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We propose a sparse grid stochastic collocation method for long-timesimulations of stochastic differential equations (SDEs) driven by white noise.The method uses pre-determined sparse quadrature rules for the forcing term andconstructs evolving set of sparse quadrature rules for the solution variablesin time. We carry out a restarting scheme to keep the dimension of randomvariables for the forcing term, therefore also the number of quadrature points,independent of time. At each restart, a sparse quadrature rule for the currentsolution variables is constructed based on the knowledge of moments and theprevious quadrature rules via a minimization procedure. In this way, the methodallows us to capture the long-time solutions accurately using small degrees offreedom. We apply the algorithm to low-dimensional nonlinear SDEs anddemonstrate its capability in long-time simulations numerically.

A Dynamical Sparse Grid Collocation Method for Differential Equations Driven by White Noise