Reflected BSDEs with monotone generator

Tomasz Klimsiak

doi:10.1214/ejp.v17-1759

Abstract

We give necessary and sufficient condition for existence and uniqueness of $\mathbb{L}^{p}$-solutions of reflected BSDEs with continuous barrier, generator monotone with respect to $y$ and Lipschitz continuous with respect to $z$, and with data in $\mathbb{L}^{p}$, $p\ge 1$. We also prove that the solutions maybe approximated by the penalization method.

Highlights

Let B be a standard d-dimensional Brownian motion defined on some probability space (Ω, F, P ) and let {Ft} denote the augmentation of the natural filtration generated by B
In the present paper we study the problem of existence, uniqueness and approximation of Lp-solutions of reflected backward stochastic differential equations (RBSDEs for short) with monotone generator of the form
If f satisfies (H2), (H3) and (Z) there exists at most one solution (Y, Z, K) of RBSDE(ξ, f, L) such that Y is of class (D) and Z ∈ β>α M β

Summary

Introduction

Let B be a standard d-dimensional Brownian motion defined on some probability space (Ω, F , P ) and let {Ft} denote the augmentation of the natural filtration generated by B. In the present paper we study the problem of existence, uniqueness and approximation of Lp-solutions of reflected backward stochastic differential equations (RBSDEs for short) with monotone generator of the form. L∗T supt≤T |Lt| are square-integrable, f satisfies the linear growth condition and is Lipschitz continuous with respect to both variables y and z These assumptions are too strong for many interesting applications. Let us mention that in the case p = 2 existence and uniqueness results are known for equations with generators satisfying even weaker regularity conditions. In our main result of the paper we give a necessary and sufficient condition for existence and uniqueness of Lp-integrable solution of RBSDE (1.1) under the assumptions that the data are Lp-integrable, f is monotone in y and Lipschitz continuous in z and (1.3) is satisfied.

Notation and hypotheses

A priori estimates

Some useful tools

Nonintegrable solutions of reflected BSDEs