An Upper Bound for Conditional Second Moment of the Solution of a SDE

Abstract

Let be a filtration on some probability space and let denote the class of all -adapted -valued stochastic processes M such that for all t>s ≥ 0 and the process is continuous (the conditional expectations are extended, so we do not demand that . It is shown that each is a locally square integrable martingale w. r. t. . Let X be the strong solution of the equation where , t is a continuous increasing process with -measurable values at all times, and Q is an -valued random function on , continuous in and -progressive at fixed x . Suppose also that there exists an -measurable in nonnegative random process Ψ such that, for all Then where