V.N. Kolokoltsov, M.S. Troeva. Fractional McKean–Vlasov and HJB-Isaaсs equations ... P. 87-100

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–Isaaсs equations of stochastic control and games. This approach allows us to develop a unified analysis of these equations. We obtain the well-posedness results for these equations in the sense of classical solutions, and their continuous dependence on the initial data is proved. The obtained results are extended to the case of generalized fractional equations.

Keywords: fractional McKean–Vlasov type equations, fractional HJB-Isaaсs equations, mild solutions, classical solutions,  Caputo–Djrbashian fractional derivative, generalized fractional derivatives

Received April 30, 2021

Revised June 21, 2021

Accepted July 19, 2021

Funding Agency: The work of V.N. Kolokoltsov (Sections 1, 4 and 5) was supported by the Russian Science Foundation (project no. 20-11-20119), and the work of M.S.Troeva (Sections 2, 3 and 6) was supported by the Ministry of Science and Higher Education of the Russian Federation (project no. FSRG-2020-0006).

Vassili Nikitich Kolokoltsov, Dr. Phys.-Math. Sci., National Research University Higher School of Economics, Moscow, 109028 Russia; Saint-Petersburg State University, Saint Petersburg, 198504 Russia; Federal Research Center “Computer Science and Control”, RAS, Moscow, 119333 Russia, e-mail: kolokoltsov59@mail.ru

Marianna Stepanovna Troeva, Cand. Sci. (Phys.-Math.), Research Institute of Mathematics, North-Eastern Federal University, Yakutsk, 677000 Russia, e-mail: troeva@mail.ru

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Cite this article as: V.N. Kolokoltsov, M.S. Troeva. Fractional McKean–Vlasov and HJB–Isaaсs equations, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2021, vol. 27, no. 3, pp. 87–100; Proceedings of the Steklov Institute of Mathematics  (Suppl.), 2021, Vol. 315, Suppl. 1, pp. S165–S177.