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Well-Posedness for Path-Distribution Dependent Stochastic Differential Equations with Singular Drifts

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  • Xiao-Yu Zhao

    (Tianjin University)

Abstract

Well-posedness is derived for singular path-distribution dependent stochastic differential equations (SDEs) with non-degenerate noise, where the drift is allowed to be singular in the current state, but maintains local Lipschitz continuity in the historical path, and the coefficients are Lipschitz continuous with respect to a weighted variation distance in the distribution variable. Notably, this result is new even for classical path-dependent SDEs where the coefficients are distribution independent. Moreover, by strengthening the local Lipschitz continuity to Lipschitz continuity and replacing the weighted variation distance with the Wasserstein distance, we also obtain well-posedness.

Suggested Citation

  • Xiao-Yu Zhao, 2024. "Well-Posedness for Path-Distribution Dependent Stochastic Differential Equations with Singular Drifts," Journal of Theoretical Probability, Springer, vol. 37(4), pages 3654-3687, November.
    • Handle: RePEc:spr:jotpro:v:37:y:2024:i:4:d:10.1007_s10959-024-01356-y
    DOI: 10.1007/s10959-024-01356-y
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    References listed on IDEAS

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    1. Wang, Feng-Yu, 2018. "Distribution dependent SDEs for Landau type equations," Stochastic Processes and their Applications, Elsevier, vol. 128(2), pages 595-621.
    2. Ren, Panpan, 2023. "Singular McKean–Vlasov SDEs: Well-posedness, regularities and Wang’s Harnack inequality," Stochastic Processes and their Applications, Elsevier, vol. 156(C), pages 291-311.
    3. 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.
    4. Wang, Feng-Yu & Yuan, Chenggui, 2011. "Harnack inequalities for functional SDEs with multiplicative noise and applications," Stochastic Processes and their Applications, Elsevier, vol. 121(11), pages 2692-2710, November.
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