International Journal of Pure and Applied Mathematics · 156 A. Climescu-Haulica, A.F. Gualtierotti Λ that is deﬁned, and thus computable without knowing which of the PN or PS+N

International Journal of Pure and Applied Mathematics————————————————————————–Volume 2 No. 2 2002, 153-212

LIKELIHOOD RATIO DETECTION OF RANDOM

SIGNALS: THE CASE OF CAUSALLY FILTERED

AND WEIGHTED WIENER AND POISSON NOISES

A. Climescu-Haulica1, A.F. Gualtierotti2 §

1Communications Research CentreOttawa K2H 8S2, CANADA

e-mail: [email protected], 21 Maladiere, CH-1022

Chavannes-pres-Renens, SWITZERLANDe-mail: [email protected]

Abstract: This paper contains first of all conditions for absolute continuity aswell as a new likelihood ratio formula for detecting a random signal whose lawis unknown, and which is obscured by noise that is modeled as the output of acausal filter of weighted Wiener and Poisson processes. Secondly, the derivationpresented reveals, through its reproducing kernel Hilbert space framework, thereasons that make the method work, as well as its limitations. The tools usedare the Cramer-Hida representation and stochastic calculus not assuming the“usual conditions.”

AMS Subject Classification: 60G35, 94A13Key Words: absolute continuity, likelihood ratio, detection, random signal,non-Gaussian signal and noise

1. Problem Description

Underwater acoustics is about sonars, that is, techniques that use waves ofmechanical vibration to transmit and receive information, under water, for suchwaves that are propagated most easily in that environment. Detection is about

Received: June 1, 2002 c© 2002, Academic Publications Ltd.

§Correspondence author

154 A. Climescu-Haulica, A.F. Gualtierotti

spotting the presence of information in a background of noise. One would thusexpect that detection processes would transit through the wave equation whoseprototype is given by Dp = dN, where p is pressure, N is noise, and D is anoperator of the form

D = ∆ − 1

c2∂2

∂t2+

b

c2∂

∂t∆,

where b, c > 0, and ∆ = ∂2

∂x2 + ∂2

∂y2 + ∂2

∂z2 . b is viscosity, and c, velocity.

One would solve the equation and compute the law of the process. Therewould be a process for noise only, and a process for the signal obscured bynoise. Computation of the likelihood ratio would follow. The crucial difficultyis that b and c are themselves stochastic fields of very complex structure and that“determination of the probability distributions of the wave fields in a stochasticmedium (also in a weakly inhomogeneous stochastic medium) is impossible”(Sobczyk [21, p. 152]).

A second approach that has been attempted could be defined as phenomeno-logical in that an acoustic field is defined to be a linear superposition of indi-vidual fields randomly produced by similar sources (Middleton [19]). The ran-domness is described by various analytically tractable processes. Unfortunatelysuch methods produce explicit laws only for the first order, implying reliance onindependent observations and consequently loss of information. They further-more require estimation of parameters for which there seldom will be adequateinformation.

A third approach is basically statistical and it is that which is pursuedhere. It consists in positing a very general signal-in-noise model, then derivinga generic likelihood ratio function, since the likelihood ratio is the best detectorin most practical instances (Helstrom [12]), and then discretizing the likelihoodratio to fit it to data. It should be noted that discretizing must follow derivationof the likelihood, as the law of the signal-in-noise process is typically unknown.

Indeed, “in the operation of a sonar system the operator is repeatedly facedwith the problem of detecting a signal which is obscured by noise. This signalmay be an echo resulting from a transmitted signal over which the operator hassome control, or it may have its origin in some external source. These two modesof operation are commonly distinguished as active and passive sonar. Similarsituations arise in radar surveillance and in seismic exploration . . . Signals comein all shapes and forms. In active sonar systems one may use simple sinusoidalsignals of fixed duration and modulations thereof. There are impulsive signalssuch as those made with explosions or thumpers. At the other extreme onemay make use of pseudorandom signals. In passive systems, the signals whosedetection are sought may be noise in the conventional meaning of the word;

LIKELIHOOD RATIO DETECTION OF RANDOM... 155

noise produced by propellers or underwater swimmers, for example. . . .

. . . When echoes are produced by extended targets such as submarines,there are two distinct ways in which the echo structure is affected. First, thereis the interference between reflections from the different structural features onthe hull of the submarine. This interference leads to a target strength thatfluctuates rapidly with changes in the aspect. Secondly, there is the elongationof the composite echo due to the distribution of reflecting features along thesubmarine. . . . A final source of pulse distortion is the Doppler shifts producedby the relative motions between the source, the medium, and the targets. Sincethe source, the medium and the target (or detector in passive listening) mayeach have a different vector velocity relative to the bottom, the variety of effectsmay be quite large” (Horton [14, 1.1]).

Previous work on this topic has mostly been a (difficult) mathematical tra-vail with models whose bearing on really applied problems has been somewhattenuous. A recent survey on what is available can be found in Kailath et al [16].One will find below a method to obtain likelihood ratios for a fairly general andrealistic family of noise models.

1.1. The Detection Problem

“Detection of stochastic signals in Gaussian or non-Gaussian noise is a validmodel for many important signal detection problems. In some of these prob-lems, the noise is very nonstationary and the signal cannot be represented asa set of narrowband components. Many examples can be given of applicationswhere such problems arise; they abound in such areas as sonar and radar”(Baker et al [2, p. 1]).

The notation shall be S for the signal sent, N for the noise, and S + N

for the signal received when a signal was sent. This additive model can be arestriction, but it has been shown that this is not the case when the noise ispurely Gaussian (Baker et al [1]).

There are four operations that one must perform successfully to be able toclaim that a likelihood ratio detection problem is solved. To enumerate them,let PN and PS+N be, respectively, the probabilities induced on L2 [0, T ] by Nand S +N. [0, T ] is the time span of observation for detection. One must thenfirstly ascertain that the likelihood ratio exists for the problem at hand, whichis not always the case as Slepian [20] noticed already in 1958! Technically, onemust check that PS+N is absolutely continuous with respect to PN .

Secondly, when the likelihood ratio exists, one must produce a functional


Λ that is defined, and thus computable without knowing which of the PN orPS+N regimes obtains, for every observable noise or signal in noise path. Onemust have in particular that, for measurable A,

PS+N (A) =

∫

A

Λ (f)PN (df) .

The functional Λ being available, one must thirdly be able to solve for Λ0,

for every predefined probability of false alarm α ∈ ]0, 1[ , the equation1

α = PN (f ∈ L [0, T ] : Λ (f) > Λ0) ,

and then compute the probability of detection 1 − β, where

β = PS+N (f ∈ L [0, T ] : Λ (f) ≤ Λ0) .

Fourthly, given observations of f at times 0 ≤ t1 < · · · < tn ≤ T, one

must be able to obtain approximations Λn to Λ and Λ(n)0 to Λ0 such that,

simultaneously,

PN

(f ∈ L [0, T ] : Λn (f (t1) , . . . , f (tn)) > Λ

(n)0

)≈ α,

PS+N

(f ∈ L [0, T ] : Λn (f (t1) , . . . , f (tn)) > Λ

(n)0

)≈ 1 − β.

1.2. Content of Paper

This paper contains the derivation, under minimal assumptions, of a likelihooddetection formula for a random signal whose law is unknown, and which isblurred by a noise which is the output of a causal filters, that filters a weightedsum of Wiener and Poisson processes. The usefulness of such models is discussedat length in Baker et al [2], [3], [4].

Two remarks about the derivation should be made. As one is using stochas-tic calculus on D [0, T ] , for processes which are adapted to the filtration gener-ated by the evaluation maps, and defined simultaneously for couples of proba-bility measures not known, a priori, to be mutually absolutely continuous, theusual assumption of the “usual conditions” of stochastic calculus being met isnot warranted. That is why the reference used for stochastic calculus is VonWeizsacker et al [23]. Secondly, most of the derivation is made under assump-tions that are somewhat more general than those used in the end to obtainthe likelihood ratio: the main reason for so doing is that one can thus better

1f ∈ L [0, T ] means thatR

f2 < ∞.


see where one meets the limits of the method that is used. In particular, oneeventually explicitly sees to what extent the chosen framework is essential fora “realistic” modelling of signal detection problems.

The authors of this paper are happy to aknowledge a debt to Dr. C.R.Baker, of UNC, Chapel Hill, who saw early on (in 1980 already) that, in orderto obtain realistic detection models, one should couple the Cramer-Hida rep-resentation with stochastic calculus, and to Dr. J. Memin, of Rennes 1, whowas the first to obtain the proper form of the likelihood ratio for Wiener noise(Memin [18]) and whose methods have proven of use here also.

2. The Model

2.1. The Noise Nα

The noise Nα is defined as the integral of a non-anticipative deterministic ker-nel with respect to a process with orthogonal increments, and may be lookedat as a filtered white noise with independent, weighted Gaussian and Poissoncomponents.

2.1.1. The Integrator

As usual, one assumes that (Ω,A, P ) is the reference probability space, and thatall processes considered are defined on that space, and adapted to a filtrationA of A, which satisfies the “usual conditions”.

A generalized Brownian motion is a Brownian motion for which the variancefunction is a non-negative, monotone non-decreasing, and continuous function.It is the type of Brownian motion that emerges from the Cramer-Hida repre-sentation (Cramer [8] and Hida [13]). Its paths are almost surely continuous,and those that are not may be taken as being continuous to the right (VonWeizsacker et al [23, 4.3.5, p. 71]). It is denoted B1 in the sequel, and β1

represents its variance function:

V [B1 (·, t)] = E[B2

1 (·, t)]

= β1 (t) , 0 ≤ t ≤ T.

One has that (Von Weizsacker [23, Examples, p. 148]), for fixed t ∈ [0, T ] ,almost surely, with respect to P,

〈B1〉 (ω, t) = β1 (t) .


B2 denotes a Poisson process. Then β2 (t) , which stands forE [B2 (·, t)] , is finite, and continuous for t ≥ 0 (Todorovic [22, 2.4.1, p. 41]).Let

B2 (ω, t) = B2 (ω, t) − β2 (t) ,

where B2 is a square integrable martingale. One has that (Von Weizsacker [23,Examples, p. 148]), for fixed t ∈ [0, T ] , almost surely, with respect to P,

〈B2〉 (ω, t) = β2 (t) .

Furthermore, for fixed t ∈ [0, T ] , almost surely, with respect to P,

[B2] (ω, t) = B2 (ω, t) .

It is assumed that B1 and B2 are independent.Let then 0 ≤ α ≤ 1, and set2:

βα (t) = αβ1 (t) + (1 − α) β2 (t) ,

Bα (ω, t) =√αB1 (ω, t) +

√1 − α B2 (ω, t) .

Bα is then a square integrable martingale, and, for fixed t ∈ [0, T ] , almostsurely, with respect to P,

〈Bα〉 (ω, t) = βα (t) ,

[Bα] (ω, t) = αβ1 (t) + (1 − α)B2 (ω, t) .

2.1.2. The Integrand

Let F denote a Borel measurable function over the rectangle [0, T ]× [0, T ] thathas the following properties:

– for t and x in [0, T ] , fixed, but arbitrary, such that x > t, F (t, x) = 0,

– for t ∈ [0, T ] , fixed, but arbitrary,∫ t

0 F2 (t, x) βα (dx) <∞,

– the map t 7→ [F (t, ·)]α ∈ L2 [βα] is continuous (where [F (t, ·)]α is theequivalence class of F (t, ·) in L2 [βα]),

– [F (t, ·)]α, t ∈ [0, T ] generates L2 [βα] .

Remark. The conditions that F must satisfy are those that insure that Nα

has a canonical representation of multiplicity one, in the sense of Cramer-Hida[8], [13]. A discussion of the nature of the restriction on the noise process thatis thus introduced may be found in Ephremides [10].

2To have a weighted sum of the type αB1 + (1 − α) B2 one would need to divide Bα by√α +

√1 − α but this would only add complexity to the notation.


2.1.3. Noise Model and Properties

One sets, as an integral with respect to a process with orthogonal increments(Todorovic [22, 7.4, p. 160]), and for t ∈ [0, T ] fixed, but arbitrary,

Nα (ω, t) =

∫ t

0F (t, x)Bα (ω, dx) .

Then, for t ∈ [0, T ] fixed, but arbitrary, E [Nα (·, t)] = 0, and the covarianceCNα of Nα is, furthermore, given by the following equality:

CNα (s, t) =

∫ s∧t

0F (s, x)F (t, x) βα (dx) , (s, t) ∈ [0, T ] × [0, T ] .

As a consequence of the assumptions on F, the function t 7→CNα (t, t) is continuous for t ∈ ]0, T [ . Nα is thus continuous in quadratic mean(Todorovic [22, 6.21, p. 133]) and its covariance is continuous (Todorovic [22,6.2.2, p. 133]). Furthermore, the paths of Nα are, almost surely, with respectto P, in L2 [0, T ] .

