Abstract

Let $\bF=\left(\cF(t), \:t\in \bR_+\right)$ be a filtration on some probability space, and $X$ be the strong solution of the equation $X(t)=\X+\int_0^tQ(s,X(s))\rd\iota(s)+\int_0^t\sigma(s,X(s-)) \rd Y(s),$ where $\X$ is an $\cF(0)$-measurable $\bR^d$-valued random variable, $\iota$ is a continuous increasing process with $\cF(0)$-measurable values at all times, $Y$ is an $\bR^m$-valued locally square integrable martingale with respect to $\bF$ subjected to some mild additional demands, $Q$ and $\sigma$ are continuous in $x\in\bR^d$ random functions on $\bR_+\times\bR^d$ (the former $\bR^d$-valued and $\bF$-progressive in $(\omega,t)\in\Omega\times\bR_+$, the latter ($d\times m)$-matrix-valued and $\bF$-predictable). Suppose also that there exists an $\cF(0)\otimes\cB_+$-measurable in $(\omega,t)$ nonnegative random process $\psi$ such that, for all $t,x$, $x^\top Q(t,x) \le-\psi(t)|x|^2$ and $\int_0^t\psi(s)\rd\iota(s)<\infty.$ Under these assumptions, $\E(|X(t)|^2|\cF(0)$ is evaluated from above.

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