Numerical analysis of a time discretized method for nonlinear filtering problem with L\'evy process observations

In this paper, we consider a nonlinear filtering model with observations driven by correlated Wiener processes and point processes. We first derive a Zakai equation whose solution is a unnormalized probability density function of the filter solution. Then we apply a splitting-up technique to decompose the Zakai equation into three stochastic differential equations, based on which we construct a splitting-up approximate solution and prove its half-order convergence. Furthermore, we apply a finite difference method to construct a time semi-discrete approximate solution to the splitting-up system and prove its half-order convergence to the exact solution of the Zakai equation. Finally, we present some numerical experiments to demonstrate the theoretical analysis.


Introduction
The aim of a nonlinear filtering problem is to seek the conditional expectation, which is the best estimate of the unobserved state of a stochastic dynamical system given its partial observation.The observation is usually described as a nonlinear stochastic differential equation driven by a noise process.In many applications, such as biology [1], physics [2], target tracking [3] and weather forecast [4], the noise can be characterized by a standard Wiener process.However, in some applications, such as the number of customers arriving at a supermarket [5] and the number of births in a given period of time [6], the noise is governed by a point process.In some other applications such as the credit risk models [7,8], mathematical finance [9,10] and insurance [11,12], the noise can be described by a mixture of a Wiener process and a point process, which is usually called Lévy process.
There have been a few theoretical and numerical studies on nonlinear filtering problems driven by Lévy processes.Qiao and Duan [13] studied a nonlinear filtering model where both the state and observation involve point processes.They simultaneously derived the Zakai equations and Kushner-Stratonovich equations and proved their well-posedness.Fernando and Hausenblas [14] investigated a nonlinear filter model with correlated point processes for the state and observation.They provided sufficient conditions for the wellposedness of the corresponding Zakai equation.Frey etc. [8] used the PDF filter method to approximate a nonlinear filter model driven by point processes and independent Wiener processes.The PDF filter method is designed to directly approach the conditional density function, which satisfies a stochastic partial differential equation, namely Zakai equation [15].In [8] the authors applied a spectral Galerkin method to set up a spatial semi-discrete equation and proved that its solution converges to the exact solution of the Zakai equation.However, they did not provide the convergence order.They also used the Euler-Maruyama scheme and a splitting-up method, to discretize temporal variables.
In this paper, we use the splitting-up method to investigate the numerical approximation of a nonlinear filtering model where the observation is driven by the mixture of point processes and correlated Wiener processes.The splittingup method [16][17][18] is a well-known strategy for solving Zakai equations.It decomposes the Zakai equation into a system consisting of deterministic PDEs and stochastic differential equations (SDEs) [4,16,17,19,20].Our contribution in this paper is twofold.First, we decompose the Zakai equation into three equations: an SDE driven by the Wiener process, a second order parabolic equation satisfying the uniform elliptic condition, and an SDE driven by a point process.Through the solution operators of the three equations and their a priori estimates, we construct a splitting-up approximation and prove that it converges to the Zakai solution with first order accuracy.We note that in some references [18,21,22] concerning the nonlinear filtering models with correlated noises, the decomposed second-order parabolic equation is possibly degenerate, which may cause difficulty for numerical implementations.Our second contribution is the derivation of the half-order convergence of the time semidiscrete approximation.To the best of our knowledge, this is the first time a convergence order of a numerical method for nonlinear filtering problems with jump processes has been provided.This paper is organized as follows.In section 2, we introduce a nonlinear filtering model with the mixed noise of point process and correlated Wiener process and then derive the corresponding Zakai equation.In section 3, we apply a splitting-up method to construct a splitting-up approximate solution to the Zakai equation and establish a priori estimates for the splitting-up solution and show that the convergence is of half order.In section 4, we use finite difference methods to construct a time semi-discrete approximation and prove that the semi-discrete solution converges to the exact solution with half order.Finally in section 5, we present some numerical experiments to illustrate our theoretical analysis.

A nonlinear filtering model with jump observations and its Zakai equation
In this section, we first introduce a nonlinear filtering model whose observations are driven by Lévy processes.Then we derive the corresponding Zakai equation which characterizes the development of the density function of the filtering solution process.Finally, we investigate the regularity of the solution of the Zakai equation.