Let H (Nα) denote the reproducing kernel Hilbert space (RKHS) of Nα.

One has that (Grenander [11, p. 97])

H (Nα) =

f (t) =

∫ t

0F (t, x) f (x)βα (dx) , f ∈ L2 [βα]

.

For the inner product 〈 · , · 〉H(Nα) of H (Nα) , one has furthermore that

〈f , g〉H(Nα) = 〈f, g〉L2[βα]

whenever f and g ∈ L2 [βα] , and

f (t) =

∫ t

0F (t, x) f (x)βα (dx) , g (t) =

∫ t

0F (t, x) g (x)βα (dx) .

The covariance operator RNα : L2 [0, T ] −→ L2 [0, T ] is computed using theformula:

RNα

([f ]L2[0,T ]

)=

[∫ T

0CNα (·, x) f (x) dx

]

L2[0,T ]

, f ∈ L2 [0, T ] ,

where square brackets denote again equivalence classes. This operator is nonnegative, self-adjoint, and continuous, with finite trace (Balakrishnan [5, p.125]). It can thus be written as

RNα =∞∑

i=1

λi [ei ⊗ ei] ,


where, for an orthonormal family en, n ∈ IN ,

RNαen = λien, [en ⊗ en] f = 〈f, en〉L2[0,T ] en, λi ≥ 0,∞∑

n=1

λn <∞.

In an obvious way, one may write, in L2 [P ] ,

[Nα (·, t)]L2[P ] =[N (1)

α (·, t)]

L2[P ]+[N (2)

α (·, t)]

L2[P ],

with

N (1)α (ω, t) =

√α

∫ t

0F (t, x)B1 (ω, dx) ,

N (2)α (ω, t) =

√1 − α

∫ t

0F (t, x) B2 (ω, dx) .

N(1)α and N

(2)α are independent, and thus, on L2 [0, T ] ,

PNα = PN

(1)α⋆ P

N(2)α,

where the star denotes convolution, and, for instance, PNα is the measure in-duced on L2 [0, T ] by P and the maps

ω 7→ 〈Nα (ω, ·) , f〉L2[0,T ], f ∈ L2 [0, T ] .

If S(1)α ,S(2)

α ,Sα denote, respectively, the supports, in L2 [0, T ] , of PN

(1)α, P

N(2)α,

and PNα , one then has that

Sα = S(1)α + S(2)

α .

Remark. R denoting range, let

K1 = R(R

12

N(1)α

), K = R

(R

12Nα

).

The expression given above, relating the supports of PN

(1)α, P

N(2)α, and PNα ,

and a result of Ito [15] yield then that, whenever K1 = L2 [0, T ] , then

Sα = L2 [0, T ] .


If now U : L2 [βα] −→ L2 [0, T ] denotes the operator for which Uf is theequivalence class, in L2 [0, T ] , of 〈F (t, ·) , f〉L2[βα], one furthermore has that

U = R12NαJ⋆, where J : L2 [0, T ] −→ L2 [βα]

is a partial isometry onto L2 [βα] , with initial spaceK. The operator J is unitaryas soon as K = L2 [0, T ] . A sufficient condition is that K1 be L2 [0, T ] , that is,that P

N(1)α

have full support.

Let finally K be the range of R12Nα, and define, on K, the inner product

〈R12Nαf,R

12Nαg〉

K= 〈f, g〉L2[0,T ].

Then L2 [βα] and K are unitarily equivalent, and thus so are H (Nα) andK. As a consequence, one has, mutatis mutandis,

K = K1 +K

2 .

2.2. The Signal S

Let S denote a random signal, adapted to A. It is assumed that, almost surely,with respect to P,

S (ω, ·) ∈ H(N (1)

α

).

As it can be seen further on in the paper, the method used does work forS (ω, ·) ∈ H (Nα) only when β1 = β2. Nevertheless the assumptions which aremade, though less “natural” and elegant than the latter, for the problem athand, cover that case also. They have however the advantage of unmasking therole of each assumption.

It can be shown (Baker et al [1, Theorem 3, Step 3, p. 170]) that thefollowing representation obtains:

S (ω, t) = α

∫ t

0F (t, x) s (ω, x)β1 (dx) ,

for some predictable s, with paths in L2 [β1] . Thus one always has that

P(ω ∈ Ω : ||s (ω, ·)||2L2[β1]

<∞)

= 1.

Also, in what follows, s will usually be progressively measurable, exceptwhen a predictability assumption is required, and then the assumption will beexplicit. One sees here that the latter is not a restriction.


Finally, still in what follows, Xα will represent the process Sα + Nα andYα a process such that, for t ∈ [0, T ] , fixed, but arbitrary, almost surely, withrespect to P,

Yα (ω, t) = α

∫ t

0s (ω, x) β1 (dx) +Bα (ω, t) .

2.3. Summary List of Assumptions

Here is a list of recurrent assumptions which will be called upon in order toshorten the statement of many propositions. D denotes the σ-field of D [0, T ]generated by the evaluation maps

ev (f, t) = f (t) , t ∈ [0, T ] , f ∈ D [0, T ] ,

Dt that which is generated by the evaluation maps “up to time t,” and D =Dt, t ∈ [0, T ] .

A0. The basic probability space is (Ω,A, P ) , and the basic filtration isA. For A the “usual assumptions” obtain.

A1. B(·)α is a process, defined on an appropriate probability space, with

respect to an appropriate filtration, represented by the symbol “·” (whichcan be absent!). It has the following defining ingredients:

– 0 < α < 1;

– B(·)α =

√α B

(·)1 +

√1 − α B

(·)2 ;

– B(·)1 is generalized Brownian motion with variance function β1 : it

has continuous paths, almost surely, and the non-continuous onesare continuous to the right β1 is continuous non-decreasing;

– B(·)2 is a Poisson process with expectation β2 and B

(·)2 = B

(·)2 − β2;

– B(·)1 and B

(·)2 are independent;

A2. s is a process, progressively measurable for A, with the property3

P

(ω ∈ Ω :

∫ T

0s2 (ω, x) β1 (dx) <∞

)= 1.

3This assumption is the consequence of the “requirement” that S (ω, ·) ∈ H“

N(1)α

”

.


A3. Yα is a process with paths in D [0, T ] . It has the property that, fort ∈ [0, T ] , fixed, but arbitrary, almost surely, with respect to P,

Yα (ω, t) = α

∫ T

0s (ω, x)β1 (dx) +Bα (ω, t) .

A4. s is a process, progressively measurable for D, with the propertythat

P

(ω ∈ Ω :

∫ T

0s2 (Yα (ω, ·) , x)β1 (dx) <∞

)= 1.

A5. Yα is a process with paths in D [0, T ] . It has the property that, fort ∈ [0, T ] , fixed, but arbitrary, almost surely, with respect to P,

Yα (ω, t) = α

∫ T

0s (Yα (ω, ·) , x) β1 (dx) +Bα (ω, t) .

A6. s is a process, progressively measurable for D, with the propertythat

P

(ω ∈ Ω :

∫ T

0s2 (Bα (ω, ·) , x)β1 (dx) <∞

)= 1.

A7. For φ, a deterministic, strictly positive and measurable function suchthat ∫ T

0φ (x) β2 (dx) <∞,

one can and does define

ln Lα,s,φ (ω, t) = −√α

∫ t

0s (ω, x)B1 (dx)

− α

2

∫ t

0s2 (ω, x) β1 (dx) +

∫ t

0ln [φ (x)]B2 (ω, dx)

+

∫ t

0[1 − φ (x)]β2 (dx) .

Remark. The terms of Lα,s,φ involving φ,B2, and β2 are basically thosethat yield the likelihood in the “pure” Poisson case (with deterministic intensity:see – Bremaud [6, T2, p. 165]).

A “likelihood ratio” L of the form

ln [L] = −∫s dBα − γ

∫s2 d [Bα] ,


or

ln [L] = −∫s dBα − δ

∫s2 d〈Bα〉,

would require, to progress along the “Girsanov’s route,” which is the one thatshall be travelled, an s with uniformly bounded jumps and, in the first case,jumps strictly smaller than one (Kallenberg [17, Lemma 23.19, p. 449]). Onthe one hand, it is unlikely that such evidence would be readily available, and,on the other, the simpler form that has been chosen for the initial likelihoodprovides sufficient evidence to confirm the fact that the “effective” part of thelikelihood is its Gaussian component.

A8.

EP [Lα,s,φ (·, T )] = 1.

The following is a useful lemma (Memin [18, 2.8. – Theoreme, p. 14]).

Lemma 1. When, respectively, A0, A2, A4 and A6 obtain, it can then

always be furthermore assumed, without the “usual assumptions,” that the

maps

t 7→∫ t

0| s | (ω, x)β1 (dx) ,

t 7→∫ t

0s2 (ω, x) β1 (dx) ,

t 7→∫ t

0| s | (Yα (ω, ·) , x)β1 (dx) ,

t 7→∫ t

0s2 (Yα (ω, ·) , x) β1 (dx)

are all continuous in the extended real line.

3. Absolute Continuity and Likelihood Ratio

for PBα and PYα

3.1. A Version of Girsanov’s Theorem

The results in this section follow from repeated use of standard results ofstochastic calculus, and in particular, of Ito’s formula (Von Weizsacker et al


[23, P. 194]). It is assumed that A0, A1, A2, and A7 obtain. The process Mis defined by the relation

M (ω, t) =

∫ t

0s (ω, x)B1 (ω, dx) .

Then

Lα,s,φ (ω, t) = 1 −√α

∫ t

0Lα,s,φ (ω, x−)M (ω, dx)

−∫ t

0Lα,s,φ (ω, x−) [1 − φ (x)] B2 (ω, dx) .

Lα,s,φ is thus a positive local martingale, and, consequently, a supermartingale.So E [Lα,s,φ (·, t)] ≤ 1, 0 ≤ t ≤ T.

To use the change of measure method, one must prove that the originalsignal-plus-noise process has, with respect to an explicitly defined, absolutelycontinuous probability measure, the same law as the original noise. It followsfrom the statements used below that one can only have, to achieve that φ ≡ 1.

In what follows, it is furthermore assumed that A8 obtains. This has theconsequence that

E [Lα,s,φ (·, t)] = 1, 0 ≤ t ≤ T,

which in turn allows one to define a probability measure Qα,s,φ by setting

Qα,s,φ (A) =

∫

A

Lα,s,φ (ω, T )P (dω) , A ∈ A.

As an immediate consequence, one has that P and Qα,s,φ, as defined above,are mutually absolutely continuous. Furthermore

dQα,s,φ

dP= Lα,s,φ (·, T ) and

dP

dQα,s,φ

=1

Lα,s,φ (·, T ).

Let the process Zα,s,φ be defined as follows. Set

Uα,s,φ (ω, t) =√α

∫ t

0s (ω, x)β1 (dx) +B1 (ω, t) ,

and

Vα,s,φ (ω, t) = B2 (ω, t) −∫ t

0φ (x)β2 (dx) .

Then,Zα,s,φ =

√α Uα,s,φ +

√1 − α Vα,s,φ,

and one subsequently has that


– the process Uα,s,φ (ω, t) is, with respect to Qα,s,φ, a generalized Brownianmotion such that

〈Uα,s,φ〉Qα,s,φ = β1,

where the notation 〈Uα,s,φ〉Qα,s,φ is chosen to remind one of the measurethat prevails;

– the process B2 is, with respect to Qα,s,φ, a Poisson process such that

E [B2 (·, t)] =

∫ t

0φ (x)β2 (dx) ;

– the process Zα,s,φ is, with respect to Qα,s,φ, a martingale;

– with respect to Qα,s,φ, Uα,s,φ and B2 are independent processes.

As a consequence of the above, one has the following proposition, which is aversion of Girsanov’s theorem. For a process X, X denotes the random elementω 7→ X (ω, ·) .

Proposition 1. It is assumed that A0, A1, A2, A7 and A8 obtain. Then,

with respect to Qα,s,1, Zα,f,1 defined by

Zα,f,1 =√α Uα,f,1 +

√1 − α Vα,f,1

is such that

Qα,s,1 Z−1α,s,1 = P B−1

α .

Remark. In what follows, one will replace Zα,s,1 with the shorter Yα.

3.2. Absolute Continuity and Radon-Nikodym Derivatives

for PBα and PYα

With the model assumed so far, only an implicit form of the likelihood ratio isavailable. That is, if PBα is the probability induced on D by P and Bα, andPYα that induced by P and Y α, obviously:

Proposition 2. It is assumed that A0, A1, A2, A7 and A8 obtain. Then,PYα and PBα are mutually absolutely continuous probability measures defined onD, and, for f ∈ D [0, T ] ,


– almost surely, with respect to PYα ,

dPBα

dPYα

[f ] = EPYα

[Lα,s,1 (·, T ) Y α = f

],

– almost surely, with respect to PBα ,

dPYα

dPBα

[f ] = EPBα

[1

Lα,s,1(·,T ) Y α = f].

