nonlinear filtering of stochastic dynamical systems with ... - Project Euclid

Adv. Appl. Prob. 47, 902–918 (2015) Printed in Northern Ireland © Applied Probability Trust 2015

NONLINEAR FILTERING OF STOCHASTIC DYNAMICAL SYSTEMS WITH LÉVY NOISES HUIJIE QIAO,∗ Southeast University JINQIAO DUAN,∗∗ Illinois Institute of Technology Abstract Nonlinear filtering is investigated in a system where both the signal system and the observation system are under non-Gaussian Lévy fluctuations. Firstly, the Zakai equation is derived, and it is further used to derive the Kushner–Stratonovich equation. Secondly, by a filtered martingale problem, uniqueness for strong solutions of the Kushner– Stratonovich equation and the Zakai equation is proved. Thirdly, under some extra regularity conditions, the Zakai equation for the unnormalized density is also derived in the case of α-stable Lévy noise. Keywords: Nonlinear filtering; Lévy process; stochastic partial differential equations with jumps; Zakai equation 2010 Mathematics Subject Classification: Primary 60G52; 60G35; 60H15

1. Introduction Consider a partially observed system (X, Y ) on a complete filtered probability space (, F, {Ft }t∈[0,T ] , P) with T > 0 a fixed time. Here Xt stands for the unobservable component of the process, referred to as the signal process, whereas Yt is the observable part, called the observation process. Given a Borel measurable function F , the nonlinear filtering problem leads to evaluating the ‘filter’E[F (Xt ) | FYt ], where FYt is the σ -algebra generated by {Ys , 0 ≤ s ≤ t} and E|F (Xt )| < ∞ for t ∈ [0, T ]. Filtering problems arise from various contexts in engineering, information science, and finance. Filtering problems for systems with Gaussian noise have been widely studied; see [3], [19], and the references therein. When X and Y are continuous diffusion processes, Rozovskii [19] formulated a nonlinear filtering theory by means of stochastic evolution systems. See [20] for more recent results under more general conditions. Nonlinear filtering problems with jump diffusion signal processes or observation processes are considered by some authors. In [14], Meyer-Brandis and Proske studied a nonlinear filtering problem with continuous diffusion signals and mixed observations, modeled by a Brownian motion and a generalized Cox process, whose jump intensity is given in terms of a Lévy measure (see Section 2.2). Later, Mandrekar et al. [12] added jumps to signal processes and obtained the corresponding Zakai equation. Popa and Sritharan [15] derived Zakai and Kushner–Stratonovich equations for nonlinear filtering problems with Itô–Lévy diffusion signal processes and continuous diffusion observation processes. In [4] and [5], Ceci and Colaneri dealt with a filtering problem of a jump diffusion process X and a correlated jump diffusion Received 28 February 2014; revision received 17 June 2014. ∗ Postal address: Department of Mathematics, Southeast University, Nanjing, Jiangsu, 211189, China. Email address: [email protected] ∗∗ Postal address: Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA. Email address: [email protected]

