Stochastic Volterra integral equations with jumps and the strong superconvergence of the Euler–Maruyama approximation

Abstract

We study the existence and uniqueness of a solution to the stochastic Volterra integral equations (SVIEs) with jumps. Moreover, we apply the Euler–Maruyama approximation for SVIEs with jumps and investigate the boundedness and the convergence of the numerical solution. Furthermore, we show that the numerical solution strongly superconverges with order 1 if the diffusion and jump coefficients satisfy $b(t,t)=0$ and $c(t,t,\xi )=0$. Otherwise, the numerical solution is strongly convergent of order $\frac{1}{2}$. The theoretical results are illustrated by numerical simulations and comprehensive examples.

Introduction

Stochastic differential equations (SDEs) with jumps offer the most flexible, numerically accessible mathematical framework to help model the evolution of financial and other random quantities through time. In particular, feedback effects can be easily modeled and jumps enable us to frame events. It is important to be able to incorporate event driven uncertainty into a model, and this can be expressed by jumps. This arises, for instance, when one works on credit risk, insurance risk, or operational risk. SDEs enable us to model the feedback effects in the presence of jumps and independence on the level of the state variable itself [1], [2], [3], [4].

When one needs to handle the mathematical description of jumps, then the Poisson process is usually used, as it has the advantage of counting events and generating an increasing sequence of jump times related to each event that it counts, and consequently, gives the number of jumps that have occurred up to any point in time. However, when events that have randomly distributed jump sizes are modeled, it is more appropriate to use SDEs driven by Poisson measures, instead of Poisson processes, see [5], [6], [7], [8], [9], [10] for discussion of recent developments on SDEs driven by Poisson random measure.

In real life, many phenomena can be mathematically formulated by so called Volterra integral equations (VIEs). Therefore, in recent years, VIEs have attracted the attention of many scholars, who have investigated the analytical and numerical solutions of VIEs, as reflected by various books and papers, for instance [11], [12], [13], [14] and the references therein.

The so called stochastic Volterra integral equations (SVIEs) have arisen as a result of evolution. SVIEs have become very common and widely used in numerous branches of science, due to their connections with mathematical finance, biology, engineering, etc. [14], [15], [16], [17].

Although SVIEs with jumps have important applications, only a few papers have been concerned with these types of equations. Controlled stochastic Volterra integral equations with jumps were studied in [18]. Under a Lipschitz condition, they proved the existence and uniqueness of solutions to SVIEs with jumps driven by Brownian motion and an independent compensated Poisson random measure.

Motivated by the above, in the present paper we consider a class of stochastic dynamical systems driven by Brownian motion and a pure jump Lèvy process and establish the existence and uniqueness of solutions for such class of stochastic systems. However, one must admit that it is in most cases impossible to provide an analytic solution to an SVIE. Therefore, applying numerical methods is always preferable. Thus, we apply the Euler–Maruyama (EM) scheme to an SVIE and establish a numerical solution for such equations. Under given conditions, we investigate the boundedness and the convergence of the EM solution to the true solution. We prove that the new numerical approach is bounded and strongly superconverges of order 1 if the diffusion and jump coefficients satisfy $b(t,t)=0$ and $c(t,t,\xi )=0$. Otherwise, it is strongly convergent of order $\frac{1}{2}$. Moreover, we provide the results of numerical experiments and extensive examples to clarify the convergence of the numerical solution and illustrate the theoretical results.

It should be pointed out that the Euler–Maruyama method has already been applied to SVIEs with Brownian motion by Liang et al. [15]. The Euler–Maruyama approximation has also been applied to stochastic differential equations, in [19], who studied the convergence in probability of the Euler–Maruyama approximate solution to the true solution of hybrid stochastic equations. However, for SVIEs with jumps, little is known about the existence and uniqueness of their solutions and the convergence of the EM method. Therefore, our goal here is to fill this gap.

The rest of this paper is organized as follows. In Section 2, we introduce some preliminaries and give the formula of the considered equation. Moreover, we establish the existence and uniqueness of the solution to an SVIE with Brownian motion and pure jumps. In Section 3, we introduce the Euler–Maruyama method and study the boundedness and the convergence of the numerical solution. Lastly, the results of some numerical experiments and some illustrative examples are presented in Section 4.