It can be shown, as in the case for which Bα = B1, that the following corollaryis valid.

Corollary. It is assumed that A0, A1, A2, A7 obtain. Then the followingis true: EP [L (·, T )] < 1, and one has that PYα is absolutely continuous withrespect to PBα .

3.3. Factorizations

Explicit expressions for the likelihood ratio require that the right-hand sides inthe conclusion of Proposition 2 to be explicitly expressed as ordinary functionsdefined on D [0, T ] . This is achieved through factorization by Yα of the differentcomponents of Lα,s,1.

When the evaluation maps, denoted ev, are taken as processes with respectto “lifted‘” probabilities of the form PU , one will use the notation evPU forev. σt (Yα) is the σ-field generated by Yα (·, s) , s ≤ t , and σt (Yα) is σt (Yα)completed with the sets of measure zero, with respect to P, belonging to At.

σ (Yα) and σ (Yα) denote the resulting filtrations.

Proposition 3. It is assumed A0 and A1 obtain. Let Yα denote a pro-

cess with paths in D [0, T ] . If Bα is adapted to σ (Yα) , there exist, defined on

(D [0, T ] ,D, PYα) , processes BYα

1 , BYα

2 , and BYαα , adapted to D, with paths in

D [0, T ] , such that, for4

BYαα =

√α BYα

1 +√

1 − α BYα

2 ,

PYα [BYα

1

]−1= P B−1

1 ,

PYα [BYα

2

]−1= P B−1

2 ,

PYα [BYα

α

]−1= P B−1

α ,

4BYα

2 = BYα

2 − β2.


and, for t ∈ [0, T ] , fixed, but arbitrary, almost surely, with respect to P,

B1 (ω, t) = BYα

1 (Yα (ω, ·) , t) ,B2 (ω, t) = BYα

2 (Yα (ω, ·) , t) ,Bα (ω, t) = BYα

α (Yα (ω, ·) , t) .

Proof. The notation used is as follows. For any process U such thatU (ω, t−) makes sense,

∆U (ω, t) = U (ω, t) − U (ω, t−) .

The process

U s (ω, t) =∑

x≤t

∆U (ω, x)

is then called the process of the jumps of U. The process U c is subsequentlydefined as

U c (ω, t) = U (ω, t) − U s (ω, t) .

Then

∆Bα (ω, t) =√

1 − α ∆B2 (ω, t) ,

Bsα (ω, t) =

√1 − α B2 (ω, t) ,

Bcα (ω, t) =

√α B1 (ω, t) −

√1 − α β2 (t) .

Consequently, as Bα is adapted to σ (Yα) , so are then Bsα, and Bc

α, and thusB1 and B2. Since D [0, T ] is a metric space, it can be checked that, as in thepurely Gaussian case (Memin [18, 2.4. – Lemme, p. 8]), there is, defined on(D [0, T ] ,D, PYα) , a process BYα

1 , adapted to D, with paths in C [0, T ] , suchthat, for t ∈ [0, T ] , fixed, almost surely, with respect to P,

B1 (ω, t) = BYα

1 (Yα (ω, ·) , t) .

It thus suffices to obtain the analogous result for B2.

As in the Gaussian case, there exists, for t ∈ [0, T ] , fixed, but arbitrary, amodification B2 (·, t) , of B2 (·, t) , which is adapted to σt (Yα) , and for whichone has that

B2 (ω, t) = BYα

2 (Yα (ω, ·) , t) ,

for some BYα

2 (·, t) adapted to Dt.


Let now t(n)i denote the fraction i

2nT, 1 ≤ i ≤ 2n, and T (n)D the set

t(n)i , 1 ≤ i ≤ 2n

.

One has that

T (n)D ⊂ T (n+1)

D ,

and that TD, defined by

TD =

∞⋃

n=1

T (n)D ,

is a dense subset of [0, T ] . By construction, the paths of BYα

2 , restricted to TD,

are, almost surely, with respect to PYα , restrictions of paths of B2, a countingprocess associated with a Poisson process. So, given n ∈ IN, and f ∈ D [0, T ] ,one defines the set T Yα

n [f ] as follows:

T Yαn [f ] =

t ∈ TD : BYα

2 (f, t) ≥ n.

The next step requires the following definitions:

T Yαn [f ] =

T , if T Yα

n [f ] = ∅ ,inf T Yα

n [f ] , if T Yαn [f ] 6= ∅

and, for t ∈ [0, T ] ,

BYα

2 (f, t) =∞∑

n=1

I[[T Yαn ,T ]] (f, t) .

One has that

f ∈ D [0, T ] : BYα

2 (f, t) = n

=f ∈ D [0, T ] : T Yα

n [f ] ≤ t < T Yα

n+1 [f ],

and thus that

ω ∈ Ω : BYα

2 (Yα (ω, ·) , t) = n

=ω ∈ Ω : T Yα

n [Yα (ω, ·)] ≤ t < T Yα

n+1 [Yα (ω, ·)].

Let

A =ω ∈ Ω : BYα

2 (Yα (ω, ·) , s) < n, s ∈ TD

.


As

ω ∈ Ω : BYα

2 (Yα (ω, ·) , s) ≥ n, s ∈ TD

=ω ∈ Ω : B2 (ω, s) ≥ n, s ∈ TD

,

it follows that

T Yαn [Yα (ω, ·)]

= IA (ω)T + IAc (ω) infs ∈ TD : BYα

2 (Yα (ω, ·) , s) ≥ n

= IA (ω)T + IAc (ω) infs ∈ TD : B2 (ω, s) ≥ n

.

Let N denote a measurable set of measure zero, with respect to P, suchthat, for ω ∈ N c,

B2 (ω, s) = B2 (ω, s) , s ∈ TD.

Then, for ω ∈ N c,

T Yαn [Yα (ω, ·)] = IA∩Nc (ω)T + IAc∩Nc (ω) inf s ∈ TD : B2 (ω, s) ≥ n .

The process B2, being separable and continuous in probability, every densesubset is a separant, so that

inf s ∈ TD : B2 (ω, s) ≥ n = inf t ∈ [0, T ] : B2 (ω, t) ≥ n .

Define thus

Tn [ω] =

T , if B2 (ω, T ) < n ,

inf t ∈ [0, T ] : B2 (ω, t) ≥ n , if B2 (ω, T ) ≥ n .

Then, almost surely, with respect to P,

T Yαn [Yα (ω, t)] = Tn [ω] ,

and, consequently, for t ∈ [0, T ] , fixed, but arbitrary,

BYα

2 (Yα (ω, ·) , t) = B2 (ω, t) .

Thus, with respect to PYα , BYα

2 is a Poisson process restricted to [0, T ] , andT Yα

n , being one of the times of discontinuity of BYα

2 , is a stopping time for D. Inthe sequel, BYα

2 will be dentoted BYα

2 , and BYα

2 will be the Poisson martingale

BYα

2 (ω, t) − β2 (t) , (ω, t) ∈ Ω × [0, T ].


Corollary. Let σYαt (Bα) be the σ-field generated by σt (Bα) and the sets of

σt (Yα) which have measure zero for P. Similarly, let σYαt

(BYα

α

)be the σ-field

generated by σt(BYα

α

)and the sets of Dt which have measure zero for PYα .

Then

σYαt (Bα) = Y −1

α

σYα

t

(BYα

α

).

Proposition 4. It is assumed that A0, A1, A4 and A5 obtain. There isthen a process BYα

α , defined on the base (D [0, T ] ,D, PYα) , adapted to D, suchthat, for t ∈ [0, T ] , fixed, but arbitrary, almost surely, with respect to PYα ,

evPYα (f, t) = α

∫ t

0s (f, x)β1 (dx) +BYα

α (f, t) ,

with A1 true for BYαα .

Proof. Define BYαα as

BYαα (f, t) = evPYα (f, t) − α

∫ t

0s (f, x)β1 (dx) .

By definition one thus has that the map

t 7→ BYαα (f, t)

is, almost surely, with respect to PYα , in D [0, T ] . But the paths of BYαα that

are not in D [0, T ] can be taken as continuous to the right, thanks to Lemma1. It is furthermore adapted to D. Finally, for t ∈ [0, T ] , fixed, but arbitrary,almost surely, with respect to P,

BYαα (Yα (ω, ·) , t) = Yα (ω, t) − α

∫ t

0s (Yα (ω, ·) , x) β1 (dx) = Bα (ω, t) .

Thus, with respect to PYα , BYαα is a Levy process. But

∆BYα

α

(Yα (ω, ·) , t) = ∆Bα (ω, t) =

√1 − α ∆B2 (ω, t) ,

so that the jump process of BYαα is, with respect to PYα , a Poisson process.

Consequently, its continuous part is a generalized Brownian motion.

Proposition 5. It is assumed that A0, A1, A4 and A5 obtain. Let thenthe process M be defined on the base (Ω,A, P ) , and for the filtration σ (Yα) ,as

M (ω, t) =

∫ t

0s (Yα (ω, ·) , x)B1 (ω, dx) .


One can then find a process MYα , defined on the base (D [0, T ] ,D, PYα) , andadapted to the filtration D, with the following properties: its paths are con-tinuous to the right and belong, almost surely, with respect to PYα , to C [0, T ] .Furthermore, for a generalized Brownian motion BYα

1 , with variance β1, definedon (D [0, T ] ,D, PYα) , and adapted to D, for t ∈ [0, T ] , fixed, but arbitrary, al-most surely with respect to PYα ,

MYα (f, t) =

∫ t

0s (f, x)BYα

1 (f, dx) ,

andMYα (Yα (ω, ·) , t) = M (ω, t) .

Proof. One first considers simple processes s of the form

s (f, t) = IA (f) I]u,v] (t) , u < v, A ∈ Du.

If one sets B = Y −1α [A] , B then belongs to σu (Yα) , and, by definition,

∫ t

0s (Yα (ω, ·) , x)B1 (ω, dx)

= IA (Yα (ω, ·)) B1 (ω, t ∧ v) −B1 (ω, t ∧ u) .

Now, from Proposition 3, one has that B1 (ω, t) = BYα

1 (Yα (ω, ·) , t) , so that,setting

MYα (f, t) =

∫ t

0s (f, x)BYα

1 (f, dx) ,

one has the result for simple processes which are products of the appropriateindicators IA and I]u,v].

Let now S denote the class of processes s, defined on D [0, T ]× [0, T ] , whichare progressively measurable for D, bounded and such that5

s Yα ·B1 =s · BYα

1

Yα,

as stated. S is a vector space containing all constants. It is closed for uniformand monotone convergence. If Sf denotes the subspace of S made of finitelinear combinations of simple processes of the form

s (f, t) = IA (f) I]u,v] (t) , u < v, A ∈ Du,

5 denotes composition and · stochastic integration.


one gets a subspace which is stable for multiplication. The monotone classtheorem then yields that S contains all bounded predictable processes, andthus all elementary processes in the sense of Von Weizsacker [23, P. 72]. Theproperties of the stochastic integral suffice then to claim that the proposition’sassertion is true.

Remark. The same proof yields, mutatis mutandis, the same result withB1 replaced with Bα, and BYα

1 replaced with BYαα .

3.4. Likelihood Ratios for PBα and PYα

This section contains the likelihood ratio formulae for the detection of the signalin Yα when the noise is Bα. They only depend, as it should be, on the signalsent, the statistics of the noise, and the received waveform. Checking that theformulae are correct is routine, given the preceding factorizations (Propositions3 to 5).

Theorem 1. It is assumed that A0, A1, A4, A5, A7 with φ = 1, andA8 obtain. Then:

– PYα and PBα are mutually absolutely continuous;

– for almost every f ∈ D [0, T ] , with respect to PYα ,

ln

[dPBα

dPYα

](f)

= −√α

∫ T

0s (f, x)BYα

1 (f, dx) − α

2

∫ T

0s2 (f, x)β1 (dx) ;

– for almost every f ∈ D [0, T ] , with respect to PYα ,

− ln

[dPBα

dPYα

](f) =

∫ T

0s (f, x) evPYα (f, dx)

− α

2

∫ T

0s2 (f, x)β1 (dx) −

√1 − α

∫ T

0s (f, x) BYα

2 (f, dx)

= ln

[dPYα

dPBα

](f) ;


– for almost every f ∈ D [0, T ] , with respect to PBα ,

− ln

[dPBα

dPYα

](f) =

∫ T

0s (f, x) evPBα (f, dx)

− α

2

∫ T

0s2 (f, x)β1 (dx) −

√1 − α

∫ T

0s (f, x) BYα,Bα

2 (f, dx)

= ln

[dPYα

dPBα

](f) ,

where BYα,Bα

2 is the representation of BYα

2 with respect to PBα . It can bechecked that the two processes are identical.

