## Electronic Communications in Probability

### The lower Snell envelope of smooth functions: an optional decomposition

Erick Trevino Aguilar

#### 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.

#### Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 12, 10 pp.

Dates
Accepted: 11 February 2018
First available in Project Euclid: 27 February 2018

https://projecteuclid.org/euclid.ecp/1519722242

Digital Object Identifier
doi:10.1214/18-ECP117

Mathematical Reviews number (MathSciNet)
MR3771770

Zentralblatt MATH identifier
1390.60139

#### Citation

Trevino Aguilar, Erick. The lower Snell envelope of smooth functions: an optional decomposition. Electron. Commun. Probab. 23 (2018), paper no. 12, 10 pp. doi:10.1214/18-ECP117. https://projecteuclid.org/euclid.ecp/1519722242

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