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Yan Li, Junhao Hu, "Numerical Analysis for Stochastic Partial Differential Delay Equations with Jumps", Abstract and Applied Analysis, vol. 2013, Article ID 128625, 8 pages, 2013. https://doi.org/10.1155/2013/128625
Numerical Analysis for Stochastic Partial Differential Delay Equations with Jumps
We investigate the convergence rate of Euler-Maruyama method for a class of stochastic partial differential delay equations driven by both Brownian motion and Poisson point processes. We discretize in space by a Galerkin method and in time by using a stochastic exponential integrator. We generalize some results of Bao et al. (2011) and Jacob et al. (2009) in finite dimensions to a class of stochastic partial differential delay equations with jumps in infinite dimensions.
The theory and application of stochastic differential equations have been widely investigated [1–7]. Liu  studied the stability of infinite dimensional stochastic differential equations. For the numerical analysis of stochastic partial differential equations, Gyöngy and Krylov  discussed the numerical approximations for linear stochastic partial differential equations in whole space. Jentzen et al.  studied the numerical simulations of nonlinear parabolic stochastic partial differential equations with additive noise. Kloeden et al.  gave the error analysis for the pathwise approximation of a general semilinear stochastic evolution equations.
By contrast, stochastic partial differential equations with jumps have begun to gain attention [11–15]. Röckner and Zhang  considered the existence, uniqueness, and large deviation principles of stochastic evolution equation with jump. In , the successive approximation of neutral SPDEs was studied. There are few papers on the convergence rate of numerical solutions for stochastic partial differential equations with jump, although there are some papers on the convergence rate of numerical solutions for stochastic differential equations with jump in finite dimensions [16, 17].
Being motivated by the papers [16, 17], we will discuss the convergence rate of Euler-Maruyama scheme for a class of stochastic partial delay equations with jump, where the numerical scheme is based on spatial discretization by Galerkin method and time discretization by using a stochastic exponential integrator. In consequence, we generalize some results of Bao et al. (2011) and Jacob et al. (2009) in finite dimensions to a class of stochastic partial delay equations with jump in infinite dimensions. The rest of this paper is arranged as follows. We give some preliminary results of Euler-Maruyama scheme in Section 2. The convergence rate is discussed in Section 3.
2. Preliminary Results
Throughout this paper, let be a complete probability space with some filtration satisfying the usual conditions (i.e., it is right continuous and contains all -null sets). Let () and () be two real separable Hilbert spaces. We denote by the family of bounded linear operators. Let and denote the family of right-continuous function and left-hand limits from to with the norm . denotes the family of almost surely bounded, -measurable, -valued random variables. For all , is regarded as -valued stochastic process.
Let be a positive constant. For given , consider the following stochastic partial differential delay equations with jumps: on with initial datum , . Here is a self-adjoint operator on . is -valued -Wiener process defined on the probability space with covariance operator . We assume that and the covariance operator of the Wiener process have the same eigenbasis of ; that is, where are the discrete spectrum of and , are the eigenvalues of . Then, is defined by where is a sequence of real-valued standard Brownian motions mutually independent of the probability space .
According to Da Prato and Zabczyk , we define stochastic integrals with respect to the -Wiener process . Let be the subspace of with the inner product . Obviously, is a Hilbert space. Denote by the family of Hilbert-Schmidt operators from into with the norm .
Let : be a predictable, -adapted process such that Then, the -valued stochastic integral is a continuous square martingale. Let be the Poisson measure which is independent of the -Wiener process . Denote the compensated or centered Poisson measure as where is known as the jump rate and is the jump distribution (a probability measure). Let be the measurable set. Denote by the space of all predictable mappings for which Then, the -valued stochastic integral is a centred square-integrable martingale.
We recall the definition of the mild solution to (1) as follows.
Definition 1. A stochastic process is called a mild solution of (1) if (i) is adapted to , , and has càdlàg path on almost surely, (ii)for arbitrary , , and almost surely for any , .
For the existence and uniqueness of the mild solution to (1) (see ), we always make the following assumptions. (H1) is a self-adjoint operator on such that has discrete spectrum with corresponding eigenbasis of . In this case generates a compact -semigroup , such that . (H2)The mappings , , and are Borel measurable and satisfy the following Lipschitz continuity condition for some constant and arbitrary and : This further implies the linear growth condition; that is, where (H3)There exists satisfying for each and . (H4)For , there exists a constant such that
We now describe our Euler-Maruyama scheme for the approximation of (1). For any , let be the orthogonal projection; that is, , , , , , and .
Consider the following stochastic differential delay equations with jumps on : This spatial approximation (14) is called the Galerkin approximation of (1). Due to the fact that , it follows that for , , .
By (H2) and (H3) and the property of the projection operator, we have that for arbitrary and . Hence, (14) admits a unique solution on .
We introduce a time discretization scheme for (14) by using a stochastic exponential integrator. For given and , the time-step size is defined by , for some sufficiently large integer . For any integer , the time discretization scheme applied to (14) produces approximations by forming where and .
The continuous-time version of this scheme associated with (14) is defined by where with denotes the integer of .
3. Convergence Rate
In this section, we shall investigate the convergence rate of the Euler-Maruyama method. In what follows, is a generic constant whose values may change from line to line.
Lemma 2. Let (H1)–(H4) hold; then there is a positive constant which depends on , and but is independent of , such that
Proof. Due to the fact that is a norm, we have from (8) that
Recall the property of the operator (see ): for , where .
By (H1) and (H2), together with the Minkowski integral inequality, we derive that By (H1), (H2), and (H3) and using the Itô isometry, we have Using Hölder inequality and (H3), for the last term of (22), we have Moreover, by using the Itô isometry and (H3), we obtain that Substituting (23) and (24) into (22), it follows that Hence, Applying the Gronwall inequality, we have Using the similar argument, the second assertion of (18) follows.
Lemma 3. Let (H1)–(H4) hold; for sufficiently small , where is constant dependent on , and , while being independent of .
Proof. For any , we have from (8) that Since is a norm, it follows that Recalling the fundamental inequality , , we get from (H1) that Therefore, By (H1), (H2), and the Minkowski integral inequality, we obtain that Together with (31), we arrive at Following the argument of (22), we derive that Substituting (32), (34), and (35) into (30), we arrive at Therefore, by Lemma 2, the required assertion (28) follows.
Now, we state our main result in this paper as follows.
Theorem 4. Let (H1)–(H4) hold, and Then, where is a constant dependent on , and , while being independent of and .
Proof. By (8) and (17), we obtain Noting that is a norm, we have By (H1) and the nondecreasing spectrum , it easily follows that By (H2), the Minkowski integral inequality, and Lemma 2, we have Applying (H1), (H2), and Lemma 3 and combining the Minkowski integral inequality and the Itô isometry yieldBy the Itô isometry and a similar argument to that of (42), we deduce that Moreover, by (31), (H2), and Lemma 2 and combining the Minkowski integral inequality and the Itô isometry, we have By (31) and the Itô isometry, we obtain that Carrying out the similar arguments to those of (43) and (45), we derive that As a result, putting (41)–(47) into (40) gives that and therefore the desired assertion follows.
This work is partially supported by National Natural Science Foundation of China Under Grants 60904005 and 11271146.
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Copyright © 2013 Yan Li and Junhao Hu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.