Second-order BSDEs with general reflection and game options under uncertainty

TL;DR

This paper extends the theory of second-order reflected backward stochastic differential equations (BSDEs) with two obstacles, establishing well-posedness and linking them to uncertain Dynkin games and American game options under volatility uncertainty.

Contribution

It generalizes second-order reflected BSDEs to two obstacles, proves well-posedness, and connects these equations to uncertain Dynkin games and pricing of American game options.

Findings

Established existence and uniqueness for doubly reflected second-order BSDEs.

Connected second-order DRBSDEs to uncertain Dynkin games.

Derived super and subhedging prices for American game options.

Abstract

The aim of this paper is twofold. First, we extend the results of [33] concerning the existence and uniqueness of second-order reflected 2BSDEs to the case of two obstacles. Under some regularity assumptions on one of the barriers, similar to the ones in [10], and when the two barriers are completely separated, we provide a complete wellposedness theory for doubly reflected second-order BSDEs. We also show that these objects are related to non-standard optimal stopping games, thus generalizing the connection between DRBSDEs and Dynkin games first proved by Cvitanic and Karatzas [11]. More precisely, we show under a technical assumption that the second order DRBSDEs provide solutions of what we call uncertain Dynkin games and that they also allow us to obtain super and subhedging prices for American game options (also called Israeli options) in financial markets with volatility uncertainty