# A Bayes Formula for Nonlinear Filtering with Gaussian and Cox Noise

Authors: Vidyadhar Mandrekar; Thilo Meyer-Brandis; Frank Proske
Journal: Journal of Probability and Statistics
 (2011)

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## Abstract

A Bayes-type formula is derived for the nonlinear filter where the observation contains both general Gaussian noise as well as Cox noise whose jump intensity depends on the signal. This formula extends the well-known Kallianpur-Striebel formula in the classical non-linear filter setting. We also discuss Zakai-type equations for both the unnormalized conditional distribution as well as unnormalized conditional density in case the signal is a Markovian jump diffusion.


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## Body

## 1. Introduction

The general filtering setting can be described as follows. Assume a partially observable process(X,Y)=(Xt,Yt)0≤t≤T∈ℝ2 defined on a probability space (Ω,ℱ,ℙ). The real valued process Xt stands for the unobservable component, referred to as the signal process or system process, whereas Yt is the observable part, called observation process. Thus information about Xt can only be obtained by extracting the information about X that is contained in the observation Yt in a best possible way. In filter theory this is done by determining the conditional distribution of Xt given the information σ-field ℱtY generated by Ys,0≤s≤t. Or stated in an equivalent way, the objective is to compute the optimal filter as the conditional expectation(1.1)EP[f(Xt)∣FtY]for a rich enough class of functionsf.In the classical nonlinear filter setting, the dynamics of the observation processYt is supposed to follow the following Itô process (1.2)dYt=h(t,Xt)dt+dWt,whereWt is a Brownian motion independent of X. Under certain conditions on the drift h(t,Xt) (see [1, 2]), Kallianpur and Striebel derived a Bayes-type formula for the conditional distribution expressed in terms of the so-called unnormalized conditional distribution. In the special case when the dynamics of the signal follows an Itô diffusion (1.3)dXt=b(t,Xt)dt+σ(t,Xt)dBt,for a second Brownian motionBt, Zakai [3] showed under certain conditions that the unnormalized conditional density is the solution of an associated stochastic partial differential equation, the so called Zakai equation.In this paper we extend the classical filter model to the following more general setting. For a general signal processX we suppose that the observation model is given as(1.4)Yt=β(t,X)+Gt+∫0t∫R0ςNλ(dt,dς),
where  (i)
Gt is a general Gaussian process with zero mean and continuous covariance function R(s,t),0≤s,t≤T, that is, independent of the signal process X;(ii)
LetℱtY (resp. ℱtX) denote the σ-algebra generated by {Ys,0≤s≤t} (resp. {Xs,0≤s≤t}) augmented by the null sets. Define the filtration (ℱt)0≤t≤T through ℱt:=ℱTX∨ℱtY. Then we assume that the process(1.5)Lt:=∫0t∫R0ςNλ(dt,dς)is a pure jumpℱt-semimartingale determined through the integer-valued random measure Nλ that has an ℱt-predictable compensator of the form(1.6)μ(dt,dς,ω)=λ(t,X,ς)dtν(dς)for a Lévy measureν and a functional λ(t,X(ω),ς). In particular, Gt and Lt are independent;(iii)
the functionβ:[0,T]×ℝ[0,T]→ℝ is such that β(t,·) is ℱtX-measurable and β(·,X(ω)) is in H(R) for almost all ω, where H(R) denotes the Hilbert space generated by R(s,t) (see Section 2).The observation dynamics consists thus of an information drift of the signal disturbed by some Gaussian noise plus a pure jump part whose jump intensity depends on the signal. Note that a jump process of the form given above is also referred to asCox process.The objective of the paper is in a first step to extend the Kallianpur-Striebel Bayes type formula to the generalized filter setting from above. When there are no jumps present in the observation dynamics (1.4), the corresponding formula has been developed in [4]. We will extend their way of reasoning to the situation including Cox noise.In a second step we then derive a Zakai-type measure valued stochastic differential equations for the unnormalized conditional distribution of the filter. For this purpose we assume the signal processX to be a Markov process with generator 𝒪t:=ℒt+ℬt given as (1.7)Ltf(x)∶=b(t,x)∂xf(x)+12σ2(t,x)∂xxf(x),Btf(x)∶=∫R0{f(x+γ(t,x)ς)-f(x)-∂xf(x)γ(t,x)ς}υ(dς),with the coefficientsb(t,x),σ(t,x),andγ(t,x) and f(x) being in C02(ℝ) for every t. Here, C02(ℝ) is the space of continuous functions with compact support and bounded derivatives up to order 2. Further, we develop a Zakai-type stochastic parabolic integropartial differential equation for the unnormalized conditional density, given it exists. In the case the dynamics of X does not contain any jumps and the Gaussian noise Gt in the observation is Brownian motion, the corresponding Zakai equation was also studied in [5]. We further refer to [6] where nonlinear filtering for jump diffusions is considered. For further information on Zakai equations in a semimartingale setting we refer to [7, 8].The remaining part of the paper is organized as follows. in Section2 we briefly recall some theory of reproducing kernel Hilbert spaces. In Section 3 we obtain the Kallianpur-Striebel formula, before we discuss the Zakai-type equations in Section 4.


