Numerical Solution of Stochastic Differential Equations: Diffusion and Jump-Diffusion Processes

Abstract

Stochastic differential equations (SDE) play an important role in a range of application areas, including biology, physics, chemistry, epidemiology, mechanics, microelectronics, economics, and finance [1]. However, most SDEs, especially nonlinear SDEs, do not have analytical solutions, so that one must resort to numerical approximation schemes in order to simulate trajectories of the solutions to the given equation. The simplest effective computational method for approximation of ordinary differential equations is the Euler’s method. The Euler–Maruyama method is the analogue of the Euler’s method for ordinary differential equations for numerical simulation of the SDEs [2]. Another numerical scheme is the Milstein method [3], which is a Taylor method, meaning that it is derived from a truncation of the stochastic (Ito) Taylor expansion of the solution. The Milstein scheme involves the derivatives. If it happens that the derivatives do not exist, then it leads to difficulties in it implementation. In that case we need a derivative-free method. In such cases one can consider an implicit schemes that avoid the use of derivatives [2]. It can be done by replacing the derivatives there by finite difference. These methods are known as Runge-Kutta schemes [4]. In this chapter, following the exposition of Kloeden and Platen [4], we describe numerical methods for solving the SDE for diffusion and jump-diffusion processes. In addition, we provide a brief review of some computational packages in R and Python for simulating stochastic diffusion and jump-diffusion processes.