A stable numerical scheme for stochastic differential equations with multiplicative noise

TL;DR

This paper presents a new explicit numerical scheme for stochastic differential equations with multiplicative noise, leveraging a direction and norm decomposition method to ensure stability and convergence.

Contribution

The paper introduces a novel decomposition-based approach for designing explicit schemes that preserve stability and achieve favorable convergence rates for stiff SDEs.

Findings

Preserves almost sure stability for any step-size

Achieves linear weak convergence rate

Attains one-half strong order of convergence

Abstract

We introduce a new approach for designing numerical schemes for stochastic differential equations (SDEs). The approach, which we have called direction and norm decomposition method, proposes to approximate the required solution $X_t$ by integrating the system of coupled SDEs that describes the evolution of the norm of $X_t$ and its projection on the unit sphere. This allows us to develop an explicit scheme for stiff SDEs with multiplicative noise that shows a solid performance in various numerical experiments. Under general conditions, the new integrator preserves the almost sure stability of the solutions for any step-size, as well as the property of being distant from $0$. The scheme also has linear rate of weak convergence for a general class of SDEs with locally Lipschitz coefficients,and one-half strong order of convergence.