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process Y in the one-dimensional case. They derived Kushner–Stratonovich equations by the innovation method in [4] and Zakai equations by the change of measure in [5]. These two methods are major approaches to investigate nonlinear filtering problems. In this paper, we combine Itô–Lévy diffusion signal processes in [15] with mixed observations in [14] and study a type of nonlinear filtering problems for the multidimensional case. Compared with the results in [4] and [5], our driven processes are more general and the assumption conditions are weaker. We derive the Zakai and Kushner–Stratonovich equations and then represent the filter by an integral. It is worthwhile to mention that the integral characterization of the unnormalized filter was considered in [7] under the same assumption to Assumption 5.1 (see Section 5). Since signals are continuous diffusion processes in [7], our Lemma 5.1 is a generalization of [7, Theorem 2.2]. Note that Grigelionis and Mikulevicius [8] studied a nonlinear filtering problem with jump diffusion signal processes and observation processes and derived the Zakai and Kushner–Stratonovich equations. However, because their observation processes are very general, the integral representation for the filter was not obtained in [8]. This paper is arranged as follows. In Section 2 we introduce Sobolev spaces and α-stable Lévy processes. Zakai and Kushner–Stratonovich equations are derived in Section 3. In Section 4, by a filtered martingale problem (FMP), uniqueness for strong solutions of the Kushner–Stratonovich equation and the Zakai equation is proved. In Section 5, we prove the existence of the unnormalized and normalized densities under some regularity conditions. The following convention will be used throughout this paper: C with or without indices will denote different positive constants (depending on the indices) whose values may change from one place to another. 2. Preliminary 2.1. Sobolev spaces Let C0 (Rn ) be the space of continuous functions f on Rn satisfying lim|x|→∞ f (x) = 0 with norm f C0 (Rn ) = supx∈Rn |f (x)|. Let C02 (Rn ) be the set of f ∈ C0 (Rn ) such that f is twice differentiable and the partial derivatives of f with order ≤ 2 belong to C0 (Rn ). Let Cck (Rn ) stand for the space of all k-times differentiable functions on Rn with compact support. Let S(Rn ) be the Schwartz space of all rapidly decreasing real-valued C ∞ functions on Rn and S (Rn ) the space of all tempered distributions on Rn . Let fˆ and f˘ be the Fourier transform and Fourier inversion transform of f ∈ S (Rn ), respectively. That is, fˆ(u) =

1 (2π)n/2

Rn

e−i u,x f (x) dx,

f˘(u) =

1 (2π )n/2

for all u ∈ Rn . We introduce the following Sobolev space: Hλ,2 (Rn ) := {f ∈ S (Rn ) : f λ,2 < ∞} for any λ ∈ R, where

f 2λ,2 :=

In particular, H0,2 (Rn ) = L2 (Rn ).

Rn

(1 + |u|2 )λ |fˆ(u)|2 du.

Rn

ei u,x f (x) dx

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2.2. Symmetric α-stable Lévy processes Definition 2.1. A process L = (Lt )t≥0 with L0 = 0 almost surely (a.s.) is a n-dimensional Lévy process if (i) L has independent increments; i.e. Lt −Ls is independent of Lv −Lu if (u, v)∩(s, t) = ∅; (ii) L has stationary increments; i.e. Lt − Ls has the same distribution as Lv − Lu if t − s = v − u > 0; (iii) Lt is stochastically continuous; (iv) Lt is right-continuous with left limit. The characteristic function of Lt is given by E[exp{i z, Lt }] = exp{t(z)},

z ∈ Rn .

The function : Rn → C is called the characteristic exponent of the Lévy process L. By the Lévy–Khintchine formula, there exist a nonnegative-definite n × n matrix Q, a measure ν on Rn satisfying ν({0}) = 0,

Rn \{0}

and γ ∈ Rn such that 1 (z) = i z, γ − z, Qz + 2

(|u|2 ∧ 1)ν(du) < ∞,

Rn \{0}

(ei z,u − 1 − i z, u 1{|u|≤1} )ν(du).

The measure ν is called the Lévy measure. Definition 2.2. For α ∈ (0, 2). An n-dimensional symmetric α-stable process Lα is a Lévy process such that its characteristic exponent is given by (z) = −C1 (n, α)|z|α

for z ∈ Rn

with C1 (n, α) := π −1/2 ((1 + α)/2) (n/2)/ ((n + α)/2). Thus, for an n-dimensional symmetric α-stable process Lα , the diffusion matrix Q = 0, the drift vector γ = 0, and the Lévy measure ν is given by ν(du) = Define (Lα f )(x) :=

C2 (n, α) du |u|n+α

with C2 (n, α) :=

α ((n + α)/2) 21−α π n/2 (1 − α/2)

Rn \{0}

(f (x + u) − f (x) − ∂x f (x), u 1{|u|≤1} )

.