A nonlinear filtering model
In this subsection, we introduce a nonlinear filtering model with noises simultaneously driven by a point and correlated Wiener processes and then discuss some basic assumptions.Let (Ω, F , P ) be a given probability space.Consider a nonlinear filtering model whose state (or signal) process X t and two observation processes Y t and Z t are given by Z t is a doubly stochastic Poisson process with density function λ(X t ), (3) where w t ∈ R m1 and v t ∈ R m2 are two standard independent Winner processes, Z t is a doubly stochastic Poisson process with a continuous density function ds is a martingale.The corresponding jump times for Z t are random variables denoted by τ 1 < τ 2 < • • • < τ n0 , where n 0 is an integer-valued random variable.
The objective of the nonlinear filtering problem is to seek an optimal estimation of X t based on observations Y t and Z t , which is characterized by the conditional expectation E[X t |Y t , Z t ].Now, we describe in detail the assumptions used in this work.
H2 g : R d × R q → R d and h : R d → R q are bounded, continuous and square integrable, σ : R d → R d×m1 is in C 2 with bounded first and second order derivatives, and b, b are in C 1 with bounded first order derivatives.
Define two families of symmetric non-negative matrix H3 There exist two constants 0 < α 1 < α 2 such that for any x ∈ R d , y ∈ R q and u ∈ R q , there hold Define a filtration associated with the observations by which is right continuous and complete.By L 2 (0, T ; V ) we denote a Hilbert space consisting of F t progressively measurable V -valued stochastic process

Zakai equation and its regularity
The main task of this subsection is to derive the Zakai equation of the nonlinear filtering model ( 1)-( 3) and study the regularity of its solution.
Assume that X t is the solution process of (1).For any φ ∈ C ∞ 0 (R d ), define π t (φ) as the conditional expectation of φ(X t ) given F t , i.e., For any t ∈ [0, T ], define According to Novikov Criterion [23, Theorem 41], η t is a nonnegative martingale if H1-H3 hold.Define a new probability measure P by virtue of the Radon-Nikodym derivative d P dP = η −1 t .The Girsanov theorem [24] implies that Ȳt = t 0 D(Y s ) −1/2 dY s is a standard Wiener process and Z t is a Poisson process with intensity equal to 1 under the probability measure P .Furthermore, in the probability space (Ω, F , P ) the three stochastic processes X t , Y t and Z t are independent of each other, and the compensated Poisson process N t := Z t − t is a martingale.
Define a stochastic process where wt = )ds is a standard Wiener process under P .From [25], we have Lemma 1 Assume H1-H3.Then Ỹt is a standard Wiener process independent of Y t under P and equation ( 1) is equivalent to Denote by Ẽ the expectation under the probability measure P .The next proposition plays an important role in the forthcoming analysis.
Proposition 2 [24,Proposition 3.15] Assume that H1-H3 hold and let U be an F t -measurable and integrable random variable.Then we have By the Kallianpur-Striebel formula [24, Proposition 3.16], we have where p(t) is the unnormalized conditional density function of Ẽ(X t |F t ).
and b 1,i denotes the i-th row of matrix B 1 (x, y).
Proof We approximate η t with η ε t = ηt 1+εηt .By Itô formula Using Itô's formula for the jump process, we have (11) Let X t satisfy (1) and for any φ Applying the product rule for semi-martingales to (10) and (12), we obtain dZs.
(13) According to Proposition 2, we only need to compute the conditional expectation based on the filtration F T .Take conditional expectation about η ε t φ(X t ) based on the observation F T , then we have Now, we show that as ε → 0, the following limits hold in the sense P -a.s., From the pointwise convergence of η ε t to η t as ε → 0, it follows that lim ε→0 Similarly, there holds Now, we consider item E 2 .Notice that for any ε > 0, there holds By Fubini's theorem, we exchange the integral order in E 2 to obtain By the dominated convergence theorem, we get Next, we study item E 3 .Notice that This estimate, together with the the pointwise convergence lim In a similar way, we obtain lim ε→0 E i = 0, P − a.s.for i = 6, 7.
By isometry formula, we have that for any ε > 0 According to [24, Lemma 3.21], we can change the order between conditional expectation and stochastic integral to obtain Using Jensen's inequality and Fubini's Theorem, for any ε > 0, we have This implies that E 4 is a martingale, c.f. [27, Theorem 4.3.1]and then the process t 0 ρs(∇φ T B 1 )dYs is a local martingale.Thus, the following difference is a local martingale 1+εηs ∇φ T B 1 , then lim ε→0 ξ ε s = 0, P − a.s.Obviously, for any φ ∈ V we have ξ ε s ≤ ∇φ B 1 ηs.Due to the dominated convergence theorem, we obtain Applying the stochastic dominated convergence theorem [23,Theorem 32], we have Hence, we obtain Finally, we investigate the term E 9 .Let Then lim This estimate, together with the stochastic Fubini's theorem [23,Theorem 64] implies that we can change the order between the stochastic integral and conditional expectation in E 9 to obtain E 9 = t 0 Ẽ(G ε s |F T )dZs.Using the same argument as above, we obtain Hence, lim The next theorem follows from Theorem 3.
Theorem 4 Assume H1-H3.Then p(t) satisfies Zakai equation: where for any φ ∈ V The differential operator B ⋆ is not bounded in the usual sense.We now study its boundedness in L(V, V ′ ).Due to H2, for any φ, ψ ∈ V , B ⋆ φ ∈ H ⊂ V ′ and satisfies where C > 0 is a constant.
The following lemma is concerned with the regularity of the second-order differential operator −L ⋆ + C.
Lemma 5 Assume H1-H3.Then there exist constants Proof The first inequality directly follows from assumption H2.Thus we only need to prove the second one.Direct computation gives, for any φ ∈ V , where From assumption H3, it follows that δ < ∞.Take a positive number α > The existence, uniqueness, and regularity of the solution to the Zakai equation ( 20), which we summarize in the next lemma., can be obtained following the approaches in [29,30], Lemma 6 Assume H1-H3.For each p 0 ∈ H, there exists a unique solution p of ( 20) Furthermore, if p 0 ∈ L 4 (Ω; H), there holds p ∈ L ∞ (0, T ; L 4 (Ω; H)).
In order to construct a stable and efficient numerical method, we take a transformation p(t) → p(t)e −µt in (20) and obtain a well-posed Zakai equation 3 A splitting-up scheme and error estimation In this section, we apply a splitting-up method to construct a temporal semidiscretized approximation of equation ( 22) and derive corresponding error estimations.

