An improved uniqueness result for a system of stochastic differential equations related to the stochastic wave equation

TL;DR

This paper enhances the understanding of strong uniqueness in a specific stochastic differential equation system related to the stochastic wave equation, establishing new conditions for uniqueness when initial conditions are not zero.

Contribution

It provides an improved strong uniqueness result for a class of SDEs with a specific power-law coefficient, extending previous findings to a broader parameter range.

Findings

Strong uniqueness holds for lpha>-1/2 with non-zero initial conditions.

The result extends previous work by relaxing conditions on the coefficient.

Short-time uniqueness is established for the system.

Abstract

We improve on the strong uniqueness results of [GLM+17], which deal with the following system of SDE. \begin{align*} dX_t&=Y_tdt \\ dY_t&=|X_{t}|^{\alpha}dB_t \end{align*} and $X_0=x_0,Y_0=y_0$. For $(x_0,y_0)\ne(0,0)$, we show that short-time uniqueness holds for $\alpha>-1/2$.