The lower Snell envelope of smooth functions: an optional decomposition

Erick Trevino Aguilar

doi:10.1214/18-ecp117

Abstract

In this paper we provide general conditions under which the lower Snell envelope defined with respect to the family $\mathcal{M} $ of equivalent local-martingale probability measures of a semimartingale $S$ admits a decomposition as a stochastic integral with respect to $S$ and an optional process of finite variation. On the other hand, based on properties of predictable stopping times we establish a version of the classical backwards induction algorithm in optimal stopping for the non-linear super-additive expectation associated to $\mathcal{M} $. This result is of independent interest and we show how to apply it in order to systematically construct instances of the lower Snell envelope with no optional decomposition. Such ‘counterexamples’ strengths the scope of our conditions.

Highlights

Let S be a fixed semimartingale and consider its class of equivalent local-martingale measures M
In this paper we provide general conditions under which the lower Snell envelope defined with respect to the family M of equivalent local-martingale probability measures of a semimartingale S admits a decomposition as a stochastic integral with respect to S and an optional process of finite variation
On the other hand, based on properties of predictable stopping times we establish a version of the classical backwards induction algorithm in optimal stopping for the non-linear super-additive expectation associated to M. This result is of independent interest and we show how to apply it in order to systematically construct instances of the lower Snell envelope with no optional decomposition

Summary

Introduction

Let S be a fixed semimartingale and consider its class of equivalent local-martingale measures M. We start our discussion with the upper Snell envelope This process is the value process of optimally stopping an underlying process H with respect to the non-linear sub-additive expectation associated to M. In our main result, we show that the lower Snell envelope will be a semimartingale with an optional decomposition, if the underlying process H is a smooth bounded function of S, say of the form f (S) with f ∈ C2 bounded. [2] investigate lower Snell envelopes for g-expectations with backward differential stochastic equations techniques They obtain a structural result which describes the lower Snell envelope as the sum of a process of bounded variation and a stochastic integral with respect to Brownian motion.

Definitions

Predictable stopping times

Backwards induction

An example of a non-semimartingale lower Snell envelope