C2 (n, α) du for f ∈ C02 (Rn ). |u|n+α

Then Lα extends uniquely to the infinitesimal generator of Lαt and by [2, Example 3.3.8, p. 166], for every f ∈ Cc∞ (Rn ), (Lα f )(x) = C1 (n, α)[−(−)α/2 f ](x). Moreover, the following result is well known (see [1]). Theorem 2.1. Let Lα be as above for α ∈ (0, 2) and L2 = , as defined on Cc∞ (Rn ) in L2 (Rn ). Then Lα , 0 < α ≤ 2, has a unique closed extension to a self-adjoint negative operator on the domain Hα,2 (Rn ).


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3. Nonlinear filtering In this section we study nonlinear filtering for a non-Gaussian signal-observation system, and derive Zakai and Kushner–Stratonovich equations. Consider the following signal-observation (Xt , Yt ) system on Rn × Rm : f1 (Xt− , u)N˜p (dt, du) dXt = b1 (Xt ) dt + σ1 (Xt ) dBt + U1 + g1 (Xt− , u)Np (dt, du), (3.1a) U\U1 f2 (t, u)N˜ λ (dt, du) dYt = b2 (t, Xt ) dt + σ2 (t) dWt + U2 + g2 (t, u)Nλ (dt, du), 0 ≤ t ≤ T, (3.1b) U\U2

where B, W are d-dimensional and m-dimensional Brownian motion, respectively, and p is a stationary Poisson point process of the class (quasi left-continuous) defined on (, F, {Ft }t∈[0,T ] , P) with values in U and characteristic measure ν1 . Here ν1 is a σ -finite measure defined on a finite-dimensional, measurable normed space (U, U) with the norm · U . Fix U1 ∈ U with ν1 (U \ U1 ) < ∞ and U1 u2U ν1 (du) < ∞. Let Np ((0, t], du) be the counting measure of pt such that ENp ((0, t], A) = tν1 (A) for A ∈ U. Denote N˜ p ((0, t], du) := Np ((0, t], du) − tν1 (du), the compensated measure of pt . Let Nλ be an integer-valued random measure and its predictable compensator is given by λ(t, Xt− , u)tν2 (du), where the function λ(t, x,u) ∈ [ι, 1), 0 < ι < 1, and ν2 is another σ -finite measure defined on U with ν2 (U\U2 ) < ∞ and U2 u2U ν2 (du) < ∞ for U2 ∈ U. That is, N˜ λ ((0, t], du) := Nλ ((0, t], du) − λ(t, Xt− , u)tν2 (du) is its compensated measure. Moreover, Bt , Wt , Np , Nλ are mutually independent. The initial value X0 is assumed to be a random variable independent of Y0 , Bt , Wt , Np , and Nλ . Under P, the initial value X0 is assumed to have a density function ρ with values in L2 (Rn ). The mappings b1 : Rn → Rn , b2 : [0, T ] × Rn → Rm , σ1 : Rn → Rn×d , σ2 : [0, T ] → m×m R , f1 : Rn × U1 → Rn , f2 : [0, T ] × U2 → Rm , g1 : Rn × (U \ U1 ) → Rn , and g2 : [0, T ] × (U \ U2 ) → Rm are all Borel measurable. We make the following assumptions, in order to guarantee existence and uniqueness for the solution of (3.1). Assumption 3.1. (i) (Hb11 ,σ1 ,f1 ). For x1 , x2 ∈ Rn , |b1 (x1 ) − b1 (x2 )| ≤ L1 |x1 − x2 |κ1 (|x1 − x2 |), U1

σ1 (x1 ) − σ1 (x2 )2 ≤ L1 |x1 − x2 |2 κ2 (|x1 − x2 |),

|f1 (x1 , u) − f1 (x2 , u)|p ν1 (du) ≤ L1 |x1 − x2 |p κ3 (|x1 − x2 |),

p

hold for = 2 and 4, where | · | denotes the length of a vector in Rn and · the Hilbert– Schmidt norm from Rd to Rn . Here L1 is a constant and κi is a positive continuous function, bounded on [1, ∞) and it satisfies lim x↓0

κi (x) = δi < ∞, log x −1

where δi ≥ 0, i = 1, 2, 3, are constants.

i = 1, 2, 3,

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H. QIAO AND J. DUAN

(ii) (Hb21 ,σ1 ,f1 ). For x ∈ Rn ,

|b1 (x)|2 + σ1 (x)2 +

U1

|f1 (x, u)|2 ν1 (du) ≤ L1 (1 + |x|)2 .

