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$\mathbb{L}^p$ $(p>1)$-solutions for BSDEs with jumps and stochastic monotone generator

Badr Elmansouri, Mohamed El Otmani

Abstract

We study multidimensional discontinuous backward stochastic differential equations in a filtration that supports both a Brownian motion and an independent integer-valued random measure. Under suitable $\mathbb{L}^p$-integrability conditions on the data, we establish the existence and uniqueness of $\mathbb{L}^p$-solutions for both cases: $p \geq 2$ and $p \in (1,2)$. The generator is assumed to be stochastically monotone in the state variable $y$, stochastically Lipschitz in the control variables $(z, u)$, and to satisfy a stochastic linear growth condition, along with an appropriate $\mathbb{L}^p$-integrability requirement.

$\mathbb{L}^p$ $(p>1)$-solutions for BSDEs with jumps and stochastic monotone generator

Abstract

We study multidimensional discontinuous backward stochastic differential equations in a filtration that supports both a Brownian motion and an independent integer-valued random measure. Under suitable -integrability conditions on the data, we establish the existence and uniqueness of -solutions for both cases: and . The generator is assumed to be stochastically monotone in the state variable , stochastically Lipschitz in the control variables , and to satisfy a stochastic linear growth condition, along with an appropriate -integrability requirement.

Paper Structure

This paper contains 13 sections, 15 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conditions on the data $(\xi,f)$:
  4. A priori estimates
  5. $\mathbb{L}^p$-estimates for $p \in [2, +\infty)$
  6. $\mathbb{L}^p$-estimates for $p =2$
  7. $\mathbb{L}^p$-estimates for $p \in (2, +\infty)$
  8. $\mathbb{L}^p$-estimates for $p \in (1,2)$
  9. Existence and uniqueness of $\mathbb{L}^p$-solutions for $p \in (1,+\infty)$
  10. $\mathbb{L}^p$-solutions for $p \in [2,+\infty)$
  11. $\mathbb{L}^p$-solutions for $p \in (1,2)$
  12. The case where the generator $f$ does not depend on the $(z,u)$-variables
  13. The general case where the generator $f$ depends on the $(z,u)$-variables

Key Result

Theorem 2

Any $\mathbb{F}$-local martingale has the decomposition where $Z \in L^2_{loc}(W)$ and $U \in G_{loc}(N)$.

Theorems & Definitions (22)

  • Remark 1
  • Theorem 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Definition 6
  • Proposition 7
  • Remark 8
  • Lemma 9
  • Proposition 10
  • ...and 12 more