## 2. Reproducing Kernel Hilbert Space and Stochastic Processes

A Hilbert spaceH consisting of real valued functions on some set T is said to be a reproducing kernel Hilbert space (RKHS), if there exists a function K on T×T with the following two properties: for every t in T and g in H,  (i)
K(·,t)∈H,(ii)
(g(·),K(·,t))=g(t). (The reproducing property)K is called the reproducing kernel of H. The following basic properties can be found in [9].  (1)
If a reproducing kernel exists, then it is unique.(2)
IfK is the reproducing kernel of a Hilbert space H, then {K(·,t),t∈T} spans H.(3)
IfK is the reproducing kernel of a Hilbert space H, then it is nonnegative definite in the sense that for all t1,…,tn in T and a1,…,an∈ℝ(2.1)∑i,j=1nK(ti,tj)aiaj≥0.The converse of (3), stated in Theorem2.1 below, is fundamental towards understanding the RKHS representation of Gaussian processes. A proof of the theorem can be found in [9].Theorem 2.1 (E. H. Moore).
A symmetric nonnegative definite functionK on T×T generates a unique Hilbert space, which we denote by H(K) or sometimes by H(K,T), of which K is the reproducing kernel.Now supposeK(s,t),s,t∈T, is a nonnegative definite function. Then, by Theorem 2.1, there is a RKHS, H(K,T), with K as its reproducing kernel. If we restrict K to T′×T′ where T′⊂T, then K is still a nonnegative definite function. Hence K restricted to T′×T′ will also correspond to a reproducing kernel Hilbert space H(K,T′) of functions defined on T′. The following result from ([9, pp. 351]) explains the relationship between these two.Theorem 2.2.
SupposeKT, defined on T×T, is the reproducing kernel of the Hilbert space H(KT) with the norm ∥·∥. Let T′⊂T and KT′ be the restriction of KT on T′×T′. Then H(KT′) consists of all f in H(KT) restricted to T′. Further, for such a restriction f′∈H(KT′) the norm ∥f′∥H(KT′) is the minimum of ∥f∥H(KT) for all f∈H(KT) whose restriction to T′ is f′.IfK(s,t) is the covariance function for some zero mean process Zt,t∈T, then, by Theorem 2.1, there exists a unique RKHS, H(K,T), for which K is the reproducing kernel. It is also easy to see (e.g., see [10, Theorem 3D]) that there exists a congruence (linear, one-to-one, inner product preserving map) between H(K) and sp¯L2{Zt,t∈T} which takes K(·,t) to Zt. Let us denote by 〈Z,h〉∈sp¯L2{Zt,t∈T}, the image of h∈H(K,T) under the congruence.We conclude the section with an important special case.