(iii) (Hb12 ,σ2 ,f2 ). It holds that σ2 (t) is invertible for t ∈ [0, T ], b2 , σ2 , σ2−1 are bounded by a positive constant L2 , and

T

U2

0

|f2 (s, u)|2 ν2 (du) ds < ∞.

By [17, Theorem 1.2], (3.1) has a pathwise unique strong solution denoted by (Xt , Yt ). Set t 1 t −1 −1 i i −1 : = exp − (σ (s)b (s, X )) dW − |σ (s)b2 (s, Xs )|2 ds 2 s t s 2 2 0 2 0 t t − log λ(s, Xs− , u)Nλ (ds, du) − (1 − λ(s, Xs , u))ν2 (du) ds . U2

0

0

U2

Throughout, we use the convention that repeated indices imply summation. Assumption 3.2. Let E exp 0

T

U2

(1 − λ(s, Xs , u))2 ν2 (du) ds λ(s, Xs , u)

< ∞.

Set Mt := −

t

0

(σ2−1 (s)b2 (s, Xs ))i

dWsi

+

t 0

U2

1 − λ(s, Xs− , u) ˜ Nλ (ds, du), λ(s, Xs− , u)

and then under Assumption 3.2, M is a locally square integrable martingale. Moreover, Mt − Mt− > −1 a.s. and

E exp 21 < M c , M c >T + < M d , M d >T

T T 1 − λ(s, Xs , u) 2 1 −1 2 |σ2 (s)b2 (s, Xs )| ds + = E exp 2 0 λ(s, Xs , u) U2 0 × λ(s, Xs , u)ν2 (du) ds < ∞, where M c and M d are continuous and purely discontinuous martingale parts of M, respectively. Thus, from [16, Theorem 6], it follows that −1 t , the Doléans–Dade exponential of M, is a martingale. Define a measure P˜ via dP˜ 1 . = dP T


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By the Girsanov theorem for Brownian motions and random measures, it follows that under the measure P˜ (3.1) is transformed as dXt = b1 (Xt ) dt + σ1 (Xt ) dBt + f1 (Xt− , u)N˜ p (dt, du) U1 + g1 (Xt− , u)Np (dt, du), U\U1 ˜ ˜ f2 (t, u)N (dt, du) + g2 (t, u)Nλ (dt, du), dYt = σ2 (t) dWt + U2

U\U2

where W˜ t := Wt +

t 0

N˜ (dt, du) := Nλ (dt, du) − dtν2 (du).

σ2−1 (s)b2 (s, Xs ) ds,

We collect some properties of t , W˜ t , and N˜ in the next lemma. Lemma 3.1. (i) t satisfies the following equation: t t t = 1 + s (σ2−1 (s)b2 (s, Xs ))i dW˜ si + 0

0

U2

s− (λ(s, Xs− , u) − 1)N˜ (ds, du).

˜ W˜ t is a Brownian motion and N˜ is a Poisson martingale measure. (ii) Under the measure P, Proof. For (i), by the Itô formula, we obtain t 1 t −1 i i s (σ2 (s)b2 (s, Xs )) dWs + s |σ2−1 (s)b2 (s, Xs )|2 ds t = 1 + 2 0 0 t + s− (λ(s, Xs− , u) − 1)Nλ (ds, du) 0

+

t

U2

0

=1+ 0

U2

t

s− (1 − λ(s, Xs− , u))ν2 (du) ds +

s (σ2−1 (s)b2 (s, Xs ))i dW˜ si +

t U2

0

1 2

t

0

s |σ2−1 (s)b2 (s, Xs )|2 ds

s− (λ(s, Xs− , u) − 1)N˜ (ds, du).

