Uniqueness of a three-dimensional stochastic differential equation

TL;DR

This paper proves the strong uniqueness of solutions for a specific three-dimensional stochastic differential equation system when the parameter alpha is between 3/4 and 1, extending previous two-dimensional results.

Contribution

It establishes the strong uniqueness of solutions for a three-dimensional SDE system with a non-Lipschitz coefficient, generalizing earlier two-dimensional findings.

Findings

Unique strong solution exists for the system when 3/4<alpha<1.

Extension of uniqueness results from 2D to 3D systems.

Conditions on initial values for solution uniqueness.

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: $dX_t=Y_tdt\quad, dY_t=Z_tdt,\quad dZ_t=|X_t|^{\alpha}dB_t$. We proved that if $(X_0,Y_0,Z_0)\neq(0,0,0)$, and $\frac{3}{4}<\alpha<1$, then the system of equations has a unique solution in the strong sense.