4. Path Requirements for Absolute and Mutual

Absolute Continuity

Mutual absolute continuity has been obtained under two conditions, namelythat the random variable Lα,s,1 (·, T ) has expectation one, and that the “signal-plus-noise” process is the solution of a stochastic differential equation. The firstcondition is not, given the context, “natural” in that “natural” conditions are interms of the finiteness of the signal’s energy, i.e., in the chosen context, that ofthe signal’s RKHS norm. As such, it is then a “path condition,” rather than an“expectation condition” This section is thus devoted firstly to the investigationof mutual absolute continuity in terms of such path conditions. In the secondpart of the section, one studies innovation representations of “signal-plus-noise”models which are the usual way to transform the received signal into the solutionof a stochastic differential equation.

4.1. Sufficient path Conditions for Absolute and

Mutual Absolute Continuity

In what follows one keeps the same assumptions that prevailed to this point.The first result is the next proposition (Proposition 6) which will be proved asa sequence of lemmas: it determines conditions for mutual absolute continuityin terms of square integrability of the “derivative” of the signal paths. As onlyassumption A4, and not assumptions A4 and A6, do follow from the RKHSrequirement, Proposition 6 must be “weakened,” and that leads to Proposition7 which however “calls on” Proposition 6. Proposition 6 requires assumptions


that one is unlikely to be able to check, but its corollary says that the Cramer-Hida framework is sufficient to insure that these assumptions obtain.

Proposition 6. It is assumed that A0, A1, A4, A5 and A6 obtain, and

furthermore that s is predictable and that both6

PBα

(f ∈ D [0, T ] :

∫ T

0|s| (f, x)β2 (dx) <∞

)= 1,

and

PYα

(f ∈ D [0, T ] :

∫ T

0|s| (f, x)β2 (dx) <∞

)= 1

obtain. Then Theorem 1 is valid.

Proof. It shall be presented in a sequence of lemmas (Lemma 5 to Lemma9), followed by a short conclusion (Epilogue to Proposition 6).

Remark. From Proposition 3, given the assumptions A0, A1, A4 and A5

of the present proposition, one has that there exist, on (D [0, T ] ,D, PYα) ,

– a generalized Brownian motion BYα

1 , adapted to D, with

V[BYα

1 (·, t)]

= β1 (t) ,

– a Poisson process BYα

2 , adapted to D, independent of BYα

1 , for which

E[BYα

2 (·, t)]

= β2 (t) ,

such that, for t ∈ [0, T ] , fixed, but arbitrary, for almost every f ∈ D [0, T ] ,with respect to PYα ,

evPYα (f, t) = α

∫ t


α (f, t) ,

with BYαα (f, t) =

√α BYα

1 (f, t) +√

1 − α BYαα (f, t) , and BYα

2 = BYα

2 − β2.

Furthermore, from Lemma 1, one has that s can be replaced by s, for whichone has the further following properties:

– the map ν (f, t) =∫ t

0 s2 (f, x)β1 (dx) is continuous in IR+,

6These assumptions on s are needed in the proof of Lemma 8.


– one has that

PBα

(f ∈ D [0, T ] : ||s (f, ·)||2L2[β1]

<∞)

= 1,

and thatPYα

(f ∈ D [0, T ] : ||s (f, ·)||2L2[β1]

<∞)

= 1,

– and, for t ∈ [0, T ] , fixed, but arbitrary, for almost every f ∈ D [0, T ] ,with respect to PYα ,

evPYα (f, t) = α

∫ t


α (f, t) .

The following steps restrict the problem to paths f ∈ D [0, T ] for whichν (f, T ) < ∞. The (strict) stopping time Tn : D [0, T ] −→ [0, T ] is defined bythe equality

Tn (f) =T, if t ∈ [0, T ] : ν (f, t) ≥ n = ∅ ,inf t ∈ [0, T ] : ν (f, t) ≥ n , if t ∈ [0, T ] : ν (f, t) ≥ n 6= ∅ .

It should be noted that limn↑∞ Tn (f) = T if and only if t < T impliesν (f, t) <∞.

Further definitions are needed, as follows:

D [0, T ] = f ∈ D [0, T ] : ν (f, T ) <∞ ,D = D ∩ D [0, T ] , D ∈ D,

PYα

(D)

= PYα

(D ∩ D [0, T ]

),

D = D ∩ D [0, T ] ,

D = D ∩ D [0, T ] .

The process evPYαn is subsequently defined on the base(

D [0, T ] , D, PYα

), with respect to the filtration D, as

evPYαn (f, t) =

evPYα (f, t) , if (f, t) ∈ [[0, Tn[[ .

evPYα (f, t) − α∫ t

Tns (f, x) β1 (dx) , if (f, t) ∈ [[Tn, T ]] .


This process can be rewritten as

evPYαn (f, t) = f (t) − I[[Tn,T ]] (f, t)

α

∫ t

0I[[Tn,T ]] (f, x) s (f, x)β1 (dx)

,

and this shows first thatev

PYαn (f, t) , t ∈ [0, T ]

∈ D [0, T ] , as f ∈ D [0, T ] ,

and then that, on [[0, Tn]] ,

evPYαn (f, t) = f (t) = ev (f, t) .

One last definition yields the progressively measurable, bounded process sn,

given by the relation

sn (f, t) = I[[0,Tn]] (f, t) s (f, t) .

Let J : D [0, T ] −→ D [0, T ] be the (injection) map defined by the relationJ (f) = f. If E is a Borel set of IR,

[ev (·, t) J ]−1 (E) =f ∈ D [0, T ] : ev (J (f) , t) ∈ E

= D [0, T ] ∩ f ∈ D [0, T ] : ev (f, t) ∈ E∈ Dt.

The restriction of sn to D [0, T ] has thus the measurability properties of sn asdefined on D [0, T ] , and it will not then be necessary to introduce one morenotation to distinguish one situation from the other. In particular, an integralof the form ∫ t

0sn

(ev

PYαn (f, ·) , x

)β1 (dx)

will be well defined for f ∈ D [0, T ] .

Define now BYαα as the restriction of BYα

α to D [0, T ] . One can check that

PYα [B

Yα

α

]−1= PYα

[BYα

α

]−1.

One may then state:

Lemma 2. For every f ∈ D [0, T ] ,

Tn

(ev

PYαn (f, ·)

)= Tn (f) ,


and, for t ∈ [0, T ] , fixed, but arbitrary, almost surely, with respect to PYα ,

evPYαn (f, t) = α

∫ t

0sn

(ev

PYαn (f, ·) , x

)β1 (dx) + BYα

α (f, t) .

Proof. Let D(n)t = f ∈ D [0, T ] : Tn (f) = t ∈ Dt. The function I

D(n)t

has

then the representation

ID

(n)t

(f) = F (ev (f, ti) , 0 ≤ ti ≤ t, i ∈ IN) ,

the map F : IR∞ −→ IR being measurable. But, as Tn (f) = t, as seen above,for i ∈ IN,

evPYαn (f, ti) = f (ti) = ev (f, ti) ,

so that

F (ev (f, ti) , 0 ≤ ti ≤ t, i ∈ IN)

= F(ev

PYαn (f, ti) , 0 ≤ ti ≤ t, i ∈ IN

),

and consequently that

ID

(n)t

(f) = ID

(n)t

(ev

PYαn (f, ·)

),

which proves the first assertion of the lemma.

The same reason (and the definition of sn) yields that

sn (f, t) = sn

(ev

PYαn (f, ·) , t

).

Finally, when t < Tn (f) , and f ∈ D [0, T ] ,

evPYαn (f, t) = evPYα (f, t)

= α

∫ t

0s (f, x)β1 (dx) + BYα

α (f, t)

= α

∫ t

0sn (f, x) β1 (dx) + BYα

α (f, t)

= α

∫ t

0sn

(ev

PYαn (f, ·) , x

)β1 (dx) + BYα

α (f, t) ,


and, when t ≥ Tn (f) ,

evPYαn (f, t) = evPYα (f, t) − α

∫ t

Tn

s (f, x)β1 (dx)

= α

∫ Tn

0s (f, x) β1 (dx) + BYα

α (f, t)

= α

∫ t

0sn (f, x)β1 (dx) + BYα

α (f, t)

= α

∫ t

0sn

(ev

PYαn (f, ·) , x

)β1 (dx) + BYα

α (f, t) .

Define now Ψn : D [0, T ] −→ IR by the relation

ln[Ψn (f)

]= −√

α

∫ T

0sn

(ev

PYαn (f, ·) , x

)BYα

1 (f, dx)

−α2

∫ T

0s2n

(ev

PYαn (f, ·) , x

)β1 (dx) .

One then has, since, by definition,∫ T

0 s2n (f, x) β1 (dx) ≤ n,

Lemma 3. EPYα

[Ψn

]= 1.

Lemma 4. For f ∈ D [0, T ] , let Ψ be defined by

ln [Ψ (f)] = −√α

∫ T

0s (f, x)BYα

1 (f, dx) − α

2

∫ T

0s2 (f, x)β1 (dx) ,

and let Ψ denote the restriction of Ψ to D [0, T ](Ψ = Ψ J

). Then,

limn→∞

Ψn (f) = Ψ (f) ,

in probability, with respect to PYα .

Proof. For (f, t) ∈ [[0, Tn]] ,

evPYαn (f, t) = f (t) ,


so that, by monotone convergence, for almost every f ∈ D [0, T ] , with respectto PYα ,

limn→∞

∫ T

0s2n

(ev

PYαn (f, ·) , x

)β1 (dx)

= limn→∞

∫ T

0I[[0,Tn]] (f, x) s

2 (f, x)β1 (dx)

=

∫ T

0s2 (f, x) β1 (dx) .

Furthermore, for almost every f ∈ D [0, T ] , with respect to PYα , for n largeenough, Tn (f) = T, so that, for that same f, for n large enough,

sup0≤t≤T

|s (f, t)| I]]Tn,T ]] (f, t)

= 0.

Consequently

limnPYα

(f ∈ D [0, T ] : sup

0≤t≤T

|s (f, t)| I]]Tn,T ]] (f, x)

> ǫ

)= 0,

and thus (continuity of the integral: see – Von Weizsacker et al [23, 5.5.3, p.98]), if

Jn (f, t) =

∫ t

0sn (f, x) BYα

1 (f, dx) −∫ t

0s (f, t) BYα

1 (f, dx) ,

then

limnPYα

(f ∈ D [0, T ] : sup

0≤t≤T

|Jn (f, t)| > ǫ

)= 0.

Lemma 5. Let the probability measure QYαn be defined on D by the following

relation:

QYαn = PYα

[ev

PYαn

]−1.

Then, for A ∈ DTn ,

QYαn (A) = PYα

(D [0, T ] ∩A

)= PYα (A) .


Proof. First, one can show, as in the continuous case (Von Weizsackeret al [23, 2.2.6, page 35]), that DTn = σ

(evTn (·, t) , t ∈ [0, T ]

). Indeed, let

θn : D [0, T ] −→ D [0, T ] be defined by the relation

ev (θn (f) , t) = evTn (f, t) = f (t ∧ Tn (f)) .

Let then f0 ∈ D [0, T ] be fixed, but arbitrary, and set t0 = Tn (f0) . If t ≤ t0,

then

ev (f0, t) = f0 (t) = f0 (t ∧ Tn (f0)) = evTn (f0, t) = ev (θn (f0) , t) .

Thus, for every ψ adapted to Dt0 , ψ (f0) = ψ (θn (f0)) . In particular,

ITn≤t0 (θn (f0)) = ITn≤t0 (f0) = 1.

Consequently, for every φ adapted to DTn ,

φ (θn (f0)) = φ (θn (f0)) ITn≤t0 (θn (f0)) .

But φITn≤t0 is adapted to Dt0 , so that

φ (θn (f0)) = φ (f0) ITn≤t0 (f0) = φ (f0) .

As φ is adapted to D, it has, for fixed, measurable F and ti ∈ [0, T ] , i ∈ IN,

the following representation:

φ (f) = F (ev (f, ti) , 0 ≤ ti ≤ T, i ∈ IN) .

Using the relation φ (f) = φ (θn (f)) , valid for f ∈ DTn , one has then that

φ (f) = φ (θn (f)) = F (ev (f θn, ti) , 0 ≤ ti ≤ T, i ∈ IN)

= F(evTn (f, ti) , 0 ≤ ti ≤ T, i ∈ IN

)

which is adapted to σ(evTn (·, t) , t ∈ [0, T ]

). This establishes that DTn is con-

tained in σ(evTn (·, t) , t ∈ [0, T ]

).

The reverse inclusion is obtained by noticing that ev is continuous to theright, so that (Von Weizsacker et al [23, p. 41]) evTn (·, t) is adapted to Dt∧Tn ,

and thus that

σ(evTn (·, t) , t ∈ [0, T ]

)⊆ σ

(∪t∈[0,T ]Dt∧Tn

)⊆ DTn .


Finally, for B Borel in IR, and A =f ∈ D [0, T ] : evTn (f, t) ∈ B

,

QYαn (A) = PYα

(f ∈ D [0, T ] : evYα

n (f, ·) ∈ A)

= PYα

(f ∈ D [0, T ] : evYα

n (f, t ∧ Tn (f)) ∈ B)

= PYα

(f ∈ D [0, T ] : evT

n (f, t) ∈ B)

= PYα (A) .

