Uniqueness of a three-dimensional stochastic differential equation

Abstract

In order to extend the study of uniqueness property of multi-dimensional systems of stochastic differential equations, in this paper, we look at the following three-dimensional system of equations, of which the two-dimensional case was well-studied before: $d{X}_t=Y_{t}dt,d{Y}_t=Z_{t}dt,d{Z}_t=|X_t{|}^{\alpha }d{B}_t$. We proved that if $(X_0,Y_0,Z_0)\ne (0,0,0)$, and $\frac{3}{4}<\alpha <1$, then the system of equations has a unique solution in the strong sense.