For (ii), we use [10, Theorem 3.17]. Set

˜ (Xt )t | FYt ], P˜ t (F ) := E[F

˜ denotes expectation under the measure P. ˜ The equation satisfied by P˜ t (F ) is called where E the Zakai equation. In order to derive the Zakai equation, we need the following lemma. ˜ N˜ ˜ ˜ Lemma 3.2. Let FW t , Ft be the σ -algebras generated by {Ws , 0 ≤ s ≤ t}, {N ((0, s], A), 0 ≤ s ≤ t, A ∈ U}, respectively. Then ˜

˜

N Y FYt = FW t ∨ Ft ∨ F0 .

Since its proof is similar to that in [19, Lemma 4, p. 228], we omit it.

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As in [18], introduce the infinitesimal generator L of Xt , (Lϕ)(x) =

1 ∂ 2 ϕ(x) ik ∂ϕ(x) i kj b1 (x) + σ (x)σ1 (x) ∂xi 2 ∂xi ∂xj 1 + [ϕ(x + g1 (x, u)) − ϕ(x)]ν1 (du) U\U1 ∂ϕ(x) i ϕ(x + f1 (x, u)) − ϕ(x) − f1 (x, u) ν1 (du) + ∂xi U1

for ϕ ∈ Cc∞ (Rn ),

and denote its domain by D(L). Now, we are ready to obtain the Zakai equation for P˜ t (F ). Theorem 3.1. (Zakai equation.) For F ∈ D(L), the Zakai equation of (3.1) is given by t t P˜ s (LF ) ds + P˜ s (F (σ2−1 (s)b2 (s, ·))i ) dW˜ si P˜ t (F ) = P˜ 0 (F ) + 0 0 t + (3.2) P˜ s− (F (λ(s, ·, u) − 1))N˜ (ds, du). U2

0

Proof. Applying the Itô formula to Xt , we have t t ∂F (Xs ) i ∂F (Xs ) ik b1 (Xs ) ds + σ1 (Xs ) dBsk F (Xt ) = F (X0 ) + ∂x ∂xi i 0 0 t + [F (Xs− + f1 (Xs− , u)) − F (Xs− )]N˜ p (ds, du) + +

0

U1

0

U\U1

t

t

F (Xs− + f1 (Xs− , u)) − F (Xs− ) −

0

+

[F (Xs− + g1 (Xs− , u)) − F (Xs− )]Np (ds, du)

1 2

0

U1 t ∂ 2 F (X

s)

∂xi ∂xj

∂F (Xs ) i f1 (Xs− , u) ν1 (du) ds ∂xi

kj

σ1ik (Xs )σ1 (Xs ) ds.

By the mutual independence of Bt , N˜ p (dt, du), Wt , and N˜ (dt, du), it is clear that t t F (Xs− ) ds + s− dF (Xs ). F (Xt )t = F (X0 ) + 0

0

Taking the conditional expectation on both sides of this equation, we obtain t ˜ (X0 ) | FYt ] + E ˜ ˜ (Xt )t | FYt ] = E[F F (Xs− ) ds FYt E[F 0 t ˜ +E s− dF (Xs ) FYt 0

=: I1 + I2 + I3 .

(3.3)

We now evaluate I1 , I2 , and I3 one by one. First, by the independence of X0 and FYt , we have ˜ (X0 )] = E[F ˜ (X0 ) | FY ] = P˜ 0 (F ). I1 = E[F 0

(3.4)


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By Lemma 3.1, it holds that t ˜ s F (Xs )(σ2−1 (s)b2 (s, Xs ))i dW˜ si FYt I2 = E 0 t ˜ +E s− F (Xs− )(λ(s, Xs− , u) − 1)N˜ (ds, du) FYt U2

0

=: I21 + I22 . For I21 , by Lemma 3.2 and [19, Theorem 1.4.7], we obtain t ˜ s F (Xs )(σ −1 (s)b2 (s, Xs ))i | FYs ] dW˜ si . E[ I21 = 2 0

For I22 , using the same method to that in the proof of [19, Theorem 1.4.7], we can prove by the definition of stochastic integrations with respect to random measures (see [9]) and Lemma 3.2 that t ˜ s− F (Xs− )(λ(s, Xs− , u) − 1) | FYs ]N˜ (ds, du). I22 = E[ 0