One then finishes the proof with a monotone class argument.

Lemma 6. The assumptions are those of Proposition 6. The sequenceΨn, n ∈ IN

is then uniformly integrable for PYα .

Proof. Let σPYαt

(ev

PYαn

)denote the σ-field generated by

ev

PYαn (·, s) , s ≤ t

, and the sets of Dt which have measure zero for PYα .

By Lemma 2, the following obtains, almost surely, with respect to PYα , fort ∈ [0, T ] , fixed, but arbitrary:

evPYαn (f, t) = α

∫ t

0sn

(ev

PYαn (f, ·) , x

)β1 (dx) + BYα

α (f, t) .

BYαα (·, t) is thus adapted to σ

PYαt

(ev

PYαn

).

The setup is now as follows. The underlying probability is PYα . evPYαn is

a process with paths in D [0, T ] . BYαα is a process for which A1 obtains. As

BYαα (·, t) is adapted to σ

PYαt

(ev

PYαn

), it follows from Proposition 3 that there

is a process BYαα,n which factors BYα

α through evPYαn :

BYαα (f, t) = BYα

α,n

(ev

PYαn (f, ·) , t

),

almost surely, with respect to P Yα , and for which , with respect to the prob-ability measure QYα

n , defined in Lemma 5, A1 obtains. One then sets, forf ∈ D [0, T ] , almost surely, with respect to QYα

n ,

ln [Φn (f)] = −√α

∫ T

0sn (f, x) BYα

1,n (f, dx) − α

2

∫ T

0s2n (f, x)β1 (dx) .


By Proposition 5, almost surely, with respect to PYα ,

Ψn (f) = Φn

(ev

PYαn (f, ·)

).

Furthermore, the equation

evPYαn (f, t) = α

∫ t

0sn

(ev

PYαn (f, ·) , x

)β1 (dx) + BYα

α (f, t) ,

can be rewritten as

evPYαn (f, t) = α

∫ t

0sn

(ev

PYαn (f, ·) , x

)β1 (dx) + BYα

α,n

(ev

PYαn (f, ·) , t

),

which yields

evQYαn (f, t) = α

∫ t


α,n (f, t) ,

almost surely, with respect to QYαn . Applying Lemma 3, one has that

EQ

Yαn

[Φn] = EPYα

[Φn evPYα

n

]= EPYα

[Ψn

]= 1.

The two relations

evQYαn (f, t) = α

∫ t


α,n (f, t) ,

EQ

Yαn

[Φn] = 1,

together with Proposition 2, insure that QYαn and PBα are mutually absolutely

continuous and that, almost surely with respect to QYαn ,

dPBα

dQYαn

(f) = EQ

Yαn

[Φn | evQ

Yαn = f

]= Φn (f) .

But, according to Theorem 1 (fourth item), Φn has, with respect to PBα , andfor some Poisson process BYα,Bα

2 , the following equivalent representation:

ln [Φn] (f) = −∫ T

0sn (f, x) evPBα (f, dx)

+α

2

∫ T

0s2n (f, x)β1 (dx)

+√

1 − α

∫ T

0sn (f, x) BYα,Bα

2 (f, dx) .


Define

Mn (f, t) =

∫ T

0sn (f, x) evPBα (f, dx) ,

Nn (f, t) =

∫ T

0sn (f, x) BYα,Bα

2 (f, dx) ,

Vn (f, t) =

∫ T

0s2n (f, x)β1 (dx) ,

Wn (f, t) = −Mn (f, t) +√

1 − α Nn (f, t) +α

2Vn (f, t) ,

and let K > 0 denote an arbitrary constant. One then has that

∫

Ψn>KΨn (f) PYα (df)

=

∫

Φn>KΦn (f) QYα

n (df) = PBα (Φn > K) .

But

PBα (Φn > K) = PBα (f ∈ D [0, T ] : Wn (f, T ) > ln [K])

≤ PBα

(f ∈ D [0, T ] : |Mn (f, T )| > ln [K]

3

)

+ PBα

(f ∈ D [0, T ] : |Nn (f, T )| > ln [K]

3√

1 − α

)

+ PBα

(f ∈ D [0, T ] : Vn (f, T ) >

2 ln [K]

3α

).


Now,

PBα

(f ∈ D [0, T ] : |Mn (f, T )| > ln [K]

3

)

= P

(ω ∈ Ω :

∣∣∣∣∫ T

0sn (Bα (ω, ·) , x)Bα (ω, dx)

∣∣∣∣ >ln [K]

3

)

≤ P

(ω ∈ Ω :

√α

∣∣∣∣∫ T

0sn (Bα (ω, ·) , x)B1 (ω, dx)

∣∣∣∣ >ln [K]

6

)

+ P

(ω ∈ Ω :

√1 − α

∫ T

0|sn| (Bα (ω, ·) , x)B2 (ω, dx) >

ln [K]

12

)

+ P

(ω ∈ Ω :

√1 − α

∫ T

0|sn| (Bα (ω, ·) , x) β2 (dx) >

ln [K]

12

)

and, since, for a continuous local martingale M, and constants α > 0, andK > 0, (Memin [18, 2.83. Lemme, p. 19]7)

P (ω ∈ Ω : |M (ω, t)| > α) ≤ P (ω ∈ Ω : 〈M〉 (ω, t) > K) + 2e−α2

2K ,

one has, for L > 0,

P

(ω ∈ Ω :

√α

∣∣∣∣∫ T

0sn (Bα (ω, ·) , x)B1 (ω, dx)

∣∣∣∣ >ln [K]

6

)

≤ P

(ω ∈ Ω :

∫ T

0s2n (Bα (ω, ·) , x)β1 (dx) > L

)

+ 2exp

−

[ln[K]

6

]2

2L

.

Choosing L = ln [K] , the latter exponential term becomes K− 172 . Furthermore,

as

P

(ω ∈ Ω :

∫ T

0s2 (Bα (ω, ·) , x) β1 (dx) <∞

)= 1,

7See also the remark that follows.


it follows that

limK↑∞

P

(ω ∈ Ω :

√α

∣∣∣∣∫ T

0sn (Bα (ω, ·) , x)B1 (ω, dx)

∣∣∣∣ >ln [K]

6

)= 0,

independently of n. Now, if τp’s denotes the time at which jump number p ofthe Poisson process B2 occurs, since | p ∈ IN : τp (ω) ≤ T |< ∞, whateverω ∈ Ω, one has that

∫ T

0|s| (Bα (ω, ·) , x)B2 (ω, dx) =

∑

τp≤T

|s| (Bα (ω, ·) , τp (ω)) <∞,

from which it follows that

limK↑∞

P

(ω ∈ Ω :

√1 − α

∫ T

0|sn| (Bα (ω, ·) , x)B2 (ω, dx) >

ln [K]

12

)

= 0,

independently of n. Finally, by assumption,

P

(ω ∈ Ω :

∫ T

0|s| (Bα (ω, ·) , x) β2 (dx) <∞

)= 1,

so that

limK↑∞

P

(ω ∈ Ω :

√1 − α

∫ T

0|sn| (Bα (ω, ·) , x) β2 (dx) >

ln [K]

12

)= 0,

independently of n. Consequently,

limK↑∞

PBα

(f ∈ D [0, T ] : |Mn (f, T )| > ln [K]

3

)= 0,

independently of n. Since, QYαn and PBα are mutually absolutely continuous,

stochastic integrals with respect to these probabilities are indistinguishable(Von Weizsacker et al [23, P. 245]) and thus

PBα

(f ∈ D [0, T ] : |Nn (f, T )| > ln [K]

3√

1 − α

)

= QYαn

(f ∈ D [0, T ] : |Nn (f, T )| > ln [K]

3√

1 − α

).


As Nn is adapted to DTn , on has, by Lemma 5, that

QYαn

(f ∈ D [0, T ] : |Nn (f, T )| > ln [K]

3√

1 − α

)

= PYα

(f ∈ D [0, T ] : |Nn (f, T )| > ln [K]

3√

1 − α

)

= P

(ω ∈ Ω :

∣∣∣∣∫ T

0sn (Yα (ω, ·) , x) B2 (ω, dx)

∣∣∣∣ >ln [K]

3√

1 − α

)

≤ P

(ω ∈ Ω :

∫ T

0|s| (Yα (ω, ·) , x)B2 (ω, dx) >

ln [K]

6√

1 − α

)

+ P

(ω ∈ Ω :

∫ T

0|s| (Yα (ω, ·) , x) β2 (ω, dx) >

ln [K]

6√

1 − α

).

Consequently, as above,

limK↑∞

PBα

(f ∈ D [0, T ] : |Nn (f, T )| > ln [K]

3√

1 − α

)= 0,

independently of n. The term containing Vn has similarly a limit that vanishes.Lemma 6 is thus proved.

Remark. If M is a local martingale, null at the origin, and such that itsjumps are, almost surely, uniformly bounded (|∆M | ≤ µ < ∞), almost surely,then8

P (|Mt| > K) ≤ P

(2ϕ

(µK

L

)[M ]t > L

)+ 2e−

K2

2L ,

where

ϕ (x) = −x+ ln (1 − x)+x2

.

When M is continuous, µ = 0, and this inequality allows one to bypass theassumptions on the integrability of s, with respect to β2. Thus, even for s’swith bounded jumps, there is no obvious extension of the method that worksfor the continuous case.

The following lemma is elementary, but useful.

Lemma 7. Let (Ω,A, P ) and (Ω,A, Q) denote two probability spaces, andassume that Ω0 ∈ A is such that P (Ω0) = Q (Ω0) = 1. Define then

A0 = A ∩ Ω0, and, for A ∈ A, A0 = A ∩ Ω0.

8The proof for the continuous case (µ = 0), can be found in Memin [18, 2.83, Lemme, p.19].


Set finally

P0 (A0) = P (A ∩ Ω0) , and Q0 (A0) = Q (A ∩ Ω0) .

Then, whenever P0 and Q0 are mutually absolutely continuous, so are P andQ, and furthermore, almost surely, with respect to P and Q,

dQ

dP(ω) =

dQ0

dP0(ω) if ω ∈ Ω0

0 if ω 6∈ Ω0.

Epilogue to Proposition 6.

Lemmas 4 and 6 yield that

limn→∞

Ψn (f) = Ψ (f) ,

in L1

[PYα

]. From Lemma 3 one then has that EPYα

[Ψ]

= 1. But then (Propo-

sition 2), if PBα is the restriction of PBα to D [0, T ] , also produced by BYαα , PYα

and PBα are mutually absolutely continuous. So, by Lemma 7, one has thatPYα and PBα are mutually absolutely continuous. Furthermore EPYα

[Ψ] = 1.

Corollary. If β2 = β1, or if, almost surely, S (ω, ·) ∈ H (Nα) , Lemma 6 istrue without the integrability conditions on s with respect to β2, since then, fori = 1, 2, ∫ t

0|s (x)| βi (dx)

2

≤ βi ([0, T ])

∫ T

0s2 (x)βi (dx) .

But then, to be true, Proposition 6 does not require those same conditions either.

Proposition 7. It is assumed that A0, A1, A4, and A5 obtain, that s ispredictable and that both

PBα

(f ∈ D [0, T ] :

∫ T

0|s| (f, x)β2 (dx) <∞

)= 1,

and

PYα

(f ∈ D [0, T ] :

∫ T

0|s| (f, x)β2 (dx) <∞

)= 1

obtain also.


Then PYα is absolutely continuous with respect to PBα , and, almost surely,with respect to PYα ,

ln

[dPYα

dPBα

(f)

]=

∫ T


− α

2

∫ T

0s2 (f, x)β1 (dx)

−√

1 − α

∫ T

0s (f, x) BYα

2 (f, dx) .

Proof. Absolute continuity comes from the Corollary to Proposition 2. Letthen f belong to D [0, T ] , and

ln [Φn (f)] =

∫ T

0sn (f, x) evPBα (f, dx)

− α

2

∫ T

0s2n (f, x)β1 (dx)

−√

1 − α

∫ T

0sn (f, x) BYα,Bα

2 (f, dx) .

Let Tn be the stopping time of the previous proposition, and set

Cn = f ∈ D [0, T ] : Tn (f) = T .

Then, for A ∈ D, A ∩ Cn belongs to DTn , (Dellacherie et al [9, 56 Theoreme,56.1, p. 189]), and, by Lemma 8,

QYαn (A ∩Cn) = PYα (A ∩ Cn) .

As PYα

(D [0, T ]

)= 1, limn PYα (Cn) = 1, QYα

n and PBα are mutually absolutely

continuous, and, almost surely, with respect to PBα ,

dQYαn

dPBα

= Φn,


one may write, using what precedes:

PYα (A) = limnPYα (A ∩ Cn) = lim

nPYα

(A ∩ D [0, T ] ∩Cn

)

= limnPYα (A ∩Cn) = lim

nQYα

n (A ∩ Cn)

= limn

∫

A∩Cn

(f)dQYα

n

dPBα

(f)PBα (df) = limn

∫

A

ICnΦn (f)PBα (df) .

