A BAYES FORMULA FOR NON-LINEAR FILTERING WITH GAUSSIAN AND COX NOISE

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1 A BAYES FOMULA FO NON-LINEA FILTEING WITH GAUSSIAN AND COX NOISE V. MANDEKA, T. MEYE-BANDIS, AND F. POSKE Abstract. A Bayes type formula is derived for the non-linear 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 know 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 t T 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 Ft Y generated by Y s, s t. Or stated in an equivalent way, the objective is to compute the optimal filter as the conditional expectation for a rich enough class of functions f. E P f(x t F Y t In the classical non-linear filter setting, the dynamics of the observation process Y t is supposed to follow the following Itô process dy t = h(t, X t dt + dw t, 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 dy t = b(t, X t dt + σ(t, X t db t, 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. Date: April 28, 29. Key words and phrases. Non-linear Filtering, Gaussian noise, Cox noise, Bayes formula, Kallianpur- Striebel formula, Zakai euqation.

2 2 MANDEKA, MEYE-BANDIS, AND POSKE 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 t (. Y t = β(t, X + G t + ς N λ (dt, dς, where G t is a general Gaussian process with zero mean and continuous covariance function (s, t, s, t T, that is independent of the signal process X. Let Ft Y (respectively Ft X denote the σ-algebra generated by {Y s, s t} (respectively {X s, s t} augmented by the null-sets. Define the filtration (F t t T through F t := FT X F t Y. Then we assume that the process t L t := ς N λ (dt, dς 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 µ(dt, dς, ω = λ(t, X, ςdtν(dς for a Lévy measure ν and a functional λ(t, X(ω, ς. In particular, G t and L t are independent. The function β :, T,T is such that β(t, is Ft X -measurable and β(, X(ω is in H( for almost all ω, where H( denotes the Hilbert space generated by (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 (. 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 L t f(x := b(t, x x f(x + 2 σ2 (t, x xx f(x B t f(x := {f(x + γ(t, xς f(x x f(xγ(t, xς} υ(dς, with the coefficients b(t, x, σ(t, x, andγ(t, x and f(x being in C 2 ( for every t. Here, C 2 ( 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

3 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 (G 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 eproducing 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 (KHS, 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 A. ( 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,..., t n in T and a,..., a n n K(t i, t j a i a j. i,j= The converse of (3, stated in Theorem 2. below, is fundamental towards understanding the KHS representation of Gaussian processes. A proof of the theorem can be found in A. Theorem 2. (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., there is a KHS, 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. 35 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 K T on T T. Then H(K T consists of all f in H(K T restricted to T. Further, for such a restriction f H(K T the norm f H(KT is the minimum of f H(KT for all f H(K T whose restriction to T is f. If K(s, t is the covariance function for some zero mean process Z t, t T, then, by Theorem 2., there exists a unique KHS, H(K, T, for which K is the reproducing kernel. It is also easy to see (e.g., see Theorem 3D, P that there exists a congruence (linear, 3

4 4 MANDEKA, MEYE-BANDIS, AND POSKE one-to-one, inner product preserving map between H(K and sp L2 {Z t, t T} which takes K(, t to Z t. Let us denote by Z, h sp L2 {Z t, t T}, the image of h H(K, T under the congruence. We conclude the section with an important special case. 2.. A useful example. Suppose the stochastic process Z t is a Gaussian process given by Z t = t F (t, udw u, t T, where t F 2 (t, udu < for all t T and W u is Brownian motion. Then the covariance function (2. K(s, t E(Z s Z t = and the corresponding KHS is given by { (2.2 H(K = g : g(t = t t s F (t, uf (s, udu, } F (t, ug (udu, t T for some (necessarily unique g sp L2 {F (t,,t (, t T }, with the inner product where g (s = s (g, g 2 H(K = T g (ug 2(udu, F (s, ug (udu and g 2 (s = s F (s, ug 2(udu. For t T, by taking K(, t to be F (t,,t (, we see, from (2. and (2.2, that K(, t H(K. To check the reproducing property suppose h(t = t F (t, uh (u du H(K. Then (h, K(, t H(K = T h (uk(, t du = t h (uf (t, u du = h(t. Also, in this case, it is very easy to check (cf. P2, Theorem 4D that the congruence between H(K and sp L2 {Z t, t T} is given by (2.3 Z, g = T g (udw u. 3. The Filter Setting and a Bayes Formula Assume a partially observable process (X, Y = (X t, Y t t T 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: t (3. Y t = β(t, X + G t + ς N λ (dt, dς, where

