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

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 $\alpha>-1/2$.