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Coupled forward-backward stochastic differential equations with jumps in random environments

Daniel Hernández-Hernández, Joshué Helí Ricalde-Guerrero

Abstract

In this paper we obtain results for the existence and uniqueness of solutions to coupled Forward-Backward Stochastic Differential Equations (FBSDEs) with jumps defined on a random environment. This environment corresponds to a measured-valued process, similar to the one found in Conditional McKean-Vlasov Differential Equations and Mean-Field Games with Common Noise. The jump term in the FBSDE is dependent on the environment through a stochastic intensity process. We provide examples which relate our model with FBSDEs driven by Cox and Hawkes processes, as well as regime-switching Conditional McKean-Vlasov differential equations.

Coupled forward-backward stochastic differential equations with jumps in random environments

Abstract

In this paper we obtain results for the existence and uniqueness of solutions to coupled Forward-Backward Stochastic Differential Equations (FBSDEs) with jumps defined on a random environment. This environment corresponds to a measured-valued process, similar to the one found in Conditional McKean-Vlasov Differential Equations and Mean-Field Games with Common Noise. The jump term in the FBSDE is dependent on the environment through a stochastic intensity process. We provide examples which relate our model with FBSDEs driven by Cox and Hawkes processes, as well as regime-switching Conditional McKean-Vlasov differential equations.
Paper Structure (7 sections, 9 theorems, 110 equations)

This paper contains 7 sections, 9 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Notation and Basic Notions
  3. Main Results
  4. Hamiltonian systems
  5. Preliminaries from uncoupled forward and backward equations
  6. Forward-Backward SDEs with Random Environment and Jumps
  7. Some Examples

Key Result

Theorem 3.1

Assume that $\mu$ is a random environment on $(\Omega^0,\mathcal{F}^0,\mathbb{P}^0)$, and define $\mathbb{F}^{\mu,N}:=\mathbb{F}^{0,\mu}\vee \mathbb{F}^{0,N}$, where $\mathbb{F}^{0,N}$ is the filtration generated by Let $\lambda : \Omega \times [0,T] \times \mathbf{R} \to \mathbb{R}^l$ satisfy the following: Then, there exists an admissible set-up $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P},X_0,\

Theorems & Definitions (29)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • Proposition 4.1
  • Remark 4.2
  • ...and 19 more