5 G t is a Gaussian process with zero mean and continuous covariance function (s, t, s, t T, that is independent of the signal process X. The function β :, T,T is such that β(t, is Ft X -measurable and β(, X(ω is in H( for almost all ω, where H( denotes the Hilbert space generated by (s, t (see Section 2. Let Ft Y (respectively Ft X denote the σ-algebra generated by {Y s, s t} (respectively {X s, s t} augmented by the null-sets. Define the filtration (F t t T through F t := FT X F t Y. Then we assume that the process t L t := ς N λ (dt, dς 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 µ(dt, dς, ω = λ(t, X, ςdtν(dς 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 T (3.2 log 2 (λ(s, X, ς µ(ds, dς < a.s. T log 2 (λ(s, X, ς ds ν(dς < a.s. (3.3 and { t ( Λ t := exp log λ(s, X, ς t ( ( + log λ(s, X, ς Ñ λ (ds, dς } λ(s, X, ς + µ(ds, dς is a well defined F t -martingale. Here Ñλ(ds, dς stands for the compensated jump measure Ñ λ (ds, dς := N λ (ds, dς µ(dt, dς. emark 3.. 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.eėpf(lt LsA F X T = E P f(l t L s F X T E P A F X T, for all bounded measurable functions f, A F s, and 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 (3.4 E P f(x t F t Y. 5

6 6 MANDEKA, MEYE-BANDIS, AND POSKE 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 t T with β( = β(, X { Λ t := exp G, β t } 2 β 2 t Then the main tool is the following extension of Theorem 3. 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 Y t = G t + L t, where G s = β(s, X + G s, s t, is a Gaussian process with zero mean and covariance function, L s, s t, is a pure jump Lévy process with Lévy measure ν, and the process X s, s T has the same distribution as under P. Further, the processes G, L and X are independent under Q t. Proof. Fix t T. First note that since β( H( almost surely, we have by Theorem 2.2 that β,t H(; 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 E P Λ t Λ t = E P E P Λ t F X T E P Λ t F X T. Since for f H(; 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 =, and Q t is a probability measure. Now take s,..., s m t, r,..., r p t, t,..., t n T and real numbers λ,..., λ m, γ,..., γ p, α,..., α n and consider the joint characteristic function E Qt e i P n j= α jx tj +i P m i= λ ig e si +i P p k= γ k(l rk L rk = E P e i P n j= α jx tj +i P m i= λ ig e si +i P p k= γ k(l rk L rk Λ t Λ t = E P e i P n j= α jx tj E P e i P m i= λ ig e si Λ t FT X E P e i P p k= γ k(l rk L rk Λ t FT X. 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 =. Now, as in Theorem 3. in MM, we get by the independence of G from X that E P e i P m i= λ i e G si Λ t F X T = e P m i,l= λ iλ l (s i,s l, which is the characteristic function of a Gaussian process with mean zero and covariance function. Further, by the conditional independent increments of L we get like in the proof of Th. 6.6 in JS that e u E r δ(s,x,ς N e λ (ds,dς u P F X T = e r (e δ(s,x,ς δ(s,x,ς µ(dt,dς

