mail address: [email protected]

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SMALL-TIME EXPANSIONS FOR LOCAL JUMP-DIFFUSIONS … · SMALL-TIME EXPANSIONS FOR LOCAL JUMP-DIFFUSIONS MODELS WITH INFINITE JUMP ACTIVITY JOSE E. FIGUEROA-L OPEZ AND CHENG OUYANG