Estimate on Moments of the Solutions to Stochastic Differential Equations in the Plane

Abstract

Let $(\Omega, \mathscr{J}, P; \mathscr{J}_{s,t})$ be a probability space with a family of sub-$\sigma$-algebras indexed by $(s, t) \in \lbrack 0, \infty) \times \lbrack 0, \infty)$, satisfying the usual conditions. Let $X(s,t)$ be a solution of a stochastic differential equation in the plane with respect to the Wiener-Yeh process. Under one of the usual conditions used to guarantee existence and uniqueness of a solution to the equation, it is shown that the absolute moments of $X(s,t)$ grow at most exponentially in $st$. The estimate is based on a version of the two parameter Ito formula and on an extension of Gronwall's inequality to functions of two variables.