The Global Error in Weak Approximations of Stochastic Differential Equations

In this talk, the convergence analysis of a class of weak approximations of

solutions of stochastic differential equations is presented. This class includes

recent approximations such as Kusuoka's moment similar families method and the

Lyons-Victoir cubature on Wiener Space approach. It will be shown that the rate

of convergence depends intrinsically on the smoothness of the chosen test

function. For smooth functions (the required degree of smoothness depends on the

order of the approximation), an equidistant partition of the time interval on

which the approximation is sought is optimal. For functions that are less smooth

(for example Lipschitz functions), the rate of convergence decays and the

optimal partition is no longer equidistant. An asymptotic rate of convergence

will also be presented for the Lyons-Victoir method. The analysis rests upon

Kusuoka-Stroock's results on the smoothness of the distribution of the solution

of a stochastic differential equation. Finally, the results will be applied to

the numerical solution of the filtering problem.