Existence and uniqueness theorems for solutions of McKean–Vlasov stochastic equations

Authors:
Yuliya Mishura and Alexander Veretennikov

Journal:
Theor. Probability and Math. Statist. **103** (2020), 59-101

MSC (2020):
Primary 60H10; Secondary 60E99, 60F17, 60H05, 60J60

DOI:
https://doi.org/10.1090/tpms/1135

Published electronically:
June 16, 2021

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Abstract: New weak and strong existence and weak and strong uniqueness results for the solutions of multi-dimensional stochastic McKean–Vlasov equation are established under relaxed regularity conditions. Weak existence requires a non-degeneracy of diffusion and no more than a linear growth of both coefficients in the state variable. Weak and strong uniqueness are established under the restricted assumption of diffusion, yet without any regularity of the drift; this part is based on the analysis of the total variation metric.

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Additional Information

**Yuliya Mishura**

Affiliation:
Taras Shevchenko National University of Kyiv

Email:
myus@univ.kiev.ua

**Alexander Veretennikov**

Affiliation:
University of Leeds, United Kingdom; and National Research University Higher School of Economics, Russian Federation, and Institute for Information Transmission Problems, Moscow, Russia

Email:
alexander.veretennikov2011@yandex.ru

Keywords:
Stochastic Itô–McKean–Vlasov equation,
weak and strong existence,
weak and strong uniqueness,
relaxed regularity conditions

Received by editor(s):
December 12, 2019

Published electronically:
June 16, 2021

Additional Notes:
For the second author this research has been funded by the Russian Academic Excellence Project ’5-100’ (Proposition 1, Lemma 3) and by the Russian Science Foundation project 17-11-01098 (Theorem 2). Certain stages of this work had been fulfilled while the second author was visiting Bielefeld University within the programme SFB1283. The author appreciates this opportunity very much.

Dedicated:
In memory of A.V. Skorokhod (10.09.1930 – 03.01.2011)

Article copyright:
© Copyright 2020
Taras Shevchenko National University of Kyiv