U2

˜ s− F (Xs− )(λ(s, Xs− , u) − 1) | FYs ] is not predictable. Using the method of [13, Note that E[ Theorem 2.3], we show that it admits a predictable modification. Thus, t t I2 = P˜ s (F (σ2−1 (s)b2 (s, ·))i ) dW˜ si + P˜ s− (F (λ(s, ·, u) − 1))N˜ (ds, du). (3.5) 0

0

U2

For I3 , from the independence of B, N˜ p , and FY , it follows that t t Y ∂F (Xs ) i ∂F (Xs ) ik k Y ˜ ˜ I3 = E s b1 (Xs ) ds Ft + E s σ1 (Xs ) dBs Ft ∂x ∂x i i 0 0 t Y ˜ ˜ +E s [F (Xs− + f1 (Xs− , u)) − F (Xs− )]Np (ds, du) | Ft 0

t ˜ +E 0

U1

U\U1

s [F (Xs− + g1 (Xs− , u)) − F (Xs− )]Np (ds, du) | FYt

t 1 ∂ 2 F (Xs ) ik kj ˜ +E s σ1 (Xs )σ1 (Xs ) ds FYt 2 0 ∂xi ∂xj t ∂F (Xs ) i ˜ +E s F (Xs− + f1 (Xs− , u)) − F (Xs− ) − f1 (Xs− , u) (3.6) ∂xi 0 U1 × ν1 (du) ds FYt t t 2 ˜ s ∂F (Xs ) bi (Xs ) FYs ds + 1 ˜ s ∂ F (Xs ) σ ik (Xs )σ kj (Xs ) FYs ds = E E 1 1 1 ∂xi 2 0 ∂xi ∂xj 0 t ˜ +E s [F (Xs− + g1 (Xs− , u)) − F (Xs− )]ν1 (du) ds | FYt 0 U\U1 t ∂F (Xs ) i ˜ f1 (Xs− , u) FYs + E s F (Xs− + f1 (Xs− , u)) − F (Xs− ) − ∂x i 0 U1 × ν1 (du) ds

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H. QIAO AND J. DUAN

t

= 0

=

t

˜ s LF (Xs ) | FYs ] ds E[ P˜ s (LF ) ds,

(3.7)

0

where we have used the property of random measures (see [9]). Combining (3.3) with (3.4), (3.5), and (3.7), we obtain the Zakai equation (3.2). Thus, by Theorem 3.1, P˜ t (1) satisfies the following equation: t t P˜ s ((σ2−1 (s)b2 (s, ·))i ) dW˜ si + P˜ s− (λ(s, ·, u) − 1)N˜ (ds, du) P˜ t (1) = 1 +

0

0

U2

t

=1+ P˜ s (1)Ps ((σ2−1 (s)b2 (s, ·))i ) dW˜ si 0 t + P˜ s− (1)Ps− (λ(s, ·, u) − 1)N˜ (ds, du).

(3.8)

U2

0

Set M˜ t :=

t 0

Ps ((σ2−1 (s)b2 (s, ·))i ) dW˜ si +

t 0

U2

Ps− (λ(s, ·, u) − 1)N˜ (ds, du).

Since ι < λ(t, x, u) < 1, by (Hb12 ,σ2 ,f2 ), the Jensen inequality, and Assumption 3.2, we obtain ˜ t , the Doléans–Dade exponential of M. ˜ Thus, P˜ t (1) > 0. P˜ t (1) = E (M) Besides, set Pt (F ) := E[F (Xt ) | FYt ], and then, by the Kallianpur–Striebel formula, the following holds: Pt (F ) = E[F (Xt ) | FYt ] =

˜ (Xt )t | FYt ] P˜ t (F ) E[F = . Y ˜ t | Ft ] E[ P˜ t (1)

(3.9)

By using (3.8), the Zakai equation (3.2), and the Itô formula, together with a similar argument as in the proof of Theorem 3.1, we obtain the Kushner–Stratonovich equation satisfied by Pt (F ). Theorem 3.2. (Kushner–Stratonovich equation.) For F ∈ D(L), Pt (F ) solves the following equation: t t Ps (LF ) ds + (Ps (F (σ2−1 (s)b2 (s, ·))i ) Pt (F ) = P0 (F ) + 0