### 2.1. A Useful Example

Suppose the stochastic processZt is a Gaussian process given by (2.2)Zt=∫0tF(t,u)dWu,0≤t≤T,where∫0tF2(t,u)du<∞ for all 0≤t≤T and Wu is Brownian motion. Then the covariance function  (2.3)K(s,t)≡E(ZsZt)=∫0t∧sF(t,u)F(s,u)du,
and the corresponding RKHS is given by  (2.4)H(K)={g:g(t)=∫0tF(t,u)g*(u)du,0≤t≤T}
for some (necessarily unique) g*∈sp¯L2{F(t,·)1[0,t](·),0≤t≤T}, with the inner product(2.5)(g1,g2)H(K)=∫0Tg1*(u)g2*(u)du,where(2.6)g1(s)=∫0sF(s,u)g1*(u)du,g2(s)=∫0sF(s,u)g2*(u)du.For0≤t≤T, by taking K(·,t)* to be F(t,·)1[0,t](·), we see, from (2.3) and (2.4), that K(·,t)∈H(K). To check the reproducing property suppose h(t)=∫0tF(t,u)h*(u)du∈H(K). Then(2.7)(h,K(⋅,t))H(K)=∫0Th*(u)K(⋅,t)*du=∫0th*(u)F(t,u)du=h(t).Also, in this case, it is very easy to check (cf. [11], Theorem 4D) that the congruence between H(K) and sp¯L2{Zt,t∈T} is given by  (2.8)〈Z,g〉=∫0Tg*(u)dWu.


## 2.1. A Useful Example

Suppose the stochastic processZt is a Gaussian process given by (2.2)Zt=∫0tF(t,u)dWu,0≤t≤T,where∫0tF2(t,u)du<∞ for all 0≤t≤T and Wu is Brownian motion. Then the covariance function  (2.3)K(s,t)≡E(ZsZt)=∫0t∧sF(t,u)F(s,u)du,
and the corresponding RKHS is given by  (2.4)H(K)={g:g(t)=∫0tF(t,u)g*(u)du,0≤t≤T}
for some (necessarily unique) g*∈sp¯L2{F(t,·)1[0,t](·),0≤t≤T}, with the inner product(2.5)(g1,g2)H(K)=∫0Tg1*(u)g2*(u)du,where(2.6)g1(s)=∫0sF(s,u)g1*(u)du,g2(s)=∫0sF(s,u)g2*(u)du.For0≤t≤T, by taking K(·,t)* to be F(t,·)1[0,t](·), we see, from (2.3) and (2.4), that K(·,t)∈H(K). To check the reproducing property suppose h(t)=∫0tF(t,u)h*(u)du∈H(K). Then(2.7)(h,K(⋅,t))H(K)=∫0Th*(u)K(⋅,t)*du=∫0th*(u)F(t,u)du=h(t).Also, in this case, it is very easy to check (cf. [11], Theorem 4D) that the congruence between H(K) and sp¯L2{Zt,t∈T} is given by  (2.8)〈Z,g〉=∫0Tg*(u)dWu.


