Stochastic Differential Equation for Brox Diffusion

Abstract

This paper studies the weak and strong solutions to the stochastic differential equation $dX(t)=-\frac{1}{2}\dot{W}(X(t))dt+d\mathcal{B}(t)$, where $(\mathcal{B}(t),t\ge 0)$ is a standard Brownian motion and $W(x)$ is a two sided Brownian motion, independent of $\mathcal{B}$. It is shown that the Itô-McKean representation associated with any Brownian motion (independent of $W$) is a weak solution to the above equation. It is also shown that there exists a unique strong solution to the equation. Itô calculus for the solution is developed. For dealing with the singularity of drift term ${\int }_0^T\dot{W}(X(t))dt$, the main idea is to use the concept of local time together with the polygonal approximation $W_{\pi }$. Some new results on the local time of Brownian motion needed in our proof are established.