0 −1 i − Ps (F )Ps ((σ2 (s)b2 (s, ·)) )) dW¯ si t Ps− (F λ(s, ·, u)) − Ps− (F )Ps− (λ(s, ·, u))

+

0

where W¯ t := W˜ t −

t 0

U2

Ps− (λ(s, ·, u))

N¯ (ds, du),

Ps (σ2−1 (s)b2 (s, ·)) ds is the innovation process and

N¯ (dt, du) = Nλ (dt, du) − Pt− (λ(t, ·, u))ν2 (du) dt. The Kushner–Stratonovich equation (3.10) corresponds to [8, Equation (3.2)].

(3.10)


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4. Uniqueness for the Kushner–Stratonovich equation and the Zakai equation In this section we first show uniqueness for strong solutions to the Kushner–Stratonovich equation by means of an FMP (see [11]), and then apply the relation between the Kushner– Stratonovich equation and the Zakai equation to prove uniqueness for the Zakai equation. In this section we assume that g1 (x, u) = 0, g2 (t, u) = 0, and λ(t, x, u) = λ(t, u). And then (3.1) is written as

σ (X )0 Bt b1 (Xt ) Xt dt + 1 t d = d Yt Wt b2 (t, Xt ) 0σ2 (t)

f1 (Xt− , u) ˜ 0 + Np (dt, du) + (4.1) N˜ λ (dt, du). 0 U1 U2 f2 (t, u) Lemma 4.1. Suppose that (Hb11 ,σ1 ,f1 ), (Hb21 ,σ1 ,f1 ), and (Hb12 ,σ2 ,f2 ) are satisfied. Then the infinitesimal generator of (4.1) is given by LX,Y H (x, y) =

∂H (x, y) i 1 ∂ 2 H (x, y) ik kj b1 (x) + σ (x)σ1 (x) ∂xi 2 ∂xi ∂xj 1 ∂H (x, y) i + H (x + f1 (x, u), y) − H (x, y) − f1 (x, u) ν1 (du) ∂xi U1 ∂H (x, y) l 1 ∂ 2 H (x, y) lk kq b2 (t, x) + σ (t)σ2 (t) ∂yl 2 ∂yl ∂yq 2 ∂H (x, y) l + H (x, y + f2 (t, u)) − H (x, y) − f2 (t, u) λ(t, u)ν2 (du) ∂yl U2 +

for H ∈ D(LX,Y ). By the Itô formula and the definition of the infinitesimal generator, it is easy to prove the above lemma. Therefore, we omit its proof. Before introducing an FMP, we define several notations. Let P (Rn ) denote the set of the probability measures on Rn and M + (Rn ) denote the set of positive bounded Borel measures on Rn . For a process π -valued in P (Rn ) or M + (Rn ), πt (F ) ≡ Rn F (x)πt (·, dx), F ∈ C02 (Rn ). Definition 4.1. A process (π, U ) defined on a probability space (, F, {Ft }t∈[0,T ] , P), with càdlàg trajectories and values in P (Rn ) × Rm , is a solution of the FMP (LX,Y , X0 , Y0 ) if π is X,Y ), FU t -adapted and, for all H ∈ D(L t πs (LX,Y H (·, Us )) ds πt (H (·, Ut )) − 0

is a

(P, FU t )-martingale

and E[π0 (H (·, U0 ))] = E[H (X0 , Y0 )].

Definition 4.2. Uniqueness for the FMP (LX,Y , X0 , Y0 ) means that if (π, U ) is a solution of the FMP (LX,Y , X0 , Y0 ), for each t ∈ [0, T ] there exists a Borel measurable P (Rn )-valued function H˜ t satisfying πt = H˜ t (U ),

Pt = H˜ t (Y ),

and (π, U ) has the same distribution as (P, Y ).

