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Well-posedness of density dependent SDE driven by α-stable process with Hölder drifts

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  • Wu, Mingyan
  • Hao, Zimo

Abstract

In this paper, we show the weak and strong well-posedness of density dependent stochastic differential equations driven by α-stable processes with α∈(1,2). The existence part is based on Euler’s approximation as Hao et al. (2021), while, the uniqueness is based on the Schauder estimates in Besov spaces for nonlocal Fokker–Planck equations. For the existence, we only assume the drift being continuous in the density variable. For the weak uniqueness, the drift is assumed to be Lipschitz in the density variable, while for the strong uniqueness, we also need to assume the drift being β0-order Hölder continuous in the spatial variable, where β0∈(1−α/2,1).

Suggested Citation

  • Wu, Mingyan & Hao, Zimo, 2023. "Well-posedness of density dependent SDE driven by α-stable process with Hölder drifts," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 416-442.
    • Handle: RePEc:eee:spapps:v:164:y:2023:i:c:p:416-442
    DOI: 10.1016/j.spa.2023.07.016
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    References listed on IDEAS

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    1. Benazzoli, Chiara & Campi, Luciano & Di Persio, Luca, 2020. "Mean field games with controlled jump–diffusion dynamics: Existence results and an illiquid interbank market model," Stochastic Processes and their Applications, Elsevier, vol. 130(11), pages 6927-6964.
    2. Xia, Pengcheng & Xie, Longjie & Zhang, Xicheng & Zhao, Guohuan, 2020. "Lq(Lp)-theory of stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 5188-5211.
    3. Kühn, Franziska & Schilling, René L., 2019. "Strong convergence of the Euler–Maruyama approximation for a class of Lévy-driven SDEs," Stochastic Processes and their Applications, Elsevier, vol. 129(8), pages 2654-2680.
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