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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Additional Information
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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References
- R. Bass, Diffusions and Elliptic Operators, Springer, New York, 1998. MR 1483890
- S. Benachour, B. Roynette, D. Talay, P. Vallois, Nonlinear self-stabilizing processes. I: Existence, invariant probability, propagation of chaos, Stochastic Processes Appl. 75 (1998), no. 2, 173–201. MR 1632193
- M. Bossy and D. Talay, A stochastic particle method for the McKean–Vlasov and the Burgers equation, Math. Comp. 66 (1997), 157–192. MR 1370849
- T. S. Chiang, McKean–Vlasov equations with discontinuous coefficients, Soochow J. Math. 20 (1994), no. 4, 507–526. MR 1309485
- D. Crisan and J. Xiong, Approximate McKean–Vlasov representations for a class of SPDEs, Stochastics 82 (2010), no. 1, 53–68. MR 2677539
- R. Dobrushin, Vlasov equations, Funct. Anal. Appl. 13 (1979), 115–123. MR 541637
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- I. I. Gihman and A. V. Skorohod, Stochastic differential equations, Springer, Berlin, 1972. MR 0346904
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- B. Jourdain, Diffusions with a nonlinear irregular drift coefficient and probabilistic interpretation of generalized Burgers’ equations, ESAIM Probab. Statist. 1 (1995/97), 339–355. MR 1476333
- B. Jourdain and S. Méléard, Propagation of chaos and fluctuations for a moderate model with smooth initial data, Ann. Inst. H. Poincaré Probab. Statist. 34 (1998), no. 6, 727–766. MR 1653393
- M. Kac, Foundations of kinetic theory, In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. III, University of California Press, Berkeley and Los Angeles, 1956, pp. 171–197. MR 0084985
- R. Z. Khasminskii, Stochastic Stability of Differential Equations, 2nd ed., Springer, Berlin, 2012. MR 2894052
- N. V. Krylov, On Ito’s stochastic integral equations, Theory Probab. Appl. 14 (1969), 330–336. MR 0270462
- N. V. Krylov, On the selection of a Markov process from a system of processes and the construction of quasi-diffusion processes, Izvestiya: Mathematics 7 (1973), no. 3, 691–709. MR 0339338
- N. V. Krylov, Controlled diffusion processes, 2nd ed., Springer, Berlin, 2009. MR 2723141
- N. V. Krylov, Introduction to the Theory of Random Processes, AMS, Providence, RI, 2002. MR 1885884
- T. G. Kurtz, Weak and strong solutions of general stochastic models, Electron. Commun. Probab. 19 (2014), 1–16. MR 3254737
- H. P. McKean, A class of Markov processes associated with nonlinear parabolic equations, Proc. Nat. Acad. Sci. U.S.A. 56 (1966), 1907–1911. MR 221595
- O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasi-linear Equations of Parabolic Type, AMS, RI, 1968. MR 0241821
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- A. Yu. Veretennikov, On strong solutions and explicit formulas for solutions of stochastic integral equations, Math. USSR Sb. 39 (1981), 387–403. MR 568986
- A. Yu. Veretennikov, Parabolic equations and Ito’s stochastic equations with coefficients discontinuous in the time variable, Math. Notes 31 (1982), 278–283.
- A. Yu. Veretennikov, On ergodic measures for McKean–Vlasov stochastic equations, In Monte Carlo and quasi-Monte Carlo methods 2004, Springer, Berlin, 2006, pp. 471–486. MR 2208726
- A. Yu. Veretennikov and N. V. Krylov, On explicit formulas for solutions of stochastic equations, Math. USSR Sb. 29 (1976), no. 2, 239–256. MR 0410921
- A. A. Vlasov, The vibrational properties of an electron gas, Physics-Uspekhi 10 (1968), no. 6, 721–733.
- T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ. 11 (1971), 155–167. MR 278420
- A. K. Zvonkin and N. V. Krylov, Strong solutions of stochastic differential equations, In Proceedings of the School and Seminar on the Theory of Random Processes (Druskininkai, 1974), Part II (Russian), Inst. Fiz. i Mat. Akad. Nauk Litovsk. SSR, Vilnius, 1975, pp. 9–88. MR 0426154
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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
