Preservation of quadratic invariants of stochastic differential equations via Runge–Kutta methods

In this paper, we give conditions for stochastic Runge–Kutta (SRK) methods to preserve quadratic invariants. It is shown that SRK methods preserving quadratic invariants are symplectic. Based on both convergence order conditions and quadratic invariant-preserving conditions, we construct some SRK schemes preserving quadratic invariants with strong and weak convergence order with the help of computer algebra, respectively. Numerical experiments are executed to verify our theoretical analysis and show the superiority of these schemes.