© 1987 by Institute of Mathematics and its Applications
In-Probability Approximation and Simulation of Nonlinear Jump-Diffusion Stochastic Differential Equations
1Faculty of Mathematical Studies, University of Southampton SO95NH
2School of Electrical Engineering and Science, Royal Military College of Science Shrivenham, Swindon, Wilts, SN68LA
The problem of numerical solution of a wide class of Itôstochastic differential equations simultaneously driven by Wiener and inhomogeneous Poisson processes is considered. The equations serve as models for systems affected by random disturbances of impulsive type as well as white noise. The method of discretization and Taylor expansion is used in conjunction with an ‘in-probability’ criterion for error measurement. An error-analysis theorem determining the orders of the stochastic truncation errors is presented. This theorem assesses the results of previous similar work in the special case where no jump component is present, as well as providing guidelines for the approximation of the jump-diffusion processes. For the latter case it is shown that, via this approach and unlike the no-jump case, the error can be improved over that of the basic stochastic Cauchy-Euler scheme only to the extent of increasing the order of an additive component of the total error.
A first-order algorithm with this property is derived and results on the rate of convergence of both schemes and computable evaluations, implementation, and simulations are presented. Simulation results demonstrate significant superiority of the proposed algorithm over the basic scheme in stability and efficiency.