Journal
SIAM JOURNAL ON NUMERICAL ANALYSIS
Volume 48, Issue 3, Pages 922-952Publisher
SIAM PUBLICATIONS
DOI: 10.1137/09076636X
Keywords
stochastic Runge-Kutta method; stochastic differential equation; multicolored rooted tree analysis; strong approximation; numerical method; commutative noise; diagonal noise; additive noise
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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 Ito 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 Ito 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.
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