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

Yuliya S. Mishura, Alexander Yu. Veretennikov

Abstract

New weak and strong existence and weak and strong uniqueness results for multi-dimensional stochastic McKean--Vlasov equations are established under relaxed regularity conditions. Weak existence is a variation of Krylov's weak existence for Itô's SDEs under the nondegeneracy of diffusion and no more than a linear growth in the state variable; this part is designed to fill in the existing gap, as earlier such results for McKean-Vlasov equations were not written. Weak and strong uniqueness is established under the restricted assumption of diffusion depending only on time and the state variable, yet without any regularity of the drift in the state variable and also under a linear growth condition on the drift; this part is based on the analysis of the total variation metric.

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

Abstract

New weak and strong existence and weak and strong uniqueness results for multi-dimensional stochastic McKean--Vlasov equations are established under relaxed regularity conditions. Weak existence is a variation of Krylov's weak existence for Itô's SDEs under the nondegeneracy of diffusion and no more than a linear growth in the state variable; this part is designed to fill in the existing gap, as earlier such results for McKean-Vlasov equations were not written. Weak and strong uniqueness is established under the restricted assumption of diffusion depending only on time and the state variable, yet without any regularity of the drift in the state variable and also under a linear growth condition on the drift; this part is based on the analysis of the total variation metric.

Paper Structure

This paper contains 11 sections, 10 theorems, 222 equations.

Table of Contents

  1. Introduction
  2. Weak existence
  3. Main results
  4. Proof of Proposition \ref{['pro22']}
  5. Proof of Theorem \ref{['thm1']}
  6. Strong solutions; strong and weak uniqueness
  7. On strong existence
  8. Weak uniqueness
  9. Proof of Theorem \ref{['thm5a']}
  10. Strong uniqueness
  11. Appendix

Key Result

Lemma 1

In terms of notation (e200), let the Borel coefficients $b(t,x,y)$ and $\sigma(t,x,y)$ for each $x$ satisfy with some locally bounded Borel function $C(x), \, x\in \mathbb R^d$, and let $\mu_t(dy)$ be a marginal distribution of any solution $X_t$ of the equation (e1). Then the functions $\widetilde{b}(t,x):= B[t, x, \mu_t]$ and $\widetilde{\sigma}(t,x):= \Sigma[t, x, \mu_t]$ are Borel measurable

Theorems & Definitions (23)

  • Definition 1
  • Lemma 1
  • proof
  • Theorem \oldthetheorem
  • Remark 1
  • Proposition 1
  • Remark 2
  • Remark 3
  • Proposition 2
  • Remark 4
  • ...and 13 more