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Fractional McKean–Vlasov and Hamilton–Jacobi–Bellman–Isaacs Equations

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Abstract

We study a class of abstract nonlinear fractional pseudo-differential equations in Banach spaces that includes both the McKean–Vlasov type equations describing nonlinear Markov processes and the Hamilton–Jacobi–Bellman–Isaacs (HJB–Isaacs) equations of stochastic control and games. This approach allows us to develop a unified analysis of these equations. We establish their well-posedness in the sense of classical solutions and prove the continuous dependence of the solutions on the initial data. The obtained results are extended to the case of generalized fractional equations.

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Funding

The research of the first author (Sections 1, 4, and 5) was supported by the Russian Science Foundation (project no. 20-11-20119), and the research of the second author (Sections 2, 3, and 6) was supported by the Ministry of Science and Higher Education of the Russian Federation (project no. FSRG-2020-0006).

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Authors and Affiliations

  1. National Research University Higher School of Economics, 101000, Moscow, Russia

    V. N. Kolokoltsov

  2. St. Petersburg State University, 199034, St. Petersburg, Russia

    V. N. Kolokoltsov

  3. Federal Research Center “Informatics and Control” of the Russian Academy of Sciences, 119333, Moscow, Russia

    V. N. Kolokoltsov

  4. North-Eastern Federal University, 677027, Yakutsk, Russia

    M. S. Troeva

Authors
  1. V. N. Kolokoltsov
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  2. M. S. Troeva
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Correspondence to V. N. Kolokoltsov or M. S. Troeva.

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The authors declare that they have no conflicts of interest.

Additional information

Translated from Trudy Instituta Matematiki i Mekhaniki UrO RAN, Vol. 27, No. 3, pp. 87 - 100, 2021 https://doi.org/10.21538/0134-4889-2021-27-3-87-100.

Translated by I. Tselishcheva

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Kolokoltsov, V.N., Troeva, M.S. Fractional McKean–Vlasov and Hamilton–Jacobi–Bellman–Isaacs Equations. Proc. Steklov Inst. Math. 315 (Suppl 1), S165–S177 (2021). https://doi.org/10.1134/S0081543821060134

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