A Bayes Formula for Nonlinear Filtering with Gaussian and Cox Noise

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.


Introduction
The general filtering setting can be described as follows.Assume a partially observable process (X, Y ) = (X t , Y t ) 0≤t≤T ∈ R 2 defined on a probability space (Ω, F, P).The real valued process X t stands for the unobservable component, referred to as the signal process or system process, whereas Y t is the observable part, called observation process.Thus information about X t can only be obtained by extracting the information about X that is contained in the observation Y t in a best possible way.In filter theory this is done by determining the conditional distribution of X t given the information σ-field F Y t generated by Y s , 0 ≤ s ≤ t.Or stated in an equivalent way, the objective is to compute the optimal filter as the conditional expectation t ] for a rich enough class of functions f .
In the classical non-linear filter setting, the dynamics of the observation process Y t is supposed to follow the following Itô process where W t is a Brownian motion independent of X.Under certain conditions on the drift h(t, X t ) (see [KS], [K]), 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 for a second Brownian motion B t , Zakai ( [Z]) showed under certain conditions that the unormalized 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 process X we suppose the observation model is given as where • G t 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.
Then we assume that the process is a pure jump F t -semimartingale determined through the integer valued random measure N λ that has an F t -predictable compensator of the form for a Lévy measure ν and a functional λ(t, X(ω), ς).In particular, G t and L t are independent.
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 as Cox 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 (1.1) the corresponding formula has been developped in ( [MM]).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 process X to be a Markov process with generator O t := L t + B t given as with the coefficients b(t, x), σ(t, x), andγ(t, x) and f (x) being in C 2 0 (R) for every t.Here, C 2 0 (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 integro partial 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 G t in the observation is Brownian motion, the corresponding Zakai equation was also studied in ( [MP]).For further information on Zakai equations in a semimartingale setting we also refer to ( [G1]) and ( [G2]).
The remaining part of the paper is organized as follows.in Section 2 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.

Reproducing Kernel Hilbert Space and Stochastic Processes
A Hilbert space H 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, . (The reproducing property) K is called the reproducing kernel of H.The following basic properties can be found in [A].
(1) If a reproducing kernel exists, then it is unique.
(2) If K is the reproducing kernel of a Hilbert space H, then {K(•, t), t ∈ T} spans H.
(3) If K is the reproducing kernel of a Hilbert space H, then it is nonnegative definite in the sense that for all t 1 , . . ., t n in T and a The converse of (3), stated in Theorem 2.1 below, is fundamental towards understanding the RKHS representation of Gaussian processes.A proof of the theorem can be found in [A].
Theorem 2.1 (E.H. Moore).A symmetric nonnegative definite function K 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 suppose K(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 ( [A]; pp.351) explains the relationship between these two.
Theorem 2.2.Suppose K T , defined on T × T, is the reproducing kernel of the Hilbert space H(K T ) with the norm • .Let T ⊂ T, and K T be the restriction of is the covariance function for some zero mean process Z t , 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 Theorem 3D, [P1]) that there exists a congruence (linear, one-to-one, inner product preserving map) between H(K) and sp L 2 {Z t , t ∈ T} which takes We conclude the section with an important special case.
2.1.A useful example.Suppose the stochastic process Z t is a Gaussian process given by and the corresponding RKHS is given by (2.2) where For 0 ≤ t ≤ T , by taking K(•, t) * to be F (t, •)1 [0,t] (•), we see, from (2.1) and (2.2), that K(•, t) ∈ H(K).To check the reproducing property suppose h(t) Also, in this case, it is very easy to check (cf.[P2], Theorem 4D) that the congruence between H(K) and sp L 2 {Z t , t ∈ T} is given by

