Second order reflected backward stochastic differential equations

TL;DR

This paper introduces second order reflected backward stochastic differential equations (2RBSDEs), establishing their existence, uniqueness, and applications to American option super-hedging under volatility uncertainty.

Contribution

It extends the theory of 2BSDEs to include reflection on cdlg obstacles, linking to optimal stopping and financial hedging strategies.

Findings

Proved existence and uniqueness of reflected 2BSDE solutions.

Linked reflected 2BSDEs to nonclassical optimal stopping problems.

Demonstrated application to super-hedging American options.

Abstract

In this article, we build upon the work of Soner, Touzi and Zhang [Probab. Theory Related Fields 153 (2012) 149-190] to define a notion of a second order backward stochastic differential equation reflected on a lower c\`adl\`ag obstacle. We prove existence and uniqueness of the solution under a Lipschitz-type assumption on the generator, and we investigate some links between our reflected 2BSDEs and nonclassical optimal stopping problems. Finally, we show that reflected 2BSDEs provide a super-hedging price for American options in a market with volatility uncertainty.