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Stochastic Differential Equations with Singular Coefficients: The Martingale Problem View and the Stochastic Dynamics View

Author

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  • Elena Issoglio

    (Universitá di Torino)

  • Francesco Russo

    (Unité de Mathématiques appliquées, ENSTA Paris, Institut Polytechnique de Paris)

Abstract

We consider stochastic differential equations (SDEs) with (distributional) drift in negative Besov spaces and random initial condition and investigate them from two different viewpoints. In the first part we set up a martingale problem and show its well-posedness. We then prove further properties of the martingale problem, such as continuity with respect to the drift and the link with the Fokker–Planck equation. We also show that the solutions are weak Dirichlet processes for which we evaluate the quadratic variation of the martingale component. In the second part we identify the dynamics of the solution of the martingale problem by describing the proper associated SDE. Under suitable assumptions we show equivalence with the solution to the martingale problem.

Suggested Citation

  • Elena Issoglio & Francesco Russo, 2024. "Stochastic Differential Equations with Singular Coefficients: The Martingale Problem View and the Stochastic Dynamics View," Journal of Theoretical Probability, Springer, vol. 37(3), pages 2352-2393, September.
    • Handle: RePEc:spr:jotpro:v:37:y:2024:i:3:d:10.1007_s10959-024-01325-5
    DOI: 10.1007/s10959-024-01325-5
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    References listed on IDEAS

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    1. Gozzi, Fausto & Russo, Francesco, 2006. "Weak Dirichlet processes with a stochastic control perspective," Stochastic Processes and their Applications, Elsevier, vol. 116(11), pages 1563-1583, November.
    2. Russo, Francesco & Vallois, Pierre, 1995. "The generalized covariation process and Ito formula," Stochastic Processes and their Applications, Elsevier, vol. 59(1), pages 81-104, September.
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