In this thesis, 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 of Wiener Space approach. It is 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 is
also established 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 are applied to the numerical
solution of the filtering problem and the pricing of asian options.