Runge-Kutta methods for the strong approximation of solutions of stochastic differential equations

摘要

Some new stochastic Runge-Kutta (SRK) methods for the strong approximation of solutions of stochastic differential equations (SDEs) with improved efficiency are introduced. Their convergence is proved by applying multicolored rooted tree analysis. Order conditions for the coefficients of explicit and implicit SRK methods are calculated. As the main novelty, order 1.0 strong SRK methods with significantly reduced computational complexity for It? as well as for Stratonovich SDEs with a multidimensional driving Wiener process are presented where the number of stages is independent of the dimension of the Wiener process. Further, an order 1.0 strong SRK method customized for It? SDEs with commutative noise is introduced. Finally, some order 1.5 strong SRK methods for SDEs with scalar, diagonal, and additive noise are proposed. All introduced SRK methods feature significantly reduced computational complexity compared to well-known schemes.