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

Mishura, Yuliya; Veretennikov, A. Yu.

doi:10.48550/arxiv.1603.02212

Cited by 58 publications

(95 citation statements)

References 18 publications

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“…In the pioneering work by Funaki [Fu84] the martingale problem for a non-linear PDE is clearly formulated. After that global well-posedness of DDSDE has been studied a lot in the literature (see [MV16] [Wa18] [RZ21] and references therein). In the case where there is a common environmental noise influencing each particles, this suggests particle systems with common noise like (1.7) and there are also a lot of work concerning the mean field limit of particle systems with common noise and the limiting DDSDE (see e.g.…”

Section: (M+1)mentioning

confidence: 99%

Singular kinetic equations and applications

Hao,

Zhang,

Zhu

et al. 2021

Preprint

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In this paper we study singular kinetic equations on R 2d by the paracontrolled distribution method introduced in [GIP15]. We first develop paracontrolled calculus in the kinetic setting, and use it to establish the global well-posedness for the linear singular kinetic equations under the assumptions that the products of singular terms are well-defined. We also demonstrate how the required products can be defined in the case that singular term is a Gaussian random field by probabilistic calculation. Interestingly, although the terms in the zeroth Wiener chaos of regularization approximation are not zero, they converge in suitable weighted Besov spaces and no renormalization is required. As applications the global well-posedness for a nonlinear kinetic equation with singular coefficients is obtained by the entropy method. Moreover, we also solve the martingale problem for nonlinear kinetic distribution dependent stochastic differential equations with singular drifts. Contents 1. Introduction 1.1. Main results 1.2. Sketch of proofs and structure of the paper 1.3. Further relevant literature 1.4. Notations and conventions 2. Preliminaries 2.1. Weighted anisotropic Besov spaces 2.2. Paraproduct calculus 3. Kinetic semigroups and commutator estimates 3.1. Kinetic semigroup estimates 3.2. Kinetic Hölder spaces and characterization 3.3. Schauder's estimates 3.4. Commutator estimates 3.5. Renormalized pairs 4. Linear kinetic equations with distribution drifts 4.1. Paracontrolled solutions 4.2. Well-posedness for (4.1) 5. Well-posedness of singular mean field equations 6. Nonlinear martingale problem with singular drifts 7. Existence of renormalized pairs in probabilistic sense 7.1. Statement of main result 7.2. Examples for (A β ) 7.3. Proof of Theorem 7.2 Appendix A. Characterizations for B s,a p,q (ρ) Appendix B. Proof of Lemma 7.4 References

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Section: (M+1)mentioning

confidence: 99%

Singular kinetic equations and applications

Hao,

Zhang,

Zhu

et al. 2021

Preprint

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“…There are many results on the strong and weak well-posedness of McKean-Vlasov SDEs, see [7,8,11,12,17,18,22,23,25,34] and references therein. As stated in the introduction, we only consider the strong well-posedness for the following SDEs, where the drift is linear and the diffusion is Hölder continuous in the space variable.…”

Section: Strong Well-posednessmentioning

confidence: 99%

Log-Harnack Inequality and Exponential Ergodicity for Distribution Dependent CKLS and Vasicek Model

Bai

Huang

2021

Preprint

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In this paper, Wang's log-Harnack inequality and exponential ergodicity are derived for two types of distribution dependent SDEs: one is the CKLS model, where the diffusion coefficient is a power function of order θ with θ ∈ [ 1 2 , 1); the other one is Vasicek model, where the diffusion coefficient only depends on distribution. Both models in the distribution independent case are used to characterize the interest rate in finance.

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“…[26], where the author invokes martingale techniques to construct unique weak solutions in the case of a bounded drift b, which is Lipschitz continuous in the law variable. We also refer to [20], [35] in the case of weak solutions. As for path-dependent coefficients see [31].…”

Section: Introductionmentioning

confidence: 99%

Stability, uniqueness and existence of solutions to McKean-Vlasov SDEs: a multidimensional Yamada-Watanabe approach

Kalinin,

Meyer-Brandis,

Proske

2021

Preprint

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We establish stability and pathwise uniqueness of solutions to Wiener noise driven McKean-Vlasov equations with random coefficients, which are allowed to be non-Lipschitz continuous. In the case of deterministic coefficients we also obtain the existence of unique strong solutions. By using our approach, which is based on an extension of the Yamada-Watanabe ansatz to the multidimensional setting and which does not rely on the construction of Lyapunov functions, we prove first moment and pathwise α-exponential stability of solutions for α > 0. Furthermore, we are able to compute Lyapunov exponents explicitly.

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

Cited by 58 publications

References 18 publications

Singular kinetic equations and applications

Singular kinetic equations and applications

Log-Harnack Inequality and Exponential Ergodicity for Distribution Dependent CKLS and Vasicek Model

Stability, uniqueness and existence of solutions to McKean-Vlasov SDEs: a multidimensional Yamada-Watanabe approach

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