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One-Dimensional McKean-Vlasov Stochastic Variational Inequalities and Coupled BSDEs with Locally Holder Noise Coefficients

Ning Ning, Jing Wu, Jinwei Zheng

Abstract

In this article, we investigate three classes of equations: the McKean-Vlasov stochastic differential equation (MVSDE), the MVSDE with a subdifferential operator referred to as the McKean-Vlasov stochastic variational inequality (MVSVI), and the coupled forward-backward MVSVI. The latter class encompasses the FBSDE with reflection in a convex domain as a special case. We establish the well-posedness, in terms of the existence and uniqueness of a strong solution, for these three classes in their general forms. Importantly, we consider stochastic coefficients with locally Holder continuity and employ different strategies to achieve that for each class.

One-Dimensional McKean-Vlasov Stochastic Variational Inequalities and Coupled BSDEs with Locally Holder Noise Coefficients

Abstract

In this article, we investigate three classes of equations: the McKean-Vlasov stochastic differential equation (MVSDE), the MVSDE with a subdifferential operator referred to as the McKean-Vlasov stochastic variational inequality (MVSVI), and the coupled forward-backward MVSVI. The latter class encompasses the FBSDE with reflection in a convex domain as a special case. We establish the well-posedness, in terms of the existence and uniqueness of a strong solution, for these three classes in their general forms. Importantly, we consider stochastic coefficients with locally Holder continuity and employ different strategies to achieve that for each class.
Paper Structure (11 sections, 11 theorems, 190 equations)

This paper contains 11 sections, 11 theorems, 190 equations.

Table of Contents

  1. Introduction
  2. McKean-Vlasov stochastic differential equations
  3. McKean-Vlasov stochastic variational inequality
  4. McKean-Vlasov forward-backward stochastic variational system
  5. Organization of the paper
  6. Preliminaries
  7. Notations
  8. Properties
  9. Well-posedness of the MVSDE
  10. Well-posedness of the MVSVI
  11. Well-posedness of the MVFBSVS

Key Result

Theorem 2.1

The Yosida-Moreau approximation function $\psi^n$ defined in eqn:Yosida-Moreau approximation and $J_n$ defined in eqn:Jnx satisfy that, for any $x,y\in \mathbb{R}$, Here, $\Pi_{\overline{D}}(x)$ denotes the projection of $x$ onto $\overline{D}$, where

Theorems & Definitions (22)

  • Theorem 2.1: barbu2010nonlinear
  • Theorem 2.2: rockafellar1970maximal
  • Theorem 2.3: yamada1971uniqueness
  • Lemma 2.4
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • Definition 4.2
  • ...and 12 more