Stochastic differential equation for Brox diffusion

This paper studies the weak and strong solutions to the stochastic differential equation dX(t)=−[fomula presented]Ẇ(X(t))dt+dB(t), where (B(t),t≥0) is a standard Brownian motion and W(x) is a two sided Brownian motion, independent of 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 ∫₀^TẆ(X(t))dt, the main idea is to use the concept of local time together with the polygonal approximation W_π. Some new results on the local time of Brownian motion needed in our proof are established.