## 3. The Filter Setting and a Bayes Formula

Assume a partially observable process(X,Y)=(Xt,Yt)0≤t≤T∈ℝ2 defined on a probability space (Ω,ℱ,ℙ). The real valued process Xt stands for the unobservable component, referred to as the signal process, whereas Yt is the observable part, called observation process. In particular, we assume that the dynamics of the observation process is given as follows:(3.1)Yt=β(t,X)+Gt+∫0t∫R0ςNλ(dt,dς),
where  (i)
Gt is a Gaussian process with zero mean and continuous covariance function R(s,t),0≤s,t≤T, that is, independent of the signal process X;(ii)
the functionβ:[0,T]×ℝ[0,T]→ℝ is such that β(t,·) is ℱtX-measurable and β(·,X(ω)) is in H(R) for almost all ω, where H(R) denotes the Hilbert space generated by R(s,t) (see Section 2);(iii)
letℱtY (resp., ℱtX) denote the σ-algebra generated by {Ys,0≤s≤t} (resp., {Xs,0≤s≤t}) augmented by the null sets. Define the filtration (ℱt)0≤t≤T through ℱt:=ℱTX∨ℱtY. Then we assume that the process(3.2)Lt∶=∫0t∫R0ςNλ(dt,dς)is a pure jumpℱt-semimartingale determined through the integer-valued random measure Nλ that has an ℱt-predictable compensator of the form(3.3)μ(dt,dς,ω)=λ(t,X,ς)dtν(dς)for a Lévy measureν and a functional λ(t,X(ω),ς);(iv)
the functionalλ(t,X,ς) is assumed to be strictly positive and such that(3.4)∫0T∫R0log⁡2(λ(s,X,ς))μ(ds,dς)<∞a.s.,∫0T∫R0log⁡2(λ(s,X,ς))dsν(dς)<∞a.s.,(3.5)Λt∶=exp⁡{∫0t∫R0log⁡(1λ(s,X,ς))Ñλ(ds,dς)+∫0t∫R0(log⁡(1λ(s,X,ς))-1λ(s,X,ς)+1)μ(ds,dς)}is a well-definedℱt-martingale. Here Ñλ(ds,dς) stands for the compensated jump measure(3.6)Ñλ(ds,dς)∶=Nλ(ds,dς)-μ(dt,dς).Remark 3.1.
Note that the specific from of the predictable compensatorμ(dt,dς,ω)  implies that  Lt  is a process with conditionally independent increments with respect to the  σ-algebra  ℱTX, that is,
(3.7)EP[f(Lt-Ls)1A∣FTX]=EP[f(Lt-Ls)∣FTX]EP[1A∣FTX],
for all bounded measurable functions f, A∈ℱs, and 0≤s<t≤T (see, e.g., in [12, Theorem 6.6]). Also, it follows that the processes G is independent of the random measure Nλ(ds,dς).Given a Borel measurable functionf, our nonlinear filtering problem then comes down to determine the least square estimate of f(Xt), given the observations up to time t. In other words, the problem consists in evaluating the optimal filter(3.8)EP[f(Xt)∣FtY].
