A class of numerical algorithms for stochastic differential equations with randomly varying truncations

Abstract

This work develops a novel class of numerical approximation algorithms for highly nonlinear stochastic differential equations. It is inspired by a stochastic approximation/optimization algorithm. The idea is the generation of random-varying truncation bounds. The algorithms are suited in case the coefficients have faster than linear growth resulting in the finite explosion time in implementing the usual Euler-Maruyama scheme, and are easier to be implemented in contrast to the existing approaches. In this paper, weak convergence and weak convergence rates of the algorithms are established using the martingale problem formulation. Some numerical examples are presented for demonstration. Finally, remarks on algorithms with additional random switching and algorithms with decreasing step sizes are given.