7 for r u T. So that for one increment one obtains E P e iγ(lu Lr Λ t FT X { u ( ( = E P exp iγς + log Ñ λ (ds, dς r λ(s, X, ς u ( ( } + iγς + log r λ(s, X, ς λ(s, X, ς + µ(dt, dς FT X { u ( = E P exp (e iγς+log λ(s,x,ς iγς log µ(dt, dς r λ(s, X, ς u ( ( } + iγς + log r λ(s, X, ς λ(s, X, ς + µ(dt, dς FT X { u } = E P exp (e iγς+log λ(s,x,ς λ(t, X, ςdtν(dς FT X r λ(s, X, ς { ( = exp (u r e iγς } 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: { E P e i P p p k= γ k(l rk L rk Λ t FT X ( = exp (r k r k e iγ k ς } ν(dς. All together we end up with k= E Qt e i P n j= α jx tj +i P m i= λ i e G si +i P p k= γ k(l rk L rk = E P e i P n j= α jx tj e P m i,l= λ iλ l (s i,s l ep p k= (r k r k (e iγ k ς ν(dς, which completes the proof. emark 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 adon-nikodym derivative dp dq t = (Λ t (Λ t is Q t -a.s. by condition (3.2 and an argument like in (MM, p. 857 given through { t (Λ t = exp log (λ(s, X, ς Ñ(ds, dς t } + (log (λ(s, X, ς λ(s, X, ς + ds ν(dς { (Λ t = exp G, β t } 2 β 2 t. 7

8 8 MANDEKA, MEYE-BANDIS, AND POSKE Here Ñ(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 FT X -measurable integrable function g(t, X E P g(t, X F Y E Qt g(t, X(Λt (Λ t = t Ft Y E Qt (Λt (Λ t Ft Y. ( From Lemma 3.2 we know that the processes Gs, (L s s t, and (X s s T are s t 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 Ft Y can be computed as E Qt φ(x, G, L Ft Y (ω = φ(x(ˆω, G(ω, L(ω Q t (dˆω Ω = φ(x(ˆω, G(ω, L(ω P(dˆω = φ(x(ˆω, G(ω, L(ω. Ω where (ω, ˆω Ω Ω and the index ˆP denotes integration with respect to ˆω. Consequently, we get the following Bayes formula for the optimal filter Theorem 3.4. Under the above specified conditions, for any FT X -measurable integrable function g(t, X E P g(t, X F Y Ω t = g(t, X(ˆωα t(ω, ˆωα t(ω, ˆω P(dˆω Ω α t(ω, ˆωα t (ω, ˆω P(dˆω = g(t, X(ˆωα t (ω, ˆωα t(ω, ˆω α t (ω, ˆωα t (ω, ˆω, where { t α t (ω, ˆω = exp log (λ(s, X(ˆω, ς Ñ(ω, ds, dς t } + (log (λ(s, X(ˆω, ς λ(s, X(ˆω, ς + ds ν(dς { α t(ω, ˆω = exp G(ω, β(, ˆω t } 2 β(, ˆω 2 t. 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. egarding 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 L t f(x := b(t, x x f(x + 2 σ2 (t, x xx f(x

9 B t f(x := {f(x + γ(t, xς f(x x f(xγ(t, xς} υ(dς, for f C 2(. Here, C2 ( 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 ( for every t and that υ(dς is a Lévy measure with second moment. The signal process X t, t T, is then assumed to be a solution of the martingale problem corresponding to O t, i.e. f(x t t (O u f(x u du is a Ft X -martingale with respect to P for every f C 2(. Further, we restrict the Gaussian process G of the observation process in (3. to belong to the special case presented in Section 2., i.e. G t = t F (t, s dw s, where W t is Brownian motion and F (t, s is a deterministic function such that t F 2 (t, s ds, 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 β(t, X = W t := t t F (t, sh(s, X s ds. h(s, X s ds + W t we get G, β t = t h(s, X s d W s and β 2 t = t h2 (s, X s ds, and α t(ω, ˆω in Theorem 3.4 becomes { t α t(ω, ˆω = exp h(s, X s (ˆω d W s (ω t } h 2 (s, X s (ˆω ds. 2 Note that in this case W s, s t, is a Brownian motion under Q t. For f C 2( we now define the unnormalized filter V t(f = V t (f(ω by V t (f(ω := f(x t (ˆωα t (ω, ˆωα t(ω, ˆω P(dˆω = f(xt (ˆωα t (ω, ˆωα t(ω, ˆω. Ω Then this unnormalized filter obeys the following dynamics Theorem 4.. (Zakai equation I Under the above specified assumptions, the unnormalized filter V t (f satisfies the equation ( ( (4. dv t (f( (ω = V t O t f( (ω dt + V t h(t, f( (ω d W t (ω ( (4.2 + V t (λ(t,, ς f( (ω Ñ(ω, dt, dς. 9