Let now

ln [Φ (f)] =

∫ T


− α

2

∫ T

0s2 (f, x)β1 (dx)

−√

1 − α

∫ T

0s (f, x) BYα

2 (f, dx) .

When Tn (f) = T,∫ T

0 s2 (f, x)β1 (dx) ≤ n. Consequently, letting C =

D [0, T ] , one may write, as, on Cn, IC = 1 :

ICn (f)Φn (f) = ICn (f) eIC(f) ln[Φn(f)].

The following definitions will shorten some unwieldly expressions:

Mn (f, t) =

∫ t


sn,p (f, t) = sn (f, t) − sn+p (f, t) ,

Mn,p (f, t) =

∫ t

0sn,p (f, x) evPBα (f, dx) ,

M (1)n,p (ω, t) =

∫ t

0sn,p (Bα (ω, ·) , x)B1 (ω, dx) ,

M (2)n,p (ω, t) =

∫ t

0|sn,p (Bα (ω, ·) , x)|B2 (ω, dx) ,

M (3)n,p (ω, t) =

∫ t

0|sn,p (Bα (ω, ·) , x)|β2 (dx) .


One has that

PBα (f ∈ D [0, T ] : IC (f) |Mn (f, T ) −Mn+p (f, T )| > K)

= PBα (f ∈ D [0, T ] : IC (f) |Mn,p (f, T )| > K)

≤ P

(ω ∈ Ω :

√α IC (Bα (ω, ·))

∣∣∣M (1)n,p (ω, T )

∣∣∣ >K

3

)

+ P

(ω ∈ Ω :

√1 − α IC (Bα (ω, ·))M (2)

n,p (ω, T ) >K

3

)

+ P

(ω ∈ Ω :

√1 − α IC (Bα (ω, ·))M (3)

n,p (ω, T ) >K

3

).

Because of Remark 2.8 Memin [18, P. 19],

P

(ω ∈ Ω :

√α IC (Bα (ω, ·))

∣∣∣M (1)n,p (ω, T )

∣∣∣ >K

3

)

is dominated by

P(ω ∈ Ω : α IC (Bα (ω, ·)) 〈M (1)

n,p〉 (ω, T ) > L)

+ 2exp

− K2

18L

.

But, with respect to P,

〈M (1)n,p〉 (ω, T )

=

∫ T

0I]]Tn(Bα(ω,·)),Tn+p(Bα(ω,·))]] (ω, x) s

2 (Bα (ω, ·) , x) β1 (dx) ,

and, since, for Bα (ω, ·) ∈ C = D [0, T ] ,

∫ T

0s2 (Bα (ω, ·) , x) β1 (dx) <∞,

then

limn,p↑∞

P

(ω ∈ Ω :

√α IC (Bα (ω, ·))

∣∣∣M (1)n,p (ω, T )

∣∣∣ >K

3

)= 0.

Given the assumptions on the integrability of |s| , a similar argument yieldsthat

limn,p↑∞

P

(ω ∈ Ω :

√1 − α IC (Bα (ω, ·))M (2)

n,p (ω, T ) >K

3

)= 0,


and that

limn,p↑∞

P

(ω ∈ Ω :

√1 − α IC (Bα (ω, ·))M (3)

n,p (ω, T ) >K

3

)= 0.

Thus, with respect to PBα , the sequence

IC (f)

∫ T

0sn (f, x) evPBα (f, dx) , n ∈ IN

has a limit in probability, which will be denoted JBα (f) .

Now, for f ∈ D [0, T ] ,

limn↑∞

∫ T

0s2n (f, x)β1 (dx) =

∫ T

0s2 (f, x)β1 (dx) <∞,

and, for IC (f)∫ T

0 sn (f, x) BYα,Bα

2 (f, dx) , one basically repeats the argumentsalready given. As, trivially, almost surely, with respect to PBα , limn ICn = IC ,

PBα − limn

ICn (f) ln [Φn (f)] = JBα (f)

− α

2IC (f)

∫ T

0s2 (f, x) β1 (dx) −

√1 − α IC (f)

∫ T

0s (f, x) BYα,Bα

2 (f, dx) .

The exponential of this limit shall be denoted ΦPBα .

As PYα is absolutely continuous with respect to PBα , on one hand,

∫ T

0sn (f, x) evPYα (f, dx) =

∫ T


and, on the other,

∫ T

0sn (f, x) BYα,Bα

2 (f, dx) =

∫ T

0sn (f, x) BYα

2 (f, dx) ,

so that, with respect to PYα , Φn has the following representation:

ln [Φn (f)] =

∫ T

0sn (f, x) evPYα (f, dx)

− α

2

∫ T

0s2n (f, x)β1 (dx) −

√1 − α

∫ T

0sn (f, x) BYα

2 (f, dx) .


But the assumptions made and in particular A4 now imply that the limit inprobability, with respect to PYα , of the sequence Φn, n ∈ IN is Φ. So, withrespect to PYα , ΦPBα = Φ.

To finish the proof, one must still check that the sequenceICnΦn, n ∈ IN is uniformly integrable with respect to PBα , which insuresthat

limn↑∞

EPBα[ICnΦn] = EPBα

[ΦPBα

].

But, since QYαn and PBα are mutually absolutely continuous, as seen in the proof

of Lemma 6, and that Φn = dQYαn

dPBαis one of the Radon-Nikodym derivatives,

setting

Dn = f ∈ D [0, T ] : ICn (f)Φn (f) > K ,Dn = f ∈ D [0, T ] : Φn (f) > K ,

one has that

∫

Dn

ICn (f)Φn (f)PBα (df) ≤∫

Dn

Φn (f)PBα (df) = QYαn (Dn)

= PYα [ev

PYαn

]−1(Dn) = PYα

(f ∈ D [0, T ] : Φn evPYα

n (f) > K).

But, on [[0, Tn]] , evPYαn = ev, so that

PYα

(f ∈ D [0, T ] : Φn evPYα

n (f) > K)

= PYα

(f ∈ D [0, T ] : Φn (f) > K

)

= PYα

(f ∈ D [0, T ] : f ∈ D [0, T ] : Φn (f) > K ∩ D [0, T ]

).

Now, assumptions A0, A1, A4 and A5 yield Proposition 4, which allows oneto write

evPYα (f, t) = α

∫ t


α (f, t) .

So, using the representation of Φn with respect to PYα , one may then legiti-mately write:


∫

Dn

ICn (f)Φn (f)PBα (df)

≤ PYα

(√α

∫ T

0sn (f, x) BYα

1 (f, dx)

+α

2

∫ T

0s2n (f, x)β1 (dx) > ln [K]

).

The latter goes to zero as in previous arguments.

Corollary 1. When β2 = β1, or when, almost surely, S (ω, ·) ∈ H (Nα) ,the integrability assumptions on s with respect to β2 of Proposition 7 are nolonger necessary, as the argument given in the Corollary to Proposition 6 isstill valid.

Corollary 2. Given assumptions A0, A1, A4, and A5, assumption A6

is necessary and sufficient for mutual absolute continuity of PBα and PYα .

4.2. Weak Solution of a Stochastic Differential Equation

The innovations representation of the “signal-plus-noise process,” within theadopted RKHS framework, requires the seemingly unrelated, preliminary re-sults that follow. Their reason for being will thus emerge later in the paper.

A weak solution of equation

Yα (ω, t) = α

∫ t

0s (Yα (ω, ·) , x)β1 (dx) +Bα (ω, t)

is a triple Bw1 , B

w2 , P

w such that

1. Pw is a probability measure on D, such that, with respect to it,

(a) Bw1 is a generalized Brownian motion, adapted to D, with variance

VP w [Bw1 (·, t)] = β1 (t) ,

(b) Bw2 is a Poisson process, adapted to D, for which

EP w [Bw2 (·, t)] = β2 (t) ,

(c) Bw1 and Bw

2 are independent,

2. and, for fixed t ∈ [0, T ] , almost surely, with respect to Pw,

evP w

(f, t) = α

∫ T

0s (f, x)β1 (dx) +Bw

α (f, t) ,


where

Bwα (f, t) =

√α Bw

1 (f, t) +√

1 − α Bw2 (f, t) ,

Bw2 (f, t) = Bw

2 (f, t) − β2 (t) .

Lemma 8. Let Bα be a process satisfying A1. The process evPBα has then(with respect to PBα) the representation

evPBα =√α Bev

1 +√

1 − α Bev2 ,

where

PBα [Bev1 ]−1 = P B−1

1 , and PBα [Bev2 ]−1 = P B−1

2 ,

and Bev2 = Bev

2 − β2, for some probability space (Ω,A, P ) .

Proof. First, given PBα , it can always, without restriction, be assumedthat it is a measure induced from a (Ω,A, P ) space by a generalized Brownianmotion B1 and an independent Poisson process B2 “summing” to the processBα =

√αB1 +

√1 − αB2, as in assumption A1.The process Bev

2 is then definedby the equality:

Bev2 (f, t) =

1√1 − α

∑

u≤t

∆evPBα

(f, u) .

For fixed 0 ≤ t1 < · · · < tn ≤ T, and a Borel set G ∈ IRn, let

GD = f ∈ D [0, T ] : (Bev2 (f, t1) , . . . , B

ev2 (f, tn)) ∈ G .

Let GΩD = B−1

α (GD) . If ω ∈ GΩD, then

1√

1 − α

∑

u≤t1

∆Bα (ω, u) , . . . ,1√

1 − α

∑

u≤tn

∆Bα (ω, u)

∈ G,

that is

(B2 (ω, t1) , . . . , B2 (ω, tn)) ∈ G.

Bev2 is thus, with respect to PBα , a Poisson process such that

EPBα[Bev

2 (·, t)] = β2 (t) .


One shows similary that, with respect to PBα , Bev1 , defined by

Bev1 =

1√α

evPBα −

√1 − α (Bev

2 − β2),

is a generalized Brownian motion such that

EPBα[Bev

2 (·, t)] = β2 (t) .

Corollary. σt(evPBα

)= σt (Bev

1 ) ∨ σt (Bev2 )

Proposition 8. Let s be progressively measurable for D, and assume thatone has, for every f ∈ D [0, T ] ,

∫ T

0s2 (f, x)β1 (dx) <∞, and

∫ T

0|s| (f, x)β2 (dx) <∞.

With the notation of Lemma 8, define, for almost every f ∈ D [0, T ] , withrespect to PBα ,

ln [Φ (f)] =√α

∫ T

0s (f, x)Bev

1 (f, dx) − α

2

∫ T

0s2 (f, x) β1 (dx) .

Then,

Yα (ω, t) = α

∫ t

0s (Yα (ω, ·) , x)β1 (dx) +Bα (ω, t)

has a weak solution if, and only if EPBα[Φ] = 1, in which case the solution is

unique.

Proof. Suppose first that EPBα[Φ] = 1. Let then Pw be defined, as a

probability, by the relation dPw = ΦdPBα . Define also Bwα as follows:

Bwα (f, t) = α

∫ t

0−s (f, x)β1 (dx) + evPBα (f, t) .

As Φ can be written in the form

ln [Φ (f)]

= −√α

∫ T

0−s (f, x)Bev

1 (f, dx) − α

2

∫ T

0−s2 (f, x)β1 (dx) ,

one may apply Girsanov’s theorem (Proposition 1) to obtain that

Pw [Bwα ]−1 = PBα

[evPBα

]−1= PBα .


As, furthermore,

evP w

(f, t) = α

∫ t


α (f, t) ,

one has then a weak solution, since, for instance, using Lemma 8, for G, a Borelset of IRn, 0 ≤ t1 < t2 < t3 < · · · < tn ≤ T,

GD = f ∈ D [0, T ] : (f (t1) , f (t2) , f (t3) , . . . , f (tn)) ∈ G ,

andBw

2 (f, t) =∑

u≤t

∆Bwα (f, t) ,

Pw (f ∈ D [0, T ] : Bw2 (f, ·) ∈ GD)

= PBα (f ∈ D [0, T ] : Bev2 (f, ·) ∈ GD) .

Suppose now that a weak solution exists. One has then, by definition,

evP w

(f, t) = α

∫ t


α (f, t) ,

which can be rewritten in the form

evP w

(f, t) = α

∫ t

0s(evP w

(f, ·) , x)β1 (dx) +Bw

α (f, t) .

One can then apply Proposition 6 to get that Pw and PBwα

are mutually abso-lutely continuous, and that, almost surely, with respect to PBw

α,

ln

[dPw

dPBwα

](f) =

∫ T

0s (f, x) evPBw

α (f, dx)

− α

2

∫ T

0s2 (f, x)β1 (dx)

−√

1 − α

∫ T

0s (f, x) B

evPw,Bw

α

2 (f, dx) ,

where BevPw

,Bwα

2 is the representation of Bw2 , with respect to PBw

α(≡ PBα). Fur-

thermore, with respect to PBwα, one has that (Lemma 8)

evPBwα =

√αBev

1 +√

1 − αBev2 .


