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Mean-field reflected BSDEs driven by a marked point process

Yiqing Lin, Kun Xu

Abstract

In this paper, we study a class of mean-field reflected backward stochastic differential equations (MFRBSDEs) driven by a marked point process. Based on a g-expectation representation lemma, we give the existence and uniqueness of MFRBSDEs driven by a marked point process under Lipschitz generator conditions. Besides, the well-posedness of this kind of BSDEs with exponential growth generator and unbounded terminal is also provided by $θ$-method.

Mean-field reflected BSDEs driven by a marked point process

Abstract

In this paper, we study a class of mean-field reflected backward stochastic differential equations (MFRBSDEs) driven by a marked point process. Based on a g-expectation representation lemma, we give the existence and uniqueness of MFRBSDEs driven by a marked point process under Lipschitz generator conditions. Besides, the well-posedness of this kind of BSDEs with exponential growth generator and unbounded terminal is also provided by -method.
Paper Structure (5 sections, 9 theorems, 126 equations)

This paper contains 5 sections, 9 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lipschitz case
  4. Unbounded terminal condition and obstacle
  5. Proofs of Lemma \ref{['estimate 1']} and Lemma \ref{['estimate 2']}

Key Result

Proposition 3.1

Let $T>0$ and let $f^1$ be a Lipschitz driver with Lipschitz constant $C$ and let $f^2$ be a driver. For $i=1,2$, let $\left(Y^i, U^i\right)$ be a solution of the BSDE associated to: For $s$ in $[0, T]$, denote $\bar{Y}_s:=Y_s^1-Y_s^2, \bar{U}_s:=U_s^1-U_s^2, \bar{\xi} := \xi^1-\xi^2$, and $\bar{f}(s):=f^1\left(s, Y_s^2, U_s^2\right)-f^2\left(s, Y_s^2, U_s^2\right)$. Let $\eta, \beta>0$ be such t

Theorems & Definitions (23)

  • Proposition 3.1
  • proof
  • Lemma 3.2: 2023arXiv231020361L lemma 4.1
  • Remark 3.3
  • Theorem 3.4
  • proof
  • Theorem 4.1
  • Remark 4.2
  • Lemma 4.3
  • Remark 4.4
  • ...and 13 more