Electronic Communications in Probability

The lower Snell envelope of smooth functions: an optional decomposition

Erick Trevino Aguilar

Full-text: Open access

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
Received: 7 June 2017
Accepted: 11 February 2018
First available in Project Euclid: 27 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1519722242

Digital Object Identifier
doi:10.1214/18-ECP117

Mathematical Reviews number (MathSciNet)
MR3771770

Zentralblatt MATH identifier
1390.60139

Subjects
Primary: 60G17: Sample path properties 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)

Keywords
lower Snell envelope optimal stopping semimartingales

Rights
Creative Commons Attribution 4.0 International License.

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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References

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