Discrete Gradient Approach to Stochastic Differential Equations with a Conserved Quantity

Abstract

General autonomous stochastic differential equations (SDEs) driven by one-dimensional Brownian motion in the Stratonovich sense with a conserved quantity $I(y)$ are considered. Relying on this conserved quantity, an equivalent “skew gradient” (SG) form of original SDEs is constructed. With the aim of constructing conserved numerical methods, direct discrete gradient approaches which approximate directly the SG system lead to two conserved methods. In indirect discrete gradient approaches, we first split the SG system into subsystems which preserve the original conserved quantity, then apply direct discrete gradient approaches to subsystems to produce two composition schemes. The mean-square convergence of order 1 for these four methods depends on the assumptions that the coefficients of SDEs are Lipschitz continuous with bounded second moments along the solution of SDEs, are twice continuous differentiable, and have derivatives that are uniformly bounded. Three numerical experiments are presented to verify our theoretical analysis and show the advantages of these methods.