10 MANDEKA, MEYE-BANDIS, AND POSKE Proof. Set g t (ˆω := f(x T (ˆω T t (O s f(x s (ˆω ds. Then, by our assumptions on the coefficients b, σ, γ and on the Lévy measure υ(dς, we have g t < C for some constant C. Since f(x t t O tf(x s ds is a martingale we obtain (4.3 g t F X(ˆω t = f(x t, t T. If we denote Γ t (ω, ˆω := α t (ω, ˆωα t(ω, ˆω, then, because Γ t (ω, ˆω is F X(ˆω t -measurable for each ω, equation (4.3 implies that By definition of g t, V t (f = f(x t (ˆωΓ t (ω, ˆω = = g t (ˆωΓ t (ω, ˆω g t (ˆωΓ t (ω, ˆω F X(ˆω t dg t (ˆω = (O t f(x t (ˆω dt. Also, Γ t = Γ t (ω, ˆω is the Doléans-Dade solution of the following linear SDE dγ t = h(t, X t (ˆωΓ t d W t (ω + (λ(t, X t (ˆω, ς Γ t Ñ(ω, dt, dς. So we get t g t (ˆωΓ t = g (ˆωΓ + (O s f(x s (ˆωΓ s ds t h(s, X s (ˆωg s (ˆωΓ s d W s (ω + + t (λ(s, X s (ˆω, ς g s (ˆωΓ s Ñ(ω, ds, dς. The first term on the right hand side equals f(x, and for the second one we can invoke Fubini s theorem to get t t t (O s f(x s (ˆωΓ s ds = (O s f(x s (ˆωΓ s ds = V s (O s f( (ω ds. For the third term we employ the stochastic Fubini theorem for Brownian motion (see for example 5.4 in LS in order to get t h(s, X s (ˆωg s (ˆωΓ s d W s (ω = = t t h(s, X s (ˆωg s (ˆωΓ s d W s (ω h(s, X s (ˆωΓ s g s (ˆω Fs X(ˆω d W s (ω

11 = = t t h(s, X s (ˆωΓ s f(x s (ˆω d W s (ω V s (h(s, f( (ω d W s (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 t (λ(s, X s (ˆω, ς g s (ˆωΓ s Ñ(ω, ds, dς t ( = (λ(s,, ς f( (ω Ñ(ω, ds, dς, which completes the proof. V s If one further assume that the filter has a so called unormalized conditional density u(t, x then we can derive a stochastic integro-pde determining u(t, x which for the Brownian motion case was first established in Z and usually is referred to as Zakai equation. Definition 4.2. We say that a process u(t, x = u(ω, t, x is the unnormalized conditional density of the filter if (4.4 V t (f(ω = f(xu(ω, t, x dx for all bounded continuous functions f :. From now on we restrict the intergo part B t of the operator O t to be the one of a pure jump Lévy process, i.e. γ =, and we assume the initial value X (ω of the signal process to posses a density denoted by ξ(x. Then the following holds Theorem 4.3. (Zakai equation II Suppose the unnormalized conditional density u(t, x of our filter exists. Then, provided a solution exists, u(t, x solves the following stochastic integro-pde (4.5 du(t, x = Ot u(t, xdt + h(t, xu(t, xd W t (ω + (λ(t, x, ς u(t, xñ(ω, dt, dς u(, x = ξ(x. Here Ot := L t + Bt is the adjoint operator of O t given through L t f(x := x (b(t, xf(x + 2 ( xx σ 2 (t, xf(x Bt f(x := {f(x ς f(x + x f(xς} υ(dς, for f C 2 (. For sufficient conditions on the coefficients under which there exists a classical solution of (4.5 see for example MP; in M the existence of solutions in a generalized sense of stochastic distributions is treated.

