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Stochastic differential equation for Brox diffusion

Author

Listed:
  • Hu, Yaozhong
  • Lê, Khoa
  • Mytnik, Leonid

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.

Suggested Citation

  • Hu, Yaozhong & Lê, Khoa & Mytnik, Leonid, 2017. "Stochastic differential equation for Brox diffusion," Stochastic Processes and their Applications, Elsevier, vol. 127(7), pages 2281-2315.
    • Handle: RePEc:eee:spapps:v:127:y:2017:i:7:p:2281-2315
    DOI: 10.1016/j.spa.2016.10.010
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    References listed on IDEAS

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    1. Diel, Roland, 2011. "Almost sure asymptotics for the local time of a diffusion in Brownian environment," Stochastic Processes and their Applications, Elsevier, vol. 121(10), pages 2303-2330, October.
    2. Hu, Yaozhong & Le, Khoa, 2013. "A multiparameter Garsia–Rodemich–Rumsey inequality and some applications," Stochastic Processes and their Applications, Elsevier, vol. 123(9), pages 3359-3377.
    3. Shi, Zhan, 1998. "A local time curiosity in random environment," Stochastic Processes and their Applications, Elsevier, vol. 76(2), pages 231-250, August.
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    Cited by:

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    2. De Angelis, Tiziano & Germain, Maximilien & Issoglio, Elena, 2022. "A numerical scheme for stochastic differential equations with distributional drift," Stochastic Processes and their Applications, Elsevier, vol. 154(C), pages 55-90.

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