Conservative methods for stochastic differential equations with a conserved quantity
This work addresses the need for structure-preserving numerical methods in stochastic dynamics, offering a new approach for systems with conserved quantities.
The paper proposes novel conservative numerical methods for stochastic differential equations with a conserved quantity, achieving mean-square and weak order 1 for commutative noises, and analyzes the impact of quadrature formulas on convergence.
This paper proposes a novel conservative method for numerical computation of general stochastic differential equations in the Stratonovich sense with a conserved quantity. We show that the mean-square order of the method is $1$ if noises are commutative and that the weak order is also $1$. Since the proposed method may need the computation of a deterministic integral, we analyse the effect of the use of quadrature formulas on the convergence orders. Furthermore, based on the splitting technique of stochastic vector fields, we construct conservative composition methods with similar orders as the above method. Finally, numerical experiments are presented to support our theoretical results.