The Filter Setting and a Bayes Formula
Assume a partially observable process (X, Y ) = (X t , Y t ) 0≤t≤T ∈ R 2 defined on a probability space (Ω, F, P).The real valued process X t stands for the unobservable component, referred to as the signal process, whereas Y t is the observable part, called observation process.In particular, we assume that the dynamics of the observation process is given as follows: where • G t 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.
for almost all ω, where H(R) denotes the Hilbert space generated by R(s, t) (see Section 2).
• Let F Y t (respectively F X t ) denote the σ-algebra generated by {Y s , 0 ≤ s ≤ t} (respectively {X s , 0 ≤ s ≤ t}) augmented by the null-sets.Define the filtration Then we assume that the process is a pure jump F t -semimartingale determined through the integer valued random measure N λ that has an F t -predictable compensator of the form for a Lévy measure ν and a functional λ(t, X(ω), ς).• The functional λ(t, X, ς) is assumed to be strictly positive and such that and is a well defined F t -martingale.Here N λ (ds, dς) stands for the compensated jump measure N λ (ds, dς) := N λ (ds, dς) − µ(dt, dς).
Remark 3.1.Note that the specific from of the predictable compensator µ(dt, dς, ω) implies that L t is a process with conditionally independent increments with respect to the σ-algebra F X T , i.e.
, for all bounded measurable functions f , A ∈ F s , and 0 ≤ s < t ≤ T (see for example Th. 6.6 in [JS]).Also, it follows that the processes G is independent from the random measure N λ (ds, dς).
Given a Borel measurable function f , our non-linear filtering problem then comes down to determine the least square estimate of f (X t ), given the observations up to time t.In other words, the problem consists in evaluating the optimal filter In this section we want to derive a Bayes formula for the optimal filter (3.4) by an extension of the reference measure method presented in [MM] for the purely Gaussian case.For this purpose, define for each 0 ≤ t ≤ T with β(•) = β(•, X) Then the main tool is the following extension of Theorem 3.1 in [MM] Lemma 3.2.Define dQ := Λ t Λ t dP.
Then Q t is a probability measure, and under Q t we have that where G s = β(s, X) + G s , 0 ≤ s ≤ t, is a Gaussian process with zero mean and covariance function R, L s , 0 ≤ s ≤ t, is a pure jump Lévy process with Lévy measure ν, and the process X s , 0 ≤ s ≤ T has the same distribution as under P. Further, the processes G, L and X are independent under Q t .
Proof.Fix 0 ≤ 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 N λ (ds, dς) it follows that ]. Since for f ∈ H(R; t) the random variable G, f t is Gaussian with zero mean and variance f 2 t , it follows again by the independence of G from X and the martingale property of Λ t that E P [Λ t Λ t ] = 1, and Q t is a probability measure.Now take 0 ≤ s 1 , ..., s m ≤ t, 0 ≤ r 1 , ..., r p ≤ t, 0 ≤ t 1 , ..., t n ≤ T and real numbers λ 1 , ..., λ m , γ 1 , ..., γ p , α 1 , ..., α n and consider the joint characteristic function Here, for computational convenience, the part of the characteristic function that concerns L is formulated in terms of increments of L (where we set r 0 = 0).Now, as in Theorem 3.1 in [MM], we get by the independence of G from X that which is the characteristic function of a Gaussian process with mean zero and covariance function R. Further, by the conditional independent increments of L we get like in the proof of Th. 6.6 in [JS] that dt,dς)  for 0 ≤ r ≤ u ≤ T .So that for one increment one obtains 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): All together we end up with which completes the proof.
Remark 3.3.Note that in case G 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 Λ t Λ t is a martingale and dQ is a probability measure.Now, the inverse Radon-Nikodym derivative Here N (ds, dς) := N λ (ds, dς) − dtν(dς) is now a compesnsated Poisson random measure under Q t .Then we have by the Bayes formula for conditional expectation for any F X T -measurable integrable function g(T, X) From Lemma 3.2 we know that the processes G s 0≤s≤t , (L s ) 0≤s≤t , and (X s ) 0≤s≤T are independent under Q t and that the distribution of X is the same under Q t as under P.
Hence conditional expectations of the form E Qt φ(X, G, L) |F Y t can be computed as where (ω, ω) ∈ Ω×Ω and the index P denotes integration with respect to ω.Consequently, we get the following Bayes formula for the optimal filter Theorem 3.4.Under the above specified conditions, for any F X T -measurable integrable function g(T, X) Ω α t (ω, ω)α t (ω, ω) P(dω) where

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 process X, we assume its dynamics to be Markov.More precisely, we consider the parabolic integro-differential operator O t := L t + B t , where Here, C 2 0 (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 C 2 0 (R) for every t and that υ(dς) is a Lévy measure with second moment.The signal process X t , 0 ≤ t ≤ T , is then assumed to be a solution of the martingale problem corresponding to O t , i.e.
is a F X t -martingale with respect to P for every f ∈ C 2 0 (R).Further, we restrict the Gaussian process G of the observation process in (3.1) to belong to the special case presented in Section 2.1, i.e.
where W t is Brownian motion and F (t, s) is a deterministic function such that t 0 F 2 (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 Further, with s, X s ) ds, and α t (ω, ω) in Theorem 3.4 becomes Note that in this case W s , 0 ≤ s ≤ t, is a Brownian motion under Q t .
For f ∈ C 2 0 (R) we now define the unnormalized filter Then this unnormalized filter obeys the following dynamics Theorem 4.1.(Zakai equation I) Under the above specified assumptions, the unnormalized filter V t (f ) satisfies the equation Then, by our assumptions on the coefficients b, σ, γ and on the Lévy measure υ(dς), we have -measurable for each ω, equation (4.3) implies that By definition of g t , dg t (ω) = (O t f )(X t (ω)) dt.
Also, Γ t = Γ t (ω, ω) is the Doléans-Dade solution of the following linear SDE The first term on the right hand side equals f (X 0 ), and for the second one we can invoke Fubini's theorem to get For the third term we employ the stochastic Fubini theorem for Brownian motion (see for example 5.14 in [LS]) in order to get Further it holds again integration by parts and by substitution that s. by condition (3.2) and an argument like in ([MM], p. 857) given through