12 2 MANDEKA, MEYE-BANDIS, AND POSKE Proof. By (4. and (4.4 we have for all f C ( t f(xu(t, x dx = f(xξ(x dx + u(s, xosf(x dxds + t t Now, using integration by parts, we get (4.6 u(s, xl sf(x dx = + u(s, xh(s, xf(x dxd W s (ω u(s, x (λ(s, x, ς f(x dxñ(ω, ds, dς f(xo su(s, x dx. Further it holds again integration by parts and by substitution that (4.7 u(s, xbsf(x dx = f(xbsu(s, x dx. Fubini together with (4.6 and (4.7 then yields ( t f(xu(t, x dx = f(xξ(x dx + f(x + Since this holds for all f C + ( t f(x Osu(s, x ds dx u(s, xh(s, x d W s (ω dx ( t f(x u(s, x (λ(s, x, ς Ñ(ω, ds, dς dx ( we get (4.5. Acknowledgements: This work was started when V. Mandrekar was at the CMA,University of Oslo. He thanks the CMA for support and Prof. Øksendal for interesting discussions and hospitality. eferences A N. Aronszajn (95: Theory of reproducing kernels, Trans. Amer. Math. Soc., 68, pp G Grigelionis B.: Stochastic non-linear filtering equations and semimartingales. In Lecture Notes Math. 972, eds Mitter, S.K., Moro, A., Springer-Verlag, Berlin (982. G2 Grigelionis B.: Stochastic evolution equations and densities of the conditional distributions,lecture notes in Control and Info. Sci., 983, p. 49. JS Jacod, J., Shiryaev, A.N. (22: Limit Theorems for Stochastic Processes. Springer, 2. ed., Berlin Heidelberg New York. K G. Kallianpur (98: Stochastic Filtering Theory. Springer Verlag, New York, Heidelberg, Berlin, 98. KS G. Kallianpur, C. Striebel (968: Estimations of stochastic processes : Arbitrary system process with additive white noise observation errors. Ann. Math. Statis., 39 (968, pp LS Liptser.S., Shiryaev A.N. (977: Statistics of andom Processes. Vol., Springer-Verlag, NY, 977. MM P. Mandal and V. Mandrekar (2: A Bayes formula for Gaussian noise processes and its applications, SIAM J. of Control and Optimization

13 3 M Meyer-Brandis T. (25: Stochastic Feynman-Kac Equations associated to Lévy-Itô Diffusions, Stochastic Analysis and Applications, to appear. MP Meyer-Brandis, T., Proske, F.: Explicit Solution of a Non-Linear Filtering Problem for Lévy Processes with Application to Finance, Appl. Math. Optim. 5 (24, pp P E. Parzen(96: egression Analysis of Continuous Parameter Time Series, in Proc. 4th Berkeley Sympos. Math. Statis. and Prob., Vol. I, Univ. California Press, Berkeley, Calif., 96, pp P2 E. Parzen(967: Statistical inference on time series by Hilbert space methods, I, in Time Series Analysis Papers, Holden-Day, San Francisco, Cambridge, London, Amsterdam, 967, pp Z Zakai, M. (969: On the optimal filtering of diffusion processes. Z. Wahrsch. verw. Geb., No., , (969. (Vidyadhar Mandrekar, Department of Statistics and Probability, Michigan State University, East Lansing, MI48824, USA (Thilo Meyer-Brandis, Center of Mathematics for Applications, University of Oslo, P.O. Box 53, Blindern, N 36 Oslo, Norway, (Frank Proske, Center of Mathematics for Applications, University of Oslo, P.O. Box 53, Blindern, N 36 Oslo, Norway,