Some results on general quadratic reflected BSDEs driven by a continous martingale

TL;DR

This paper investigates the well-posedness of quadratic reflected backward stochastic differential equations driven by a continuous martingale, establishing key results like comparison, uniqueness, and existence under minimal assumptions.

Contribution

It provides new intrinsic proofs for comparison theorems and extends existence results to drivers with quadratic and superlinear growth, using fixed point and perturbation methods.

Findings

Comparison and uniqueness results established

Existence proved for drivers with quadratic growth

Local Lipschitz estimates in BMO obtained

Abstract

We study the well-posedness of general reflected BSDEs driven by a continuous martingale, when the coefficient f of the driver has at most quadratic growth in the control variable Z, with a bounded terminal condition and a lower obstacle which is bounded above. We obtain the basic results in this setting : comparison and uniqueness, existence, stability. For the comparison theorem and the special comparison theorem for reflected BSDEs (which allows one to compare the increasing processes of two solutions), we give intrinsic proofs which do not rely on the comparison theorem for standard BSDEs. This allows to obtain the special comparison theorem under minimal assumptions. We obtain existence by using the fixed point theorem and then a series of perturbations, first in the case where f is Lipschitz in the primary variable Y, and then in the case where f can have slightly-superlinear growth and the case where f is monotonous in Y with arbitrary growth. We also obtain a local Lipschitz estimate in BMO for the martingale part of the solution.