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Mean square convergence of the numerical solution of random differential equations. (English) Zbl 1338.60174

Summary: This paper is devoted to the construction of an approximate solution for a random differential equation with an initial condition and defined on a partition of the time-interval. We employ a random mean value theorem to achieve our goals in this work. The implicit Runge-Kutta method is presented and the conditions for the mean square convergence are established. Finally, illustrative examples are included in which the main statistical properties such as the mean and the variance of the random approximate solution process are given. The closeness of the original and approximate solutions is measured in the sense of the \(L_2\)-norm on Banach spaces and with probability one.

MSC:

60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness
65C30 Numerical solutions to stochastic differential and integral equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
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