Stochastic Differential Equation for Brox Diffusion

TL;DR

This paper investigates solutions to a stochastic differential equation involving Brox diffusion, establishing existence, uniqueness, and Itô calculus for the process with a singular drift driven by Brownian motion.

Contribution

It introduces a novel approach using local time and polygonal approximation to handle the singular drift in Brox diffusion, proving existence and uniqueness of solutions.

Findings

Existence of a weak solution via Itô-McKean representation.

Proof of unique strong solution to the SDE.

Development of Itô calculus for the solution.

Abstract

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.