Stochastic differential equation for Brox diffusion

Yaozhong Hu, Khoa Lê and Leonid Mytnik

Stochastic Processes and their Applications, 2017, vol. 127, issue 7, 2281-2315

Abstract: This paper studies the weak and strong solutions to the stochastic differential equation dX(t)=−12Ẇ(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 ∫0TẆ(X(t))dt, the main idea is to use the concept of local time together with the polygonal approximation Wπ. Some new results on the local time of Brownian motion needed in our proof are established.

Keywords: Random environment; Brox diffusion; White noise drift; Strong solution; Uniqueness; Local time; Itô formula (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (2)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:127:y:2017:i:7:p:2281-2315

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DOI: 10.1016/j.spa.2016.10.010

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