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Discrétisation d’une équation différentielle stochastique et calcul approché d’espérances de fonctionnelles de la solution. (Discretization of a stochastic differential equation and computation of expectations of functions of the solution). (French) Zbl 0662.65129
We are interested in “well” discretizing a stochastic differential equation, in view to approximate numerically the expectations of a large class of functionals of the solution. Classical methods are not very efficient. G. N. Mil’stein [Teor. Verojatn. Primen. 23, 414-419 (1978; Zbl 0391.60060)] has proposed a new method, and conjectured its rate of convergence. Here, we prove the announced result, and we introduce new schemes, which have the same rate of convergence as Milshtein’s one, and permit to treat the multidimensional equations.

MSC:
65C99 Probabilistic methods, stochastic differential equations
65L05 Numerical methods for initial value problems
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness
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