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: $d{X}_t=Y_{t}dt$, $d{Y}_t=Z_{t}dt$, $d{Z}_t=|X_t{|}^{\alpha }d{B}_t$. We prove that if $\left(X_0,Y_0,Z_0\right)\ne \left(0,0,0\right)$ and $\frac{3}{4}<\alpha <1$, then the system of equations has a unique solution in the strong sense.