The limit of the normalized error in the approximation of stochastic differential equations.

摘要

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.