Uniqueness of a three-dimensional stochastic differential equation

Abstract

In order to extend the study of the uniqueness property of multidimensional systems of stochastic differential equations, we look at the following three-dimensional system of equations, of which the two-dimensional case has been well studied: dXt=Ytdt, dYt=Ztdt, dZt=|Xt|αdBt. We prove that if (X0,Y0,Z0)≠(0,0,0) and 34<α<1, then the system of equations has a unique solution in the strong sense.