Electronic Communications in Probability

The lower Snell envelope of smooth functions: an optional decomposition

Erick Trevino Aguilar

Full-text: Open access


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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)

lower Snell envelope optimal stopping semimartingales

Creative Commons Attribution 4.0 International License.


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

Export citation


  • [1] D. Belomestny, T. Hübner, V. Krätschmer, and S. Nolte, Minimax theorems for American options in incomplete markets without time-consistency, Preprint arXiv:1708.08904 (2017).
  • [2] X. Cheng and F. Riedel, Optimal stopping under ambiguity in continuous time, Math. Financ. Econ. 7 (2013), no. 1, 29–68.
  • [3] F. Delbaen, The structure of m-stable sets and in particular of the set of risk neutral measures, In memoriam Paul-André Meyer: Séminaire de Probabilités XXXIX, Lecture Notes in Math., vol. 1874, Springer, Berlin, 2006, pp. 215–258.
  • [4] F. Delbaen and W. Schachermayer, A general version of the fundamental theorem of asset pricing, Math. Ann. 300 (1994), no. 3, 463–520.
  • [5] M. Emery, Une propriété des temps prévisibles, Seminar on Probability, XIV (Paris, 1978/1979) (French), Lecture Notes in Math., vol. 784, Springer, Berlin, 1980, pp. 316–317.
  • [6] H. Föllmer and Yu. M. Kabanov, Optional decomposition and Lagrange multipliers, Finance Stoch. 2 (1998), no. 1, 69–81.
  • [7] H. Föllmer and A. Schied, Stochastic finance, an introduction in discrete time, 2nd ed., Walter de Gruyter, Berlin New York, 2004.
  • [8] J. Jacod and A. N. Shiryayev, Limit theorems for stochastic processes, 2nd ed., A comprehensive studies in mathematics, vol. 288, Springer, Berlin, Heidelberg, New York, Honk Kong, London, Milan, Paris, Tokyo, 2002.
  • [9] P. Protter, Stochastic integration and differential equations, Stochastic modelling and applied probability, vol. 21, Springer, Berlin Heidelberg New York, version 2.1, second ed., 2005.
  • [10] D. Revuz and M. Yor, Continuous martingales and Brownian motion, 3 ed., Springer, 2005.
  • [11] E. Treviño Aguilar, Optimal stopping under model uncertainty and the regularity of lower Snell envelopes, Quant. Finance 12 (2012), no. 6, 865–871.
  • [12] E. Treviño Aguilar, Semimartingale properties of the lower Snell envelope in optimal stopping under model uncertainty, Braz. J. Probab. Stat. 31 (2017), no. 1, 194–213.
  • [13] E. Treviño Aguilar, American options in incomplete markets: Upper and lower snell envelopes and robust partial hedging, Ph.D. thesis, Humboldt Universität zu Berlin, 2008.