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Abstract McKean–Vlasov and Hamilton–Jacobi–Bellman Equations, Their Fractional Versions and Related Forward–Backward Systems on Riemannian Manifolds

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Abstract

We introduce 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 equation of stochastic control and games. This allows for a unified analysis of these equations, which leads to an effective theory of coupled forward–backward systems (forward McKean–Vlasov evolution and backward Hamilton–Jacobi–Bellman–Isaacs evolution) that are central to the modern theory of mean-field games.

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Acknowledgments

The authors are grateful to the referee for going through this article very carefully and providing them with a number of useful comments.

Funding

The work of V. N. Kolokoltsov (Sections 1–7) was supported by the Russian Science Foundation under grant no. 20-11-20119. The work of M. S. Troeva (Sections 8–10) was supported by the Ministry of Science and Higher Education of the Russian Federation (grant no. FSRG-2020-0006).

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Authors and Affiliations

  1. Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK

    V. N. Kolokoltsov

  2. National Research University Higher School of Economics, Pokrovskii bul. 11, Moscow, 109028, Russia

    V. N. Kolokoltsov

  3. North-Eastern Federal University, ul. Belinskogo 58, Yakutsk, 677000, 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.

Additional information

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2021, Vol. 315, pp. 128–150 https://doi.org/10.4213/tm4235.

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Kolokoltsov, V.N., Troeva, M.S. Abstract McKean–Vlasov and Hamilton–Jacobi–Bellman Equations, Their Fractional Versions and Related Forward–Backward Systems on Riemannian Manifolds. Proc. Steklov Inst. Math. 315, 118–139 (2021). https://doi.org/10.1134/S0081543821050096

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  • DOI: https://doi.org/10.1134/S0081543821050096

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