In this section we want to derive a Bayes formula for the optimal filter (3.8) by an extension of the reference measure method presented in [4] for the purely Gaussian case. For this purpose, define for each 0≤t≤T with β(·)=β(·,X)(3.9)Λ′t∶=exp⁡{-〈G,β〉t-12‖β‖t2}.Then the main tool is the following extension of Theorem 3.1 in [4].Lemma 3.2.
Define(3.10)dQ∶=ΛtΛ′tdP.
Then ℚt is a probability measure, and under ℚt we have that
(3.11)Yt=G̃t+Lt,
where G̃s=β(s,X)+Gs, 0≤s≤t, is a Gaussian process with zero mean and covariance function R, Ls, 0≤s≤t, is a pure jump Lévy process with Lévy measure ν, and the process Xs, 0≤s≤T has the same distribution as under ℙ. Further, the processes G̃, L, and X are independent under ℚt.Proof.
Fix0≤t≤T. First note that since β(·)∈H(R) almost surely, we have by Theorem 2.2 that β|[0,t]∈H(R;t) almost surely. Further, by the independence of the Gaussian process G from X and from the random measure Nλ(ds,dς) it follows that
(3.12)EP[ΛtΛt′]=EP[EP[Λt∣FTX]EP[Λ′t∣FTX]].
Since for f∈H(R;t) the random variable 〈G,f〉t is Gaussian with zero mean and variance ∥f∥t2, it follows again by the independence of G from X and the martingale property of Λt that Eℙ[ΛtΛ′t]=1, and ℚt is a probability measure.
Now take0≤s1,…,sm≤t,0≤r1,…,rp≤t,0≤t1,…,tn≤T and real numbers λ1,…,λm, γ1,…,γp, α1,…,αn and consider the joint characteristic function
(3.13)EQt[ei∑j=1nαjXtj+i∑i=1mλiG̃si+i∑k=1pγk(Lrk-Lrk-1)]=EP[ei∑j=1nαjXtj+i∑i=1mλiG̃si+i∑k=1pγk(Lrk-Lrk-1)ΛtΛt′]=EP[ei∑j=1nαjXtjEP[ei∑i=1mλiG̃siΛ′t∣FTX]EP[ei∑k=1pγk(Lrk-Lrk-1)Λt∣FTX]].
Here, for computational convenience, the part of the characteristic function that concerns L is formulated in terms of increments of L (where we set r0=0). Now, as in in [4, Theorem 3.1], we get by the independence of G from X that
(3.14)EP[ei∑i=1mλiG̃siΛ′t∣FTX]=e-∑i,l=1mλiλlR(si,sl),
which is the characteristic function of a Gaussian process with mean zero and covariance function R.
Further, by the conditional independent increments ofL we get like in the proof of in [12, Theorem 6.6] that
(3.15)EP[e∫ru∫R0δ(s,X,ς)Ñλ(ds,dς)∣FTX]=e∫ru∫R0(eδ(s,X,ς)-1-δ(s,X,ς))μ(dt,dς)
for 0≤r≤u≤T. So that for one increment one obtains
(3.16)EP[eiγ(Lu-Lr)Λt∣FTX]=EP[exp⁡{∫ru∫R0(iγς+log⁡(1λ(s,X,ς)))Ñλ(ds,dς)+∫ru∫R0(iγς+log⁡(1λ(s,X,ς))-1λ(s,X,ς)+1)μ(dt,dς)}∣FTX]=EP[exp⁡{∫ru∫R0(eiγς+log⁡(1/λ(s,X,ς))-1-iγς-log⁡(1λ(s,X,ς)))μ(dt,dς)+∫ru∫R0(iγς+log⁡(1λ(s,X,ς))-1λ(s,X,ς)+1)μ(dt,dς)}∣FTX]=EP[exp⁡{∫ru∫R0(eiγς+log⁡(1/λ(s,X,ς))-1λ(s,X,ς))λ(t,X,ς)dtν(dς)}∣FTX]=exp⁡{(u-r)∫R0(eiγς-1)ν(dς)}.