Consequently

EPBwα

[ln

[dPw

dPBwα

]− ln [Φ]

]2

= (1 − α)EPBwα

[∫ T

0s (f, x)Bev

2 (f, dx) −∫ T

0s (f, x)B

evPw,Bw

α

2 (f, dx)

]2.

Now the evaluation map ev is a semimartingale with respect to Pw as well aswith respect to PBw

α. These two probability measures being mutually absolutely

continuous, [ev]Pw

= [ev]PBwα . As [ev]P

w

= Bw2 and [ev]PBw

α = Bev2 , and taking

into account the fact that Bw2 = B

evPw,Bw

α

2 , one has that EPBα[Φ] = 1.

Suppose now that a second solution(Bw

1 , Bw2 , P

w)

exists. As one must have

dP w

dPBα

= Φ,

P w = Pw.

Corollary 1. Proposition 8 will be true whenever β1 = β2, or S (ω, ·) ∈H (Nα) , for every ω ∈ Ω.

Corollary 2. If one only assumes, in Proposition 8, that

PBα

(f ∈ D [0, T ] :

∫ T

0s2 (f, x)β1 (dx) <∞

)= 1,

and that

PBα

(f ∈ D [0, T ] :

∫ T

0|s| (f, x)β2 (dx) <∞

)= 1,

one still has a solution, but it cannot any longer be claimed that it is unique.

4.3. Necessary Path Conditions for Absolute and Mutual Absolute

Continuity of PYα and PBα

To proceed one needs an observation that is stated as the following lemma.

Lemma 9. Let (Ω,A, P ) be a probability space, and let B(1) and B(2) be,with respect to P, two independent filtrations of A. Set

Bt = B(1)t ∨ B(2)

t and B = Bt, t ∈ [0, T ] .

Then, if M is a martingale for B(1), it is also a martingale for B.


The proposition that follows “embodies” the need of the RKHS assumption,but also the fact that an explicit expression for the likelihood ratio requiresmutual absolute continuity rather than simply absolute continuity.

Proposition 9. Suppose Yα is a process, defined on (Ω,A, P ) , adaptedto A, with paths in D [0, T ] , such that PYα and PBα are mutually absolutelycontinuous. When β1 = β2 ≡ β, one can then find

– a process s, defined on (D [0, T ] ,D, PYα) , predictable for D, and

– a zero-mean, generalized Brownian motion B1 and a generalized Poissonprocess B2, defined on (Ω,A, P ) , adapted to σ (Yα) , with

V [B1 (·, t)] = β (t) and E [B2 (·, t)] = β (t) ,

such that, for Bα =√αB1 +

√1 − αB2, and, for t ∈ [0, T ] , fixed, but arbitrary,

almost surely, with respect to P,

Yα (ω, t) = α

∫ t

0s (Yα (ω, ·) , x) β (dx) +Bα (ω, t) ,

with

PBα

(f ∈ D [0, T ] :

∫ T

0s2 (f, x)β (dx) <∞

)= 1,

and

PYα

(f ∈ D [0, T ] :

∫ T

0s2 (f, x)β (dx) <∞

)= 1.

Proof. By Lemma 8, evPBα =√α Bev

1 +√

1 − α Bev2 . Let

B(1)t = σt (Bev

1 ) , and B(1) =B(1)

t , t ∈ [0, T ].

B(1) is a Brownian filtration.Consider now the martingale L defined for B(1) as

L (f, t) = EPBα

[dPYα

dPBα

| B(1)t

].

It has a modification L (Von Weizsacker et al [23, 9.7.5, p. 241]) which iscontinuous to the right and has continuous paths, almost surely, with respectto PBα . L has then the representation (Von Weizsacker [23, 9.7.4, p. 239])

L (f, t) = 1 +√α

∫ t

0s (f, x)Bev

1 (f, dx) ,


where s is predictable for B(1). Furthermore

PBα

(f ∈ D [0, T ] :

∫ T

0s2 (f, x)β (dx) <∞

)= 1.

LetT (f) = inf

t ∈ [0, T ] :

[L (f, t) = 0

]or[L (f, t−) = 0

].

On[[T , T

]], the paths of L are, almost surely, with respect to PBα , equal

to zero. However, because PBα and PYα are mutually absolutely continuous,L (f, T ) > 0, almost surely, with respect to PBα . Consequently,

PBα

(f ∈ D [0, T ] : inf

t∈[0,T ]L (f, t) > 0

)= 1.

The expression ln[L (f, t)

]does thus make sense, almost surely, with respect

to PBα , and Ito’s formula then yields:

ln[L (f, t)

]=

√α

∫ t

0

s (f, x)

L (f, x)Bev

1 (f, dx) − α

2

∫ t

0

(s (f, x)

L (f, x)

)2

β (dx) ,

that is

L (f, t) = e√

αR t

0s(f,x)

L(f,x)Bev

1 (f,dx)−α2

R t

0

“

s(f,x)

L(f,x)

”2β(dx)

.

Set then

s (f, t) =s (f, x)

L (f, x).

Since∫ T

0s2 (f, x) β (dx) =

∫ T

0

(s (f, x)

L (f, x)

)2

β (dx)

≤ 1

inft∈[0,T ] L2 (f, t)

∫ T

0s2 (f, x) β1 (dx) ,

one has that

PBα

(f ∈ D [0, T ] :

∫ T

0s2 (f, x)β (dx) <∞

)= 1,

so that also

PYα

(f ∈ D [0, T ] :

∫ T

0s2 (f, x)β (dx) <∞

)= 1.


Finally, EPBα

[L (·, T )

]= 1. Consequently, there exists a weak solution to the

“formal”9 equation

Yα (ω, t) = α

∫ T

0s (Yα (ω, ·) , x) β (dx) +Bα (ω, t) .

By the Corollary to Lemma 8 and Lemma 9, L is, with respect to PBα , amartingale for D. One then defines, on (D [0, T ] ,D) , and for the filtration D,

Pwα (df) = L (f, T )PBα (df) ,

and, with respect to Pwα ,

Bwα (f, t) = −α

∫ t

0s (f, x)β (dx) + evP w

α (f, t) .

By Girsanov’s theorem, one has that

Pwα [Bw

α ]−1 = PBα .

Finally, L (·, T ) , being a martingale for D (Lemma 9), is a version ofdPYα

dPBα.

Consequently Pwα = PYα . One must then set

BYαα = Bw

α Y −1α .

Proposition 10. Suppose Yα is a process, defined on (Ω,A, P ) , adaptedto A, with paths in D [0, T ] , such that PYα is absolutely continuous with respectto PBα . When β1 = β2 ≡ β, one can then find

– a process s, defined on (D [0, T ] ,D, PYα) , progressively measurable for D,and

– a zero-mean, generalized Brownian motion B1 and a generalized Poissonprocess B2, defined on (Ω,A, P ) , adapted to σ (Yα) , with

V [B1 (·, t)] = β (t) and E [B2 (·, t)] = β (t) ,

9One should not, in particular, take the Bα of the “formal” equation as the Bα of theproposition’s conclusion!


such that, for Bα =√αB1 +

√1 − αB2, and, for t ∈ [0, T ] , fixed, but arbitrary,

almost surely, with respect to P,

Yα (ω, t) = α

∫ t

0s (Yα (ω, ·) , x) β (dx) +Bα (ω, t) ,

with

PYα

(f ∈ D [0, T ] :

∫ T

0s2 (f, x)β (dx) <∞

)= 1.

Proof. One has again, as in Proposition 9, that

L (f, t) = 1 +√α

∫ t

0s (f, x)Bev

1 (f, dx) ,

with

PBα

(f ∈ D [0, T ] :

∫ T

0s2 (f, x)β (dx) <∞

)= 1.

But now L can be equal to zero, and that is why one defines

Tn (f) =

inft ∈ [0, T ] : L (f, t) < 1

n

, if

t ∈ [0, T ] : L (f, t) < 1

n

6= ∅ ,

T , ift ∈ [0, T ] : L (f, t) < 1

n

= ∅ .

If B(1) ∈ B(1)t∧Tn

, then

PYα

(B(1)

)=

∫

B(1)

L (f, T )PBα (df)

=

∫

B(1)

E[L (·, T ) | B(1)

t∧Tn

]PBα (df) =

∫

B(1)

L (f, t ∧ Tn)PBα (df) .

Thus one has, on B(1)t∧Tn

, PYα (df) = L (f, t ∧ Tn)PBα (df) . But, as

L (·, t ∧ Tn) ≥ 1n, one has also, still on B(1)

t∧Tn, that

PBα (df) =PYα (df)

L (f, t ∧ Tn),

so that, since D [0, T ] belongs to B(1)t∧Tn

,

EPYα

[1

L (·, t ∧ Tn)

]= PBα (D [0, T ]) = 1.


The sequence Tn, n ∈ IN is increasing and bounded. It thus has a limit,denoted limn Tn, which is a stopping time. As L is continuous, almost surely,with respect to PYα , one has, by Fatou’s lemma, that

EPYα

[1

L (·, t ∧ limn Tn)

]= EPYα

[lim inf

n

1

L (·, t ∧ Tn)

]

≤ lim infn

EPYα

[1

L (·, t ∧ Tn)

]= 1,

that is,

EPYα

[1

L (·, t ∧ limn Tn)

]≤ 1.

As L (·, limn Tn) = 0, almost surely, with respect to PYα , one must have limn Tn =T, almost surely, with respect to PYα . Furthermore, as

∫ Tn

0

s (f, x)

L (f, x)

2

β (dx) ≤ n2‖s (f, ·)‖2L2[β],

it follows that

1 = PYα

(f ∈ D [0, T ] : ‖s (f, ·)‖2

L2[β] <∞)

≤ PYα

(f ∈ D [0, T ] :

∫ Tn

0

s (f, x)

L (f, x)

2

β (dx) <∞).

Consequently,

PYα

(f ∈ D [0, T ] :

∫ T

0

s (f, x)

L (f, x)

2

β (dx) <∞)

= 1.

As I[[0,Tn]]s

Lis in L2 [β] , almost surely, with respect to PBα , one can legitimately

define, on (D [0, T ] ,D, PBα) , and for the filtration D, the process Bα,n, by thefollowing relation:

Bα,n (f, t) = −α∫ t

0I[[0,Tn]] (f, x)

s (f, x)

L (f, x)β (dx) + evPBα (f, t) .

LTn being a martingale for the filtration B(1), and thus, by Lemma 9, for thefiltration D, one thus defines, on Dt, a probability Qn when setting

Qn (df) = L (f, t ∧ Tn)PBα (df) .


One must then show that, on (D [0, T ] ,D, Qn) , Bα,n is martingale for D suchthat

Qn B−1

α,n = PBα .

But, almost surely, with respect to PBα ,

L (f, t ∧ Tn) ≥ 1

n,

so that one can compute ln[L (f, t ∧ Tn)

], and consequetly apply Ito’s formula

to obtain, almost surely, with respect to PBα , the following equality:

ln[L (f, t ∧ Tn)

]=

√α

∫ t

0I[[0,Tn]] (f, x)

s (f, x)

L (f, x)B1 (f, dx)

− α

2

∫ t

0I[[0,Tn]] (f, x)

s (f, x)

L (f, x)

2

β (dx) .

ButEPBα

[L (·, t ∧ Tn)

]= 1,

because of the martingale property of L (·, t ∧ Tn) , for D, and PBα . One canthus invoke Girsanov’s theorem to assert that, the base being (D [0, T ] ,D, Qn) ,and the filtration D,

Qn B−1

α,n = PBα .

Now BTn

α,n+1 = Bα,n, so that one can legitimately define, again for the base(D [0, T ] ,D, PBα) , and the filtration D, the process

Bα (f, t) = −α∫ t

0I[[0,limn Tn]] (f, x)

s (f, x)

L (f, x)β (dx) + evPBα (f, t) .

Since limn Tn = T, almost surely, with respect to PYα , and that the latter isabsolutely continuous with respect to PBα , one has, almost surely, with respectto PYα ,

Bα (f, t) = −α∫ t

0

s (f, x)

L (f, x)β (dx) + evPYα (f, t) .

One must then finally check that, for the base (D [0, T ] ,D, PYα) and thefiltration D,

PYα B−1

α = PBα .


To that end, one recognizes that

L (f, t ∧ Tn) Bα (f, t ∧ Tn) = L (f, t ∧ Tn) Bα,n (f, t ∧ Tn) .

But, on the base (D [0, T ] ,D, Qn) , and for the filtration D, Bα,n is a martingale,and Qn (df) = L (f, t ∧ Tn)PBα (df) , so that L (·, · ∧ Tn)Bα (·, · ∧ Tn) is a martingale on the base (D [0, T ] ,D, PBα) , and for the fil-tration D, and consequently a martingale on the base (D [0, T ) ,D, Qn) , for thesame the filtration. Since B(1) ⊆ D, Tn is a stopping time for D, and thus, since,

on one hand, Dt∧Tn = B(1)t∧Tn

∨ B(2)t∧Tn

, and, on the other, Qn|Dt∧Tn= PYα|Dt∧Tn

,

PYα (df) = L (f, t ∧ Tn)PBα (df) .

