Stochastic differential equation for Brox diffusion

DOI10.1016/J.SPA.2016.10.010zbMATH Open1378.60081arXiv1506.02280OpenAlexW2963898497WikidataQ115341146 ScholiaQ115341146MaRDI QIDQ2359722

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Publication date: 22 June 2017

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Abstract: This paper studies the weak and strong solutions to the stochastic differential equation $dX(t)=-frac12dotW(X(t))dt+dmathcalB(t)$, where $(mathcalB(t),tge0)$ is a standard Brownian motion and $W(x)$ is a two sided Brownian motion, independent of $mathcalB$. 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 $in{t}_0^{T}dotW(X(t))dt$, the main idea is to use the concept of local time together with the polygonal approximation $W_{p}i$. Some new results on the local time of Brownian motion needed in our proof are established.

Full work available at URL: https://arxiv.org/abs/1506.02280

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