A splitting-up approximate solution
Consider three equations Equation ( 23) is a first-order SPDE.We denote by {Q s t , 0 ≤ s ≤ t} with Q s s = I its solution operator which is a Markov semigroup, cf [31].( 24) is a determined second-order PDE satisfying uniform elliptic condition if µ is large.Denote by {R s t , t ≥ 0} with R s s = I its strongly continuous semigroup.( 25) is a stochastic differential equation driven by a point process.The existence and uniqueness of the solution to (25) are obtained in [23].We denote its solution operator by {Γ s t , t ≥ 0} with Γ s s = I.According to [23,32], there exists a constant C = C(T ) such that for any φ ∈ L 2 (Ω; H) and 0 ≤ s < t ≤ T , there holds For any given integer N > 0, let t r = rκ (r = 0, 1, • • • , N ) be the uniformly partition of interval [0, T ] with stepsize κ = T N .By virtue of the solutions of ( 23)- (25) in each interval [t r , t r+1 ] we define a splitting-up solution p r+1 κ to (22) at each node point t r+1 by Meanwhile, we also define three solution process to equations ( 23)- (25) in each interval [t r , t r+1 ], Obviously, p r+1 κ = p 3κ (t r+1 ).Then we have that p 1κ , p 3κ ∈ L 2 (0, T ; H) and p 2κ ∈ L 2 (0, T ; V ) are right continuous and their discontinuity only occurs at node points.Furthermore, there hold p 2k (t) and p 3k (t) are F tr+1 measurable for t ∈ [t r , t r+1 ]. (28)

Convergence of splitting-up solution
In this section, we shall investigate the convergence and convergence order of the splitting-up solution.
Proof Integrating ( 23) over (t, t r+1 ) , ( 24) over (tr, t r+1 ) and ( 25 Due to Theorem 7 and (4), we have Similarly, we get Applying Itô isometry formula to I 4 , we have Since Z t − t is a martingale under measure P , we have Therefore, we have proved This estimate leads to Thus we have proved p 3 = p 1 .Similarly we prove p 2 = p 1 .Thus ξ = p 1 = p 2 = p 3 , which completes the proof.
Proof Integrating equations ( 23)-( 25) over (t i−1 , t i ) and adding up, we get Sum up this equation from i = 0 to r, we get (43) Noticing that as κ → 0, for i = 2, 3 According to Itô isometry formula, we have, as κ → 0, Taking limit in (43) in weak star sense as κ → 0, we obtain This is precisely equation (22).Then the proof of this lemma follows from Lemma 6.
Proof of Theorem 8. We integrate ( 31), ( 32) and (33) over interval [t i , t i+1 ], then take expectation and sum them up to obtain Summing up this equation in i from 0 up to r − 1, we get (46) Define Numerical analysis of a time discretized method for nonlinear filtering problem We now consider the convergence of these items in L 2 (Ω; H) as κ → 0 Notice that lim κ→0 S 3 κ = p 0 2 also follows from (46).Therefore we have S κ := } + 1 and by the uniform elliptic condition, we have Thus Hence as κ → 0 Similarly, we obtain the convergence of p 2κ and p 3κ as κ → 0. As an application of Theorem 8, we immediately obtain a convergence property for splitting-up solution p r+1 κ .
For φ ∈ V and τ ∈ [0, T ], define two processes ψ and ζ by We now estimate the two processes, which will play an important role in the subsequent analysis.
Lemma 12 Assume that H1-H3 hold and Proof By Lemma 5, we have From ( 25) it follows that From (47), we have Similarly, we obtain the estimates for ζ.
Iterating the above equation in i from i = 0 to r and applying Theorem 7, we have 3 )e Cκ + Cκ 2 Ẽ p r The proof is complete.
Remark We note that p 1κ , p 2κ and p 3κ are defined by the continuous solution operators Q s t , R s t and Γ s t for ( 23), ( 24) and ( 25), respectively.They are splitting up solutions for continuous problems, not numerical ones.In the next section, we will consider temporal discretizations of ( 23), ( 24), ( 25) and construct semi-discretized splitting-up approximations for the exact solution p.

