Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$\mathbb{L}^p$-solution of generalized BSDEs in a general filtration with stochastic monotone coefficients

Badr Elmansouri, Mohamed El Otmani

TL;DR

The paper develops existence and uniqueness results for multidimensional generalized BSDEs in a general filtration that supports a Brownian motion, proving $\mathbb{L}^p$-solvability for $p\in(1,2]$ under stochastic monotone conditions in $y$, stochastic Lipschitz in $z$, and stochastic linear growth. The authors introduce weighted function spaces to accommodate the Brownian and orthogonal martingale components and establish robust a priori estimates using a generalized Tanaka formula, Kunita–Watanabe inequalities, and BDG bounds. Existence is achieved first in $\mathbb{L}^2$ via a Yosida approximation for the $f$ independent of $z$ case and a fixed-point argument for the general case, then extended to $\mathbb{L}^p$ with $p\in(1,2)$, including a discussion on stopping times and potential extensions to $p>2$. The results fill a gap in the literature by treating GBSDEs in a general filtration with an orthogonal martingale term, broadening applicability to stochastic monotone drivers and linear growth, with implications for probabilistic representations of PDEs with Neumann-type conditions in non-Brownian filtrations.

Abstract

We study multidimensional generalized backward stochastic differential equations (GBSDEs) within a general filtration that supports a Brownian motion under weak assumptions on the associated data. We establish the existence and uniqueness of solutions in $\mathbb{L}^p$ for $p \in (1,2]$. Our results apply to generators that are stochastic monotone in the $y$-variable, stochastic Lipschitz in the $z$-variable, and satisfy a general stochastic linear growth condition.

$\mathbb{L}^p$-solution of generalized BSDEs in a general filtration with stochastic monotone coefficients

TL;DR

The paper develops existence and uniqueness results for multidimensional generalized BSDEs in a general filtration that supports a Brownian motion, proving -solvability for under stochastic monotone conditions in , stochastic Lipschitz in , and stochastic linear growth. The authors introduce weighted function spaces to accommodate the Brownian and orthogonal martingale components and establish robust a priori estimates using a generalized Tanaka formula, Kunita–Watanabe inequalities, and BDG bounds. Existence is achieved first in via a Yosida approximation for the independent of case and a fixed-point argument for the general case, then extended to with , including a discussion on stopping times and potential extensions to . The results fill a gap in the literature by treating GBSDEs in a general filtration with an orthogonal martingale term, broadening applicability to stochastic monotone drivers and linear growth, with implications for probabilistic representations of PDEs with Neumann-type conditions in non-Brownian filtrations.

Abstract

We study multidimensional generalized backward stochastic differential equations (GBSDEs) within a general filtration that supports a Brownian motion under weak assumptions on the associated data. We establish the existence and uniqueness of solutions in for . Our results apply to generators that are stochastic monotone in the -variable, stochastic Lipschitz in the -variable, and satisfy a general stochastic linear growth condition.
Paper Structure (11 sections, 12 theorems, 106 equations)

This paper contains 11 sections, 12 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conditions on the data $(\xi,f,g,\kappa)$
  4. A priori estimates and uniqueness
  5. Existence
  6. Existence of $\mathbb{L}^2$-solutions
  7. A priori estimates and uniqueness
  8. Existence
  9. The case when the coefficient $f$ is independent of $z$
  10. General case
  11. Existence of $\mathbb{L}^p$-solutions for $p \in (1,2)$

Key Result

Lemma 5

Let $(F_t)_{t \leq T}$ and $(Z_t)_{t \leq T}$ be two progressively measurable processes with values respectively in $\mathbb{R}^d$ and $\mathbb{R}^{d \times k}$ such that $\mathbb{P}$-a.s. We consider the $\mathbb{R}^d$-valued semimartingale $(X_t)_{t \leq T}$ defined by Then, for any $p \geq 1$, the process $(|X_t|^p)_{t \leq T}$ is an $\mathbb{R}$-semimartingale with the following decompositio

Theorems & Definitions (16)

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