Hybrid Spectral-Time Stepping Methods for solving Nonlinear Stochastic Differential equations

Abstract

This paper concerns solving nonlinear stochastic differential equations (SDEs) by enhancing a novel method. There are two primary steps towards constructing this method. The core idea of the first step involves approximating each Brownian sample path over the given time horizon utilising Chebyshev polynomials. The second task is solved by using the Runge-Kutta method. In fact, the procedure in step one allowed us to utilise the classical Runge-Kutta method for stochastic differential equations without necessitating any alterations to their original formulations. Comprehensive numerical experiments on various test problems reveal that the proposed hybrid method achieves significantly lower average absolute errors, calculated over 10,000 sample paths, compared to five established techniques (Euler-Maruyama, Milstein, Stochastic Runge-Kutta (Platen), block-pulse, and hat-function methods), while maintaining a similar computational cost. In fact, the convergence and stability of this hybrid method are inhibited by the original Chebyshev approximation and the classical Runge-Kutta method. Also, the hybrid method provides a flexible and effective instrument for the precise simulation of nonlinear stochastic differential equations.