| This thesis studies the limits of normalized errors in various numerical schemes for SDE's and SPDE's involving white noise. We first establish a result for convergence of the normalized error Uht=h-b&parl0;Xh t-Xt&parr0; , when X is the solution of an Ito SDE, and Xh is a strong-Ito-Taylor approximation to X of order b (see Kloeden & Platen, 1992). We find U satisfying Uh ⇒ U, where " ⇒ " denotes convergence in distribution uniformly over compact time intervals.;We also establish the asymptotically optimal adaptive mesh to use in determining the numerical scheme Xh which is &cubl0;Ft&cubr0; -adapted. For suitable mesh functions f we again establish that Uht⇒Ut as h → 0, and find U, and then determine the unique mesh function which yields the smallest EU2t under a constraint on the asymptotic expected number of mesh time steps required.;Finally, we establish the limit of the normalized error in an approximation scheme for a linear stochastic parabolic PDE on a periodic domain of finite dimension. We use the Soboloev space setting of Walsh (1986), and rely on a result of Blount for explicit tail estimates of Ornstein-Uhlenbeck processes. |
