Resolution trajectorielle et analyse numerique des equations differentielles stochastiques

Let (ξ_t) be the solution of the S.D.E. (E) of Section 1. Doss 3] has shown the existence of a difFerentiable function h and of a differentiate process parametrized by the process W,γ(W,t), such that: ξ_t = h(γ(W, t), W_t). For all continuous functions u, X_t is defined by: X_t = h(γ(u, t) u_t). We develop a scheme of approximation of X_t (Theorems 2-6 and 3-4). This scheme has th following properties:?

1)it does not explicitly involve γ or h; this property is crucial, because,generally, h is not explicitly known, and its numerical approximation would be costly.

2)it converges to X_t, provided that u has bounded quadratic variation.

3)for u = W, it coincides with a scheme proposed by Milshtein 6] for quadratic-mean approximation.

Further, we give an estimate of the error due to this scheme.