The global error in weak approximations of stochastic differential equations

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.EThOS - Electronic Theses Online ServiceGBUnited Kingdo