Moment Problems Related to the Solutions of Stochastic Differential Equations

Abstract

Our goal is to analyze the moment uniqueness or non-uniqueness of the one-dimensional distributions of the solution processes of Ito type stochastic differential equations (SDE). We recall some criteria, classical and/or new, and apply them to derive results for the solutions of linear and nonlinear SDEs. Special attention is paid to the Brownian motion, stochastic integrals and geometric Brownian motion. Another possibility is to use the moment convergence theorem (Frechet-Shohat) for finding explicitly the limit one-dimensional distributions of specific processes. Related moment problems are also outlined with the focus on functional transformations of processes and approximations of the solutions of perturbed SDEs.