Stochastic θ-Methods for a Class of Jump-Diffusion Stochastic Pantograph Equations with Random Magnitude

This paper is concerned with the convergence of stochastic θ-methods for stochastic pantograph equations with Poisson-driven jumps of random magnitude. The strong order of the convergence of the numerical method is given, and the convergence of the numerical method is obtained. Some earlier results are generalized and improved.


Introduction
Recently, the study of stochastic pantograph equations (SPEs) has many results [1][2][3]. SPEs have been extensively applied in many fields such as finance, control, and engineering. However, in general, SPEs have no explicit solutions, and the study of numerical solutions of SPEs has received a great deal of attention. Fan et al. [4] investigate the th moment asymptotical stability of the analytic solution and the numerical methods for the stochastic pantograph equation by using the Razumikhin technique. Baker and Buckwar [5] gave strong approximations to the solution obtained by a continuous extension of the -Euler scheme and proved that the numerical solution produced by the continuous -method converges to the true solution with order 1/2. Fan et al. [6] investigated the existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equations under the local Lipschitz condition and the linear growth condition. Li et al. [7] investigated the convergence of the Euler method of the stochastic pantograph equations with Markovian switching under the weaker conditions. Reference [8] studied convergence and stability of numerical methods of stochastic pantograph differential equations.
In practice, stochastic differential equations with jump and numerical methods are also discussed extensively. In [9][10][11][12][13] strong convergence and mean-square stability properties were analysed in the case of Poisson-driven jumps of deterministic magnitude. References [14,15] discussed the numerical methods of stochastic differential equations with random jump magnitudes. Motivated by the papers above, in this paper, we focus on stochastic pantograph equations with random jump magnitudes. SPEs with random jump magnitudes may be regarded as an extension of stochastic pantograph equations. Jump models arise in many other application areas and have proved successful at describing unexpected, abrupt changes of state [16][17][18]. Typically, these models do not admit analytical solutions and hence must be simulated numerically. Similar to stochastic differential equations [19][20][21], explicit solutions can hardly be obtained for SPEs with random jump magnitudes. Thus, appropriate numerical approximation schemes such as the Euler (or Euler-Maruyama) are needed to apply them in practice or to study their properties.
The paper is organised as follows. In Section 2, we introduce the SPEs with random jump magnitudes and define stochastic -methods of (1). The main result of the paper is rather technical, so we present several lemmas in Section 3 and then complete the proof in Section 4.
Throughout, we assume that the jump magnitudes have bounded moments; that is, for some ≥ 1, there is a constant = such that We further employ the following assumptions.
Assumption 1. The functions , , and ℎ satisfy the global Lipschitz condition, that is, for each = 1, 2, 3, there is a positive constant 1 such that where , ∈ R .
In fact, the global Lipschitz condition (3) implies the linear growth condition (4). Under these conditions, it can be shown similarly as in [20] that (1) has a unique solution with all moments bounded.
For ∈ [ , +1 ], we define and denote Then, we define the continuous-time approximation which interpolates the discrete numerical approximation (6). So a convergence result for ( ) immediately provides a result for .
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Lemmas
Throughout our analysis, , , = 1, 2, . . . denote generic constants, independent of ℎ. The main theorem of the paper is rather technical. We will present a number of useful lemmas in the section and then complete the proof in Section 4.

Main Results
We can now state and prove our main result of this paper.
Remark 7. Theorem 6 shows that the order of convergence in mean square is close to 1. Moreover, stochastic -methods give strong convergence rate arbitrarily close to order 1/2 under appropriate moment bounds on the jump magnitude.
This problem class is now widely used in mathematical finance. By Theorem 6, we obtain the following corollaries.
The convergent result can be extended to the case of nonlinear coefficients that are local Lipschitz [6,7,12] based on the style of analysis in [22].