The generalization to the sum of increments is straightforward, and one obtains the characteristic function of the finite dimensional distribution of a Lévy process (of finite variation):
(3.17)EP[ei∑k=1pγk(Lrk-Lrk-1)Λt∣FTX]=exp⁡{∑k=1p(rk-rk-1)∫R0(eiγkς-1)ν(dς)}.
All together we end up with
(3.18)EQt[ei∑j=1nαjXtj+i∑i=1mλiG̃si+i∑k=1pγk(Lrk-Lrk-1)]=EP[ei∑j=1nαjXtj]⋅e-∑i,l=1mλiλlR(si,sl)⋅e∑k=1p(rk-rk-1)∫R0(eiγkς-1)ν(dς),
which completes the proof.Remark 3.3.
Note that in caseG  is Brownian motion Lemma 3.2 is just the usual Girsanov theorem for Brownian motion and random measures. In this case, it follows from Cameron-Martin's result and the fact that  X  is independent of G that  ΛtΛt′  is a martingale and  dℚ  is a probability measure.Now, the inverse Radon-Nikodym derivative(3.19)dPdQt=(Λt)-1(Λ′t)-1isℚt-a.s. by condition (3.4) and an argument like in [4, p. 857] given through (3.20)(Λt)-1=exp⁡{∫0t∫R0log⁡(λ(s,X,ς))Ñ(ds,dς)+∫0t∫R0(log⁡(λ(s,X,ς))-λ(s,X,ς)+1)dsν(dς)},(Λt′)-1=exp⁡{〈G̃,β〉t-12‖β‖t2}.Here(3.21)Ñ(ds,dς)∶=Nλ(ds,dς)-dtν(dς)is now a compensated Poisson random measure underℚt. Then we have by the Bayes formula for conditional expectation for any ℱTX-measurable integrable function g(T,X)(3.22)EP[g(T,X)∣FtY]=EQt[g(T,X)(Λt)-1(Λt′)-1∣FtY]EQt[(Λt)-1(Λt′)-1∣FtY].From Lemma3.2 we know that the processes (G̃s)0≤s≤t, (Ls)0≤s≤t, and (Xs)0≤s≤T are independent under ℚt and that the distribution of X is the same under ℚt as under ℙ. Hence conditional expectations of the form Eℚt[ϕ(X,G̃,L)∣ℱtY] can be computed as (3.23)EQt[ϕ(X,G̃,L)∣FtY](ω)=∫Ωϕ(X(ω̂),G̃(ω),L(ω))Qt(dω̂)=∫Ωϕ(X(ω̂),G̃(ω),L(ω))P(dω̂)=EP̂[ϕ(X(ω̂),G̃(ω),L(ω))],where(ω,ω̂)∈Ω×Ω and the index ℙ̂ denotes integration with respect to ω̂. Consequently, we get the following Bayes formula for the optimal filter.Theorem 3.4.
Under the above specified conditions, for anyℱTX-measurable integrable function g(T,X)(3.24)EP[g(T,X)∣FtY]=∫Ωg(T,X(ω̂))αt(ω,ω̂)αt′(ω,ω̂)P(dω̂)∫Ωαt(ω,ω̂)αt′(ω,ω̂)P(dω̂)=EP̂[g(T,X(ω̂))αt(ω,ω̂)αt′(ω,ω̂)]EP̂[αt(ω,ω̂)αt′(ω,ω̂)],
where
(3.25)αt(ω,ω̂)=exp⁡{∫0t∫R0log⁡(λ(s,X(ω̂),ς))Ñ(ω,ds,dς)+∫0t∫R0(log⁡(λ(s,X(ω̂),ς))-λ(s,X(ω̂),ς)+1)dsν(dς)},αt′(ω,ω̂)=exp⁡{〈G̃(ω),β(⋅,ω̂)〉t-12‖β(⋅,ω̂)‖t2}.