Semi-discretization and error analysis
In this section, we construct a semi-discretized splitting-up scheme by discretizing ( 23)- (25) with the finite difference method and investigate its error estimate.

Numerical experiments
In this section, we apply our algorithm to a linear filtering model and a nonlinear filtering model to illustrate our theoretical results on error estimates.We use a spectral Galerkin method to discretize the spatial variable with an n-dimensional subspace whose basis functions are given by where f i−1 (x) is a Hermite polynomial of order i − 1 and φ(x) = (2π) −1/2 e −x 2 /2 .Due to [36], the spatial Galerkin discretization errors are expected to decrease exponentially with respect to the dimension n.To proceed with numerical experiments, we need to simulate the sample trajectories of the Poisson process with density function λ(X t ).For convenience, we assume the jump time τ i may occur only at t r .To determine the jump time, we first calculate a discrete sample trajectory X tr (r = 0, 1, • • • , N ) in terms of equation ( 1) and compute the density function λ(X tr ).For any i ≤ r, define Taking a time stepsize κ = 0.5 × 10 −5 and choosing λ = 3, we trace a sample trajectory using the Zakai filter and depict it in Fig. 1.In addition, we simulate a sample path for only continuous observation Y t with λ = 0 and mixed observations of Y t and Z t with λ = 3.We trace their corresponding conditional standard deviation versus time t, see Fig. 2. It shows that including information on point process observation reduces the conditional standard deviation.The dynamic behavior of the errors as varying stepsize κ i is exhibited in Fig. 3, demonstrating the half order convergence rate.the convergence order the slope is 1/2 Fig. 3 The convergence order of the splitting-up method.Numerical analysis of a time discretized method for nonlinear filtering problem Example 2. Consider a nonlinear filtering model dX t = sin(X t )dt + 2dw t , X 0 ∼ N (5, 0.01), dY t = 5.5X t dt + 0.5dw t + dv t , Y 0 = 0, Z t is a doubly stochastic Poisson process with the intensity of λX 2 t .
The corresponding Zakai equation is Set λ = 3, T = 0.5, κ = 0.5 × 10 −5 .We trace a sample signal trajectory using our Zakai filter and depict the approximations in Fig. 4. Next, we verify the convergence order in a temporal variable.We compute m = 500 reference "exact" Zakai filter solutions by fixing the stepsize κ = 2 −20 .Then we calculate m numerical Zakai filter solutions for each stepsize κ i = 2 −i , i = 14, 15, 16, 17.We plot the errors in log-log scale, cf.Fig. 5, which shows that the convergence order is of 1  2 .the convergence order the slope is 1/2 Fig. 5 The convergence order of time discretized method.

Conclusion and discussion
In this paper, we considered a nonlinear filtering model with observations involving a mixture of a Wiener process and a point process.After deriving the corresponding Zakai equation, we constructed the splitting-up scheme where the Zakai equation is decomposed into three equations: A deterministic PDE, an SDE driven by a Wiener process, and an SDE driven by a point process.Then we discretized these equations in the temporal direction by finite difference methods.By estimating the errors of these splitting up equations and the errors of the temporal discretization, we derived the half-order convergence result for the proposed numerical scheme using Milstein's fundamental theorem on numerical methods for SDEs.Our current work focuses on semidiscretizations in time.Future research on this topic includes the construction and error estimates for fully discretized numerical schemes.
).Assuming τ 0 = 0 and τ i is the present jump time; we will describe a criterion for finding the next jump time τ i+1 .Let E be a random number generated by a random variable with unit exponential distribution and independent of X tr .Then the next jump time τ i+1 is defined as the first time that T i (t r ) exceeds E.