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

Andriy Yurachkivsky

Cybernetics Department, Taras Shevchenko National University, Kyiv, Ukraine.

DOI: 10.4236/am.2013.41023                

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

Keywords

Conditional Expectation; Martingale; Stochastic Equation

Conflicts of Interest

The authors declare no conflicts of interest.

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