Stopper-Controller Games embedded in Single-Player Control Problems

TL;DR

This paper extends the known reduction of American option pricing to European options to a broader context, including non-linear stochastic processes and control problems, with implications for model uncertainty and game theory.

Contribution

It demonstrates that the payoff reduction phenomenon applies beyond linear models, encompassing non-linear stochastic processes and controller-and-stopper-games.

Findings

Reduction phenomenon holds in non-linear stochastic frameworks

Applicable to model uncertainty and control problems

Provides probabilistic and analytic analysis methods

Abstract

In 2002, Benjamin Jourdain and Claude Martini discovered that for a class of payoff functions, the pricing problem for American options can be reduced to pricing of European options for an appropriately associated payoff, all within a Black-Scholes framework. This discovery has been investigated in great detail by S\"oren Christensen, Jan Kallsen and Matthias Lenga in a recent work in 2020. In the present work we prove that this phenomenon can be observed in a wider context, and even holds true in a setup of non-linear stochastic processes. We analyse this problem from both probabilistic and analytic viewpoints. In the classical situation, Jourdain and Martini used this method to approximate prices of American put options. The broader applicability now potentially covers non-linear frameworks such as model uncertainty and controller-and-stopper-games.