## 4. Zakai-Type Equations

Using the Bayes formula from above we now want to proceed further in deriving a Zakai-type equations for the unnormalized filter. This equation is basic in order to obtain the filter recursively. To this end we have to impose certain restrictions on both the signal process and the Gaussian part of the observation process.Regarding the signal processX, we assume its dynamics to be Markov. More precisely, we consider the parabolic integrodifferential operator 𝒪t:=ℒt+ℬt, where (4.1)Ltf(x)∶=b(t,x)∂xf(x)+12σ2(t,x)∂xxf(x),Btf(x)∶=∫R0{f(x+γ(t,x)ς)-f(x)-∂xf(x)γ(t,x)ς}υ(dς),forf∈C02(ℝ). Here, C02(ℝ) is the space of continuous functions with compact support and bounded derivatives up to order 2. Further, we suppose that b(t,·), σ(t,·), and γ(t,·) are in C02(ℝ) for every t and that υ(dς) is a Lévy measure with second moment. The signal process Xt,0≤t≤T, is then assumed to be a solution of the martingale problem corresponding to 𝒪t, that is,(4.2)f(Xt)-∫0t(Ouf)(Xu)duis aℱtX-martingale with respect to ℙ for every f∈C02(ℝ).Further, we restrict the Gaussian processG of the observation process in (3.1) to belong to the special case presented in Section 2.1, that is,(4.3)Gt=∫0tF(t,s)dWs,whereWt is Brownian motion and F(t,s) is a deterministic function such that ∫0tF2(t,s)ds, 0≤t≤T. Note that this type of processes both includes Ornstein-Uhlenbeck processes as well as fractional Brownian motion. Then β(t,X) will be of the form(4.4)β(t,X)=∫0tF(t,s)h(s,Xs)ds.Further, with(4.5)W̃t∶=∫0th(s,Xs)ds+Wtwe get〈G̃,β〉t=∫0th(s,Xs)dW̃s and ∥β∥t2=∫0th2(s,Xs)ds, and αt′(ω,ω̂) in Theorem 3.4 becomes(4.6)αt′(ω,ω̂)=exp⁡{∫0th(s,Xs(ω̂))dW̃s(ω)-12∫0th2(s,Xs(ω̂))ds}.Note that in this caseW̃s,0≤s≤t, is a Brownian motion under ℚt.Forf∈C02(ℝ) we now define the unnormalized filter Vt(f)=Vt(f)(ω) by(4.7)Vt(f)(ω)∶=∫Ωf(Xt(ω̂))αt(ω,ω̂)αt′(ω,ω̂)P(dω̂)=EP̂[f(Xt(ω̂))αt(ω,ω̂)αt′(ω,ω̂)].Then this unnormalized filter obeys the following dynamics.Theorem 4.1 (Zakai equation I).
Under the above specified assumptions, the unnormalized filterVt(f) satisfies the equation
(4.8)dVt(f(⋅))(ω)=Vt(Otf(⋅))(ω)dt+Vt(h(t,⋅)f(⋅))(ω)dW̃t(ω)+∫R0Vt((λ(t,⋅,ς)-1)f(⋅))(ω)Ñ(ω,dt,dς).Proof.
Set(4.9)gt(ω̂)∶=f(XT(ω̂))-∫tT(Osf)(Xs(ω̂))ds.
Then, by our assumptions on the coefficients b, σ, γ and on the Lévy measure υ(dς), we have |gt|<C for some constant C. Since f(Xt)-∫0t𝒪tf(Xs)ds is a martingale we obtain
(4.10)EP̂[gt∣FtX(ω̂)]=f(Xt),0≤t≤T.
If we denote
(4.11)Γt(ω,ω̂)∶=αt(ω,ω̂)αt′(ω,ω̂),
then, because Γt(ω,ω̂) is ℱtX(ω̂)-measurable for each ω, (4.10) implies that
(4.12)Vt(f)=EP̂[f(Xt(ω̂))Γt(ω,ω̂)]=EP̂[EP̂[gt(ω̂)Γt(ω,ω̂)∣FtX(ω̂)]]=EP̂[gt(ω̂)Γt(ω,ω̂)].
By definition of gt,
(4.13)dgt(ω̂)=(Otf)(Xt(ω̂))dt.
Also, Γt=Γt(ω,ω̂) is the Doléans-Dade solution of the following linear SDE:
(4.14)dΓt=h(t,Xt(ω̂))ΓtdW̃t(ω)+∫R0(λ(t,Xt(ω̂),ς)-1)ΓtÑ(ω,dt,dς).
So we get
(4.15)EP̂[gt(ω̂)Γt]=EP̂[g0(ω̂)Γ0]+EP̂[∫0t(Osf)(Xs(ω̂))Γsds]+EP̂[∫0th(s,Xs(ω̂))gs(ω̂)ΓsdW̃s(ω)]+EP̂[∫0t∫R0(λ(s,Xs(ω̂),ς)-1)gs(ω̂)ΓsÑ(ω,ds,dς)].