P-a.s.,

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H. QIAO AND J. DUAN

Definition 4.3. A strong solution for (3.10) is a FYt -adapted, càdlàg, P (Rn )-valued process {πt }t∈[0,T ] such that {πt }t∈[0,T ] solves (3.10), i.e. for F ∈ D(L), t t πs (LF ) ds + (πs (F (σ2−1 (s)b2 (s, ·))i ) πt (F ) = P0 (F ) + 0

where Wˆ t := W˜ t −

t 0

0

− πs (F )πs ((σ2−1 (s)b2 (s, ·))i )) dWˆ si , πs (σ2−1 (s)b2 (s, ·)) ds.

Definition 4.4. A strong solution for (3.2) is a FYt -adapted, càdlàg, M + (Rn )-valued process {μt }t∈[0,T ] such that {μt }t∈[0,T ] solves (3.2), i.e. for F ∈ D(L), t t μs (LF ) ds + μs (F (σ2−1 (s)b2 (s, ·))i ) dW˜ si μt (F ) = P˜ 0 (F ) + 0 0 t + μs− (F )(λ(s, u) − 1)N˜ (ds, du). 0

U2

Now we prove uniqueness for solutions of the Kushner–Stratonovich equation. Theorem 4.1. Suppose that uniqueness holds for the FMP (LX,Y , X0 , Y0 ). Let {πt }t∈[0,T ] be a strong solution of (3.10). Then πt = Pt , P-a.s. for all t ∈ [0, T ]. Proof. For G ∈ Cc∞ (Rm ), applying the Itô formula to (3.1), we obtain t ∂G(Ys ) 1 t ∂ 2 G(Ys ) ik kj G(Yt ) = G(Y0 ) + σ2 (s)σ2 (s) ds b2 (s, Xs ) ds + ∂y 2 0 ∂yi ∂yj 0 t ∂G(Ys− ) j + G(Ys− + f2 (s, u)) − G(Ys− ) − f2 (s, u) λ(s, u)ν2 (du) ds ∂yj 0 U2 t t ∂G(Ys ) + σ2 (s) dWs + [G(Ys− + f2 (s, u)) − G(Ys− )]N˜ λ (ds, du). ∂y 0 0 U2 Thus, from the above formula, Definition 4.3, and the Itô formula, it follows that, for F ∈ D(L), t t ∂G(Ys ) X,Y πs [L (F (·)G(Ys ))] ds + πs (F ) σ2 (s) dWˆ s πt (F )G(Yt ) = P0 (F )G(Y0 ) + ∂y 0 0 t + G(Ys )[πs (F (σ −1 (s)b2 (s, ·))i ) − πs (F )πs ((σ −1 (s)b2 (s, ·))i )] dWˆ si 2

0

+

t 0

U2

2

πs (F )[G(Ys− + f2 (s, u)) − G(Ys− )]N˜ λ (ds, du).

Note that W¯ t is a FtY -Brownian motion (see [11]) and t ¯ ˆ (πs (σ2−1 (s)b2 (s, ·)) − Ps (σ2−1 (s)b2 (s, ·))) ds. Ws = Wt − 0

Set h(s) := πs (σ2−1 (s)b2 (s, ·)) − Ps (σ2−1 (s)b2 (s, ·)), t 2 τN := T ∧ inf t > 0 : |h(s)| ds > N , 0

(4.2)


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and then τN is a FtY -stopping time and τN → T as N → ∞ by (Hb12 ,σ2 ,f2 ). Define the probability measure τN 1 τN dPN 2 ¯ h(s) dWs − |h(s)| ds . = exp dP 2 0 0 Thus, from the Girsanov theorem, we have that Wˆ t is a FtY -Brownian motion under PN . Observe (4.2). Under PN , τN ∧t πs LX,Y (F (·)G(Ys )) ds πτN ∧t (F )G(YτN ∧t ) − 0

is a FtY -martingale. Moreover, for H ∈ D(LX,Y ), τN ∧t πs (LX,Y H (·, Ys )) ds πτN ∧t (H (·, YτN ∧t )) − 0

is a (PN , FYt )-martingale. By uniqueness for the FMP (LX,Y , X0 , Y0 ) and [11, Corollary 3.4], we know that there exists a P (Rn )-valued function H˜ t such that πt 1{t