Speaker:  Brad Franklin, UNL

Title: THE LIMIT NORMALIZED ERROR IN THE APPROXIMATION OF STOCHASTIC DIFFERENTIAL EQUATIONS

Abstract: We establish a result for convergence of the normalized error U^h =3D h^-b(X^h - X), when X is the solution of an Itô SDE,  and X^h is a strong-Itô-Taylor approximation to X of order b  (see Kloeden & Platen, 1992). We find the limit U to which U^h converges,  where convergence is "in distribution uniformly over compact time intervals."

We next establish the asymptotically optimal adaptive mesh to use for the numerical scheme X^h which is F-adapted. For suitable mesh  functions f we again establish that U^h_t -> U_t as h -> 0,  and find U, and then determine the a.s. unique mesh function which  yields the smallest EU²_t under a constraint on the asymptotic normalized expected number of mesh time steps required.