Bα is thus a local martingale on the base (D [0, T ] ,D, PYα) , for the filtrationD.

One must also check that Bα has, with respect to PYα the same law as Bα

with respect to P. But, for scalars θ1, . . . , θp, forming the vector θp, and times0 ≤ t1, . . . , tp ≤ T,

EPYα

[ei〈θp, eB

(p)

α 〉IRp

]= lim

nEPYα

[ei〈θp, eB

(p)

α,n〉IRp

]

= limnEQn

[ei〈θp, eB

(p)

α,n〉IRp

]

= EP

[ei〈θp, eB

(p)〉IRp

],

where B(p)

α , B(p)

α,n, and B(p)

are vectors with respective components Bα (·, ti) ,Bα,n (·, ti) , and Bα (ti) , for 0 ≤ ti ≤ T, 1 ≤ i ≤ p.

Corollary. It is assumed that A0, A1 and A2 obtain. One then hasthe following “innovations representation,” for t ∈ [0, T ] , almost surely, withrespect to P :

Yα (ω, t) =

∫ t

0s (Yα (ω, ·) , x) β1 (dx) +BYα

α (ω, t) ,

where

– s is defined on (D [0, T ] ,D, PYα) , and is progressively measurable for D,– BYα

α is defined on (Ω,A, P ) , adapted to σ (Yα) , and

– P [BYα

α

]−1= PBα .


5. Absolute Continuity and Likelihood Ratio for PNα and PXα .

5.1. The “Inversion” Process M

The Cramer-Hida representation “says” intuitively that the paths of Bα andNα are, probabilistically, in one-to-one correspondence. The mathematical ex-pression for this intuition is the M process whose definition and propertiesfollow.

Terms whose definition is omitted are those of Section 2. I[0,t] denotes theindicator of the interval [0, t] . The basic probability space is

(L2 [0, T ] ,B (L2 [0, T ]) , PNα) .

For t ∈ [0, T ] , fixed, but arbitrary, the following variables indexed on i areconsidered on L2 [0, T ] × [0, T ] :

Mi (f, t) =1

λi〈U[I[0,t]

], ei〉L2[0,T ]

〈f, ei〉L2[0,T ] .

One has that EPNα[Mi (·, t)] = 0, and that

EPNα[Mi (·, t)Mj (·, t)] =

1

λiλj

〈U[I[0,t]

], ei〉L2[0,T ]

〈U[I[0,t]

], ej〉L2[0,T ]

× EPNα

[〈f, ei〉L2[0,T ]〈f, ej〉L2[0,T ]

]

= δi,j 〈I[0,t], J [ei]〉L2[βα]〈I[0,t], J [ej ]〉L2[βα]

.

One needs the following facts which result from simple calculations.

- The family J [ei] , i ∈ IN is a complete orthonormal set in L2 [βα] .

-∑∞

i=1EPNα

[M2

i (·, t)]

=∣∣∣∣I[0,t]

∣∣∣∣2L2[βα]

= βα (t) .

- For t ∈ [0, T ] , fixed, but arbitrary, the series∑∞

i=1Mi (f, t) convergesalmost surely, with respect to PNα , and in L2 [PNα ] .

The inversion process M is defined by the following relation:

M (f, t) =

∞∑

i=1

Mi (f, t) .

Then one can sequentially check that the statements of the following listobtain.


– For (i, t) ∈ IN × [0, T ] , fixed, but arbitrary,

E[Bα (·, t) 〈Nα (·, ·) , ei〉L2[0,T ]

]= 〈U

[I[0,t]

], ei〉L2[0,T ]

.

– For t ∈ [0, T ] , fixed, but arbitrary,

E[M (Nα (·, ·) , t) −Bα (·, t)2

]= 0.

– Let 0 < t1 < · · · < tn ≤ T, and θ1, . . . , θn, be arbitrary constants. Then,

EPNα

[ei

Pnj=1 θjM(·,tj)

]= E

[ei

Pnj=1 θjBα(·,tj)

].

– M has, with respect to PNα , independent increments.

– For 0 < s < t ≤ T, EPNα[M (·, s)M (·, t)] = βα (s ∧ t) .

– For 0 < s < t ≤ T, EPNα

[M (·, t) −M (·, s)2

]= βα (t) − βα (s) .

– Let t ∈ [0, T ] , be fixed, but arbitrary, and let Mt be the σ-algebra gen-

erated by M (·, s) , s ≤ t , on L2 [0, T ] . Then, with respect to PNα , M

is a square integrable martingale for M = Mt , t ∈ [0, T ] .

– The process M is separable.

– With respect to PNα , the paths of M almost surely belong to D [0, T ] .

5.2. The Conditional Law of Bα given Nα

This conditional law is given by the following proposition.

Proposition 11. The assumptions being those of Section 2.1, Bα has, withrespect to Nα, a regular conditional law which is a point mass located at M.

Proof. Let F ⊆ D [0, T ] , and G ⊆ L2 [0, T ] be measurable subsets. One hasthat

P (ω ∈ Ω : Bα (ω, ·) ∈ F, Nα (ω, ·) ∈ G)

= P (ω ∈ Ω : M (Nα (ω, ·) , ·) ∈ F, Nα (ω, ·) ∈ G) .

Indeed, as, in L2 [P ] , for t ∈ [0, T ] , fixed, but arbitrary,

Bα (·, t) = M (Nα (·, ·) , t) ,


this equality is obviously true whenever

0 ≤ t1 < · · · < tp ≤ T,

Bi ∈ B [IR] , 1 ≤ i ≤ p,

F =f ∈ D [0, T ] : evt1 (f) ∈ B1, . . . , evtp (f) ∈ Bp

,

g1, . . . , gq ⊆ L2 [0, T ] ,

Bj ∈ B [IR] , 1 ≤ j ≤ q,

G =g ∈ L2 [0, T ] : 〈g, g1〉L2[0,T ] ∈ B1, . . . , 〈g, gq〉L2[0,T ] ∈ Bq

.

As such sets generate the corresponding σ-algebras, the equality is true ingeneral. But then

P (ω ∈ Ω : M (Nα (ω, ·) , ·) ∈ F, Nα (ω, ·) ∈ G)

=

∫

G

PNα (dg)P (M Nα ∈ F | Nα = g)

=

∫

G

PNα (dg)E [IF (M Nα) | Nα = g]

=

∫

G

PNα (dg) IF (M (g)) .

Corollary. EPNα

[dPYα

dPBα| Nα = g

]=

dPYα

dPBα(M (g, ·)) .

5.3. Existence and Form of the Likelihood Ratio

One has the following theorem, which in particular shows that there is no needof robust versions of the likelihood ratio in the sense of Clark [7]:

Theorem 2. One writes B for Bα, N for Nα, and Y for Yα. Other notationis as already encountered. Suppose then that

N (ω, t) =

∫ T

0F (t, x)B (ω, dx) ,

where

– assumptions A0 and A1 are valid for B with β1 = β2 = β,

– F is a non-anticipative (F (t, x) = 0, for x > t), measurable function,defined on [0, T ] × [0, T ] , whose equivalence classes generate L2 [β] ,


– S (ω, ·) ∈ H (N) , almost surely, with respect to P.

The following statements are then valid.

– PS+N is absolutely continuous with respect to PN .

– One has thatdPS+N

dPN(f) = Λ M (f) ,

where, for f ∈ L2 [0, T ] , M is the process

M (f, t) =

∞∑

k=1

1

λk

〈UI[0,t], ek〉L2[0,T ]〈f, ek〉L2[0,T ].

– With respect to PY , and for f ∈ D [0, T ] , Λ has the representation

ln[Λ (f)

]=

∫ T

0s (f, x) evPY (f, dx)

− α

2

∫ T

0s2 (f, x)β (dx)

−√

1 − α

∫ T

0s (f, x) BY

2 (f, dx) ,

with BY2 , a Poisson martingale, independent of BY

1 , and s, the predictableprocess resulting from the RKHS condition of assumption 3.

– With respect to PB , Λ can be approximated by the sequence ICnΦn, whereCn = f ∈ D [0, T ] : Tn (f) = T , Tn is the stopping time of Proposition6, and Φn is given by the following expression, which must be interpretedas that of 3.:

ln [Φn (f)] =

∫ T

0sn (f, x) evPB (f, dx)

− α

2

∫ T

0s2n (f, x)β (dx)

−√

1 − α

∫ T

0sn (f, x) BY,B

2 .


– If one can assume that

PN

(f ∈ D [0, T ] :

∫ T

0s2 (M (f, ·) , x)β (dx) <∞

)= 1,

PS+N and PN are then mutually absolutely continuous, and, mutatis mu-

tandis, the likelihood formula of 3. obtains with respect to PB = PNM−1.

A sufficient condition for the latter, in terms of S, is

E

[exp

1

2||S (·, ·)||2H(N)

]<∞.

Proof. Assumption 3, in conjunction with Baker et al [1, Theorem 3, Step3, p. 170] means that, for some appropriate s,

P

(ω ∈ Ω :

∫ T

0s2 (ω, x)β (dx) <∞

)= 1.

The Corollary to Proposition 7 yields then that PY is absolutely continuouswith respect to PB , and then, from Proposition 10, one has that Y has astochastic integral representation. The specific form of the likelihood followsthen from Proposition 7.

Now, as, mutatis mutandis (Baker et al [1, Theorem 1, p. 163]),

N = Φ B, and S +N = Φ Y ,

for any Borel set A of L2 [0, T ] , it follows, using the Corollary to Proposition11, that

PS+N (A) =

∫

A

dPY

dPB(M (f))PN (df) .

The other statements repeat earlier relevant results.

References

[1] C.R. Baker and A.F. Gualtierotti, Discrimination with respect to a Gaus-sian process, Probability, Theory and Related Fields, 71 (1986), 159-182.

[2] C.R. Baker and A.F. Gualtierotti, Likelihood ratios and signal detection fornon-Gaussian processes, In: Stochastic Processes in Underwater Acoustics,Lecture Notes in Control and Information Science, 85, Springer, Berlin(1986), 154-180.


[3] C.R. Baker and A.F. Gualtierotti, Absolute continuity and mutual in-formation for Gaussian mixtures, Stochastics and Stochastics Reports, 39

(1992), 139-157.

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[8] H. Cramer, On some classes of non-stationary stochastic processes, In:Proceedings Fourth Berkeley Symposium on Mathematical Statistics andProbability, 2 (Ed-s: L.M. Le Cam, J. Neyman and E.L. Scott), Universityof California Press, Berkeley and Los Angeles (1961), 57-78.

[9] C. Dellacherie and P.-A. Meyer, Probabilites et Potentiel, Chapitres I a IV,Hermann, Paris (1975).

[10] A. Ephremides, A property of random processes with unit multiplicity,Journal of Multivariate Analysis, 7 (1977), 525-534.

[11] U. Grenander, Abstract Inference, Wiley, New York (1981).

[12] C.W. Helstrom, Statistical Theory of Signal Detection, 2nd ed., Pergamon,Oxford (1968).

[13] T. Hida, Canonical representations of Gaussian processes and their appli-cations, Memoirs of the College of Science, University of Kyoto, Series A,XXXIII, Mathematics, No.1 (1960), 109-155.

[14] C.W. Horton, Sr., Signal Processing of Underwater Acoustic Waves, U.S.Government Printing Office, Washington, D.C. (1969).

[15] K. Ito, The topological support of Gaussian measure on Hilbert space,Nagoya Mathematical Journal, 38 (1970), 181-183.


[16] T. Kailath and V.H. Poor, Detection of stochastic processes, IEEE-IT, 44

(1998), 2230-2259.

[17] O. Kallenberg, Foundations of Modern Probability, Springer, New York(1997).

[18] J. Memin, Sur Quelques Problemes Fondamentaux de la Theorie du Fil-trage, These, Universite de Rennes (1974).

[19] D. Middleton, Non-Gaussian noise models in signal processing for telecom-munications, IEEE-IT, 45 (1999), 1129-1149.

[20] D. Slepian, Some comments on the detection of Gaussian signals in Gaus-sian noise, IEEE-IT, 4 (1958), 65-69.

[21] K. Sobczyk, Stochastic Wave Propagation, Elsevier, Amsterdam (1985).

[22] P. Todorovic, An Introduction to Stochastic Processes and Their Applica-tions, Springer, New York (1992).

[23] H. Von Weizsacker and G. Winkler, Stochastic Integrals, Vieweg, Wies-baden (1990).

Documents

International Journal of Pure and Applied Mathematics · 156 A. Climescu-Haulica, A.F. Gualtierotti Λ that is deﬁned, and thus computable without knowing which of the PN or PS+N