The first term on the right hand side equals f(X0), and for the second one we can invoke Fubini's theorem to get
(4.16)EP̂[∫0t(Osf)(Xs(ω̂))Γsds]=∫0tEP̂[(Osf)(Xs(ω̂))Γs]ds=∫0tVs(Osf(⋅))(ω)ds.
For the third term we employ the stochastic Fubini theorem for Brownian motion (see, for Example 5.14 in [13]) in order to get
(4.17)EP̂[∫0th(s,Xs(ω̂))gs(ω̂)ΓsdW̃s(ω)]=∫0tEP̂[h(s,Xs(ω̂))gs(ω̂)Γs]dW̃s(ω)=∫0tEP̂[h(s,Xs(ω̂))ΓsEP̂[gs(ω̂)∣FsX(ω̂)]]dW̃s(ω)=∫0tEP̂[h(s,Xs(ω̂))Γsf(Xs(ω̂))]dW̃s(ω)=∫0tVs(h(s,⋅)f(⋅))(ω)dW̃s(ω).
Further, one easily sees that the analogue stochastic Fubini theorem for compensated Poisson random measures holds, and we get analogously for the last term
(4.18)EP̂[∫0t∫R0(λ(s,Xs(ω̂),ς)-1)gs(ω̂)ΓsÑ(ω,ds,dς)]=∫0t∫R0Vs((λ(s,⋅,ς)-1)f(⋅))(ω)Ñ(ω,ds,dς),
which completes the proof.If one further assumes that the filter has a so-called unnormalized conditional densityu(t,x) then we can derive a stochastic integro-PDE determining u(t,x) which for the Brownian motion case was first established in [3] and usually is referred to as Zakai equation.Definition 4.2.
We say that a processu(t,x)=u(ω,t,x) is the unnormalized conditional density of the filter if
(4.19)Vt(f)(ω)=∫Rf(x)u(ω,t,x)dx
for all bounded continuous functions f:ℝ→ℝ.From now on we restrict the intergo partℬt of the operator 𝒪t to be the one of a pure jump Lévy process, that is, γ=1, and we assume the initial value X0(ω) of the signal process to possess a density denoted by ξ(x). Then the following holds.Theorem 4.3 (Zakai equation II).
Suppose the unnormalized conditional densityu(t,x) of our filter exists. Then, provided that a solution exists, u(t,x) solves the following stochastic integro-PDE:
(4.20)du(t,x)=Ot*u(t,x)dt+h(t,x)u(t,x)dW̃t(ω)+∫R0(λ(t,x,ς)-1)u(t,x)Ñ(ω,dt,dς),u(0,x)=ξ(x).
Here 𝒪t*:=ℒt*+ℬt* is the adjoint operator of 𝒪t given through
(4.21)Lt*f(x)∶=-∂x(b(t,x)f(x))+12∂xx(σ2(t,x)f(x)),Bt*f(x)∶=∫R0{f(x-ς)-f(x)+∂xf(x)ς}υ(dς),
for f∈C02(ℝ).For sufficient conditions on the coefficients under which there exists a classical solution of (4.20) see, for example, [5]; in [14] the existence of solutions in a generalized sense of stochastic distributions is treated.Proof.
By (4.8) and (4.19) we have for all f∈C0∞(ℝ)(4.22)∫Rf(x)u(t,x)dx=∫Rf(x)ξ(x)dx+∫0t∫Ru(s,x)Osf(x)dxds+∫0t∫Ru(s,x)h(s,x)f(x)dxdW̃s(ω)+∫0t∫R0∫Ru(s,x)(λ(s,x,ς)-1)f(x)dxÑ(ω,ds,dς).
Now, using integration by parts, we get
(4.23)∫Ru(s,x)Lsf(x)dx=∫Rf(x)Ls*u(s,x)dx.
Further it holds again integration by parts and by substitution that
(4.24)∫Ru(s,x)Bsf(x)dx=∫Rf(x)Bs*u(s,x)dx.
Fubini together with (4.23) and (4.24) then yields
(4.25)∫Rf(x)u(t,x)dx=∫Rf(x)ξ(x)dx+∫Rf(x)(∫0tOs*u(s,x)ds)dx+∫Rf(x)(∫0tu(s,x)h(s,x)dW̃s(ω))dx+∫Rf(x)(∫0t∫R0u(s,x)(λ(s,x,ς)-1)Ñ(ω,ds,dς))dx.
Since this holds for all f∈C0∞(ℝ), we get (4.20).



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*Source: 259091-2011-11-03.xml*