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

Cite this paper

A. Yurachkivsky, "An Upper Bound for Conditional Second Moment of the Solution of a SDE,"*Applied Mathematics*, Vol. 4 No. 1, 2013, pp. 135-143. doi: 10.4236/am.2013.41023.

A. Yurachkivsky, "An Upper Bound for Conditional Second Moment of the Solution of a SDE,"

References

[1] I. I. Gikhman and A. V. Skorokhod, “Stochastic Differential Equations and Their Applications,” Naukova Dumka, Kiev, 1982.

[2] A. N. Shiryaev, “Probability,” Springer, Berlin, 1996.

[3] R. Z. Khasminsky, “Stochastic Stability of Differential Equations,” 2nd Edition, Springer, Berlin, 2012.
doi:10.1007/978-3-642-23280-0

[4] L. J. Shen and J. T. Sun, “p-th Moment Exponential Stability of Stochastic Differential Equations with Impulse Effect,” Science China Information Sciences, Vol. 54, No. 8, 2011, pp. 1702-1711. doi:10.1007/s11432-011-4250-7

[5] R. Sh. Liptser and A. N. Shiryaev, “Theory of Martingales,” Kluwer, Dordrecht, 1989.
doi:10.1007/978-94-009-2438-3

[6] J. Jacod and A. N. Shiryayev, “Limit Theorems for Stochastic Processes,” Springer, Berlin, 1987.