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Propagation of chaos for doubly mean reflected BSDEs

Hanwu Li, Ning Ning

Abstract

In this paper, we establish propagation of chaos (POC) for doubly mean reflected backward stochastic differential equations (MRBSDEs). MRBSDEs differentiate the typical RBSDEs in that the constraint is not on the paths of the solution but on its law. This unique property has garnered significant attention since the inception of MRBSDEs. Rather than directly investigating these equations, we focus on approximating them by interacting particle systems (IPS). We propose two sets of IPS having mean-field Skorokhod problems, capturing the dynamics of IPS reflected in a mean-field way. As the dimension of the IPS tends to infinity, the POC phenomenon emerges, indicating that the system converges to a limit with independent particles, where each solves the MRBSDE. Beyond establishing the first POC result for doubly MRBSDEs, we achieve distinct convergence speeds under different scenarios.

Propagation of chaos for doubly mean reflected BSDEs

Abstract

In this paper, we establish propagation of chaos (POC) for doubly mean reflected backward stochastic differential equations (MRBSDEs). MRBSDEs differentiate the typical RBSDEs in that the constraint is not on the paths of the solution but on its law. This unique property has garnered significant attention since the inception of MRBSDEs. Rather than directly investigating these equations, we focus on approximating them by interacting particle systems (IPS). We propose two sets of IPS having mean-field Skorokhod problems, capturing the dynamics of IPS reflected in a mean-field way. As the dimension of the IPS tends to infinity, the POC phenomenon emerges, indicating that the system converges to a limit with independent particles, where each solves the MRBSDE. Beyond establishing the first POC result for doubly MRBSDEs, we achieve distinct convergence speeds under different scenarios.
Paper Structure (12 sections, 11 theorems, 189 equations)

This paper contains 12 sections, 11 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Doubly MRBSDE
  3. Our contributions
  4. Organization and notation
  5. The case of linear reflection
  6. The particle system with constant driver
  7. The particle system with non-constant driver
  8. Approximation
  9. The case of nonlinear reflection
  10. The particle system with constant driver
  11. The particle system with non-constant driver
  12. Approximation

Key Result

Theorem 2.1

Under Conditions (A1) and (A2), the IPS eq7' has a unique solution $(\{\bar{Y}^i,\bar{Z}^i\}_{1\leq i\leq N},\bar{K}^{(N)})\in (\mathcal{S}^2(\mathbb{F}^{(N)};\,\mathbb{R})\times\mathcal{H}^2(\mathbb{F}^{(N)};\,\mathbb{R}^N))^N\times\mathcal{A}^2_{BV}(\mathbb{F}^{(N)};\,\mathbb{R})$.

Theorems & Definitions (11)

  • Theorem 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Proposition 2.4
  • Theorem 2.5
  • Lemma 2.6
  • Lemma 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Proposition 3.4
  • ...and 1 more