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Time inhomogeneous Stochastic Differential Equations involving the local time of the unknown process, and associated parabolic operators

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  • Étoré, Pierre
  • Martinez, Miguel

Abstract

In this paper we study time inhomogeneous versions of one-dimensional Stochastic Differential Equations (SDE) involving the Local Time of the unknown process on curves. After proving existence and uniqueness for these SDEs under mild assumptions, we explore their link with Parabolic Differential Equations (PDE) with transmission conditions. We study the regularity of solutions of such PDEs and ensure the validity of a Feynman–Kac representation formula. These results are then used to characterize the solutions of these SDEs as time inhomogeneous Markov Feller processes.

Suggested Citation

  • Étoré, Pierre & Martinez, Miguel, 2018. "Time inhomogeneous Stochastic Differential Equations involving the local time of the unknown process, and associated parabolic operators," Stochastic Processes and their Applications, Elsevier, vol. 128(8), pages 2642-2687.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:8:p:2642-2687
    DOI: 10.1016/j.spa.2017.09.018
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    References listed on IDEAS

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    1. Trutnau, Gerald, 2011. "Pathwise uniqueness of the squared Bessel and CIR processes with skew reflection on a deterministic time dependent curve," Stochastic Processes and their Applications, Elsevier, vol. 121(8), pages 1845-1863, August.
    2. Blei, Stefan & Engelbert, Hans-Jürgen, 2013. "One-dimensional stochastic differential equations with generalized and singular drift," Stochastic Processes and their Applications, Elsevier, vol. 123(12), pages 4337-4372.
    3. Trutnau, Gerald, 2010. "Weak existence of the squared Bessel and CIR processes with skew reflection on a deterministic time-dependent curve," Stochastic Processes and their Applications, Elsevier, vol. 120(4), pages 381-402, April.
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    Cited by:

    1. Benabdallah Mohsine & Hiderah Kamal, 2018. "Strong rate of convergence for the Euler–Maruyama approximation of one-dimensional stochastic differential equations involving the local time at point zero," Monte Carlo Methods and Applications, De Gruyter, vol. 24(4), pages 249-262, December.
    2. Olivier Menoukeu-Pamen & Youssef Ouknine & Ludovic Tangpi, 2019. "Pathwise Uniqueness of Non-uniformly Elliptic SDEs with Rough Coefficients," Journal of Theoretical Probability, Springer, vol. 32(4), pages 1892-1908, December.
    3. Meier, Christian & Li, Lingfei & Zhang, Gongqiu, 2023. "Simulation of multidimensional diffusions with sticky boundaries via Markov chain approximation," European Journal of Operational Research, Elsevier, vol. 305(3), pages 1292-1308.

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