Strong and Weak Approximation Methods for Stochastic Differential Equations—Some Recent Developments
Some efficient stochastic Runge–Kutta (SRK) methods for the strong as well as for the weak approximation of solutions of stochastic differential equations (SDEs) with improved computational complexity are considered. Their convergence is analyzed by a concise colored rooted tree approach for both, Itô as well as Stratonovich SDEs. Further, order conditions for the coefficients of order 1.0 and 1.5 strong SRK methods as well as for order 2.0 weak SRK methods are given. As the main novelty, the computational complexity of the presented order 1.0 strong SRK method and the order 2.0 weak SRK method depends only linearly on the dimension of the driving Wiener process. This is a significant improvement compared to well known methods where the computational complexity depends quadratically on the dimension of the Wiener process.
KeywordsOrder Condition Stochastic Differential Equation Kutta Method Wiener Process Weak Approximation
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