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One-dimensional stochastic differential equations with generalized and singular drift

Author

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  • Blei, Stefan
  • Engelbert, Hans-Jürgen

Abstract

Introducing certain singularities, we generalize the class of one-dimensional stochastic differential equations with so-called generalized drift. Equations with generalized drift, well-known in the literature, possess a drift that is described by the semimartingale local time of the unknown process integrated with respect to a locally finite signed measure ν. The generalization which we deal with can be interpreted as allowing more general set functions ν, for example signed measures which are only σ-finite. However, we use a different approach to describe the singular drift. For the considered class of one-dimensional stochastic differential equations, we derive necessary and sufficient conditions for existence and uniqueness in law of solutions.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:12:p:4337-4372
    DOI: 10.1016/j.spa.2013.06.014
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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. Bachmann, Stefan, 2020. "On the strong Feller property for stochastic delay differential equations with singular drift," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 4563-4592.
    3. 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.
    4. Pajor-Gyulai, Zs. & Salins, M., 2017. "On dynamical systems perturbed by a null-recurrent motion: The general case," Stochastic Processes and their Applications, Elsevier, vol. 127(6), pages 1960-1997.

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