An asymptotically efficient difference formula for solving stochastic differential equations

Abstract

Time discretisations of teh vector stochastic differential equation are considered, where (y t) is a continuous scalar process whose distribution is absolutely continuous with respect to Wiener measure. Among approximations to x T that depend on (y t) only at the discretisation points, the conditional mean is asymptotically optimal in the sense that it minimises all symmetrical conditional moments of the error. A conditional central limit theorem is derived for this minimal error, and a finite difference formula is developed which yields appromimations with the same asymptotically optimal properties. This formula is necessarily more complex than the familiar Milshtein scheme (Milshtein [7]). The latter has the maximum order of convergence but its error, considered as a power serioes in the discretisation parameter h, does not have the minimal leading coefficient. The results generalise for a special class of equations with multi-dimensional forcing terms