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On the approximation of the solution of an anticipating stochastic differential equation. (Sur l’approximation de la solution d’une équation différentielle stochastique anticipative.) (French) Zbl 0949.60071
Let \(W\) be the coordinate process on the classical Wiener space and \(G\) an arbitrary random variable on this space. Given \(\alpha,b\in C^2_{l,b} (R)\) with \(\sigma\sigma'\in C^2_{l,b}(R)\) the authors prove for the anticipating stochastic differential equation with Stratonovich integral \[ X_t=G+\int^t_0 \sigma(X_s) \circ dW_t+\int^t_0b(X_s) ds,\quad t\in[0,1] \] (existence and uniqueness studied by D. Ocone and E. Pardoux in [Stochastic partial differential equations and applications. II., Lect. Notes Math. 1390, 197-204 (1989; Zbl 0703.60061)]) that, under some integrability assumption on \(\sigma,b\) and their derivatives over \(R\), the approximation by the solution of this equation with \(W\) replaced by a linear interpolation converges to \(X\) in Liouville spaces. The paper generalizes earlier results obtained by D. Feyel and A. de La Pradelle [Electron. J. Probab. 3, No. 7 (1998; Zbl 0901.60028)] in the case of a deterministic initial value.
Reviewer: R.Buckdahn (Brest)
MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H05 Stochastic integrals
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