On the approximation of the solution of an anticipating stochastic differential equation (Q1969339)

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 \textit{D. Ocone} and \textit{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 \textit{D. Feyel} and \textit{A. de La Pradelle} [Electron. J. Probab. 3, No. 7 (1998; Zbl 0901.60028)] in the case of a deterministic initial value.