Reflected backward stochastic differential equations and a class of non linear dynamic pricing rule

TL;DR

This paper introduces a new way to characterize solutions of certain reflected backward stochastic differential equations with convex quadratic drivers, linking them to dynamic monetary functionals and ensuring time consistency.

Contribution

It develops an extended $g$-Snell envelope framework for RBSDEs with convex quadratic drivers and connects it to dynamic monetary concave functionals, preserving time consistency.

Findings

New characterization of RBSDE solutions with convex quadratic drivers

Extension of $g$-Snell envelope concept for these equations

Establishment of time consistency in the associated dynamic functionals

Abstract

In that paper, we provide a new characterization of the solutions of specific reflected backward stochastic differential equations (or RBSDEs) whose driver $g$ is convex and has quadratic growth in its second variable: this is done by introducing the extended notion of $g$-Snell enveloppe. Then, in a second step, we relate this representation to a specific class of dynamic monetary concave functionals already introduced in a discrete time setting. This connection implies that the solution, characterized by means of non linear expectations, has again the time consistency property.