Stochastic Differential Equation for Brox Diffusion

This paper studies the weak and strong solutions to the stochastic differential equation $dX(t)=-\frac12 \dot W(X(t))dt+d\mathcal{B}(t)$, 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 It\^o-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. It\^o 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.