Illinois Journal of Mathematics

Optimal stopping for dynamic convex risk measures

Erhan Bayraktar, Ioannis Karatzas, and Song Yao

Full-text: Open access


We use martingale and stochastic analysis techniques to study a continuous-time optimal stopping problem, in which the decision maker uses a dynamic convex risk measure to evaluate future rewards. We also find a saddle point for an equivalent zero-sum game of control and stopping, between an agent (the “stopper”) who chooses the termination time of the game, and an agent (the “controller,” or “nature”) who selects the probability measure.

Article information

Illinois J. Math., Volume 54, Number 3 (2010), 1025-1067.

First available in Project Euclid: 3 May 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60H30: Applications of stochastic analysis (to PDE, etc.) 91A15: Stochastic games


Bayraktar, Erhan; Karatzas, Ioannis; Yao, Song. Optimal stopping for dynamic convex risk measures. Illinois J. Math. 54 (2010), no. 3, 1025--1067. doi:10.1215/ijm/1336049984.

Export citation


  • E. Bayraktar and S. Yao, Optimal stopping for non-linear expectations–-part I, Stochastic Process. Appl. 121 (2011), 185–211.
  • E. Bayraktar and S. Yao, Optimal stopping for non-linear expectations–part II, Stochastic Process. Appl. 121 (2011), 212–264.
  • V. E. Beneš, Existence of optimal strategies based on specified information, for a class of stochastic decision problems, SIAM J. Control 8 (1970), 179–188.
  • J. Bion-Nadal, Time consistent dynamic risk processes, Stochastic Process. Appl. 119 (2009), no. 2, 633–654.
  • P. Cheridito, F. Delbaen and M. Kupper, Dynamic monetary risk measures for bounded discrete-time processes, Electron. J. Probab. 11 (2006), no. 3, 57–106 (electronic).
  • 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.
  • F. Delbaen, S. Peng and E. Rosazza-Gianin, Representation of the penalty term of dynamic concave utilities, Finance Stoch. 14 (2010), 449–472.
  • N. El Karoui, Les aspects probabilistes du contrôle stochastique, Ninth Saint Flour Probability Summer School–-1979 (Saint Flour, 1979), Lecture Notes in Math., vol. 876, Springer, Berlin, 1981, pp. 73–238.
  • N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng and M. C. Quenez, Reflected solutions of backward SDE's, and related obstacle problems for PDE's, Ann. Probab. 25 (1997), no. 2, 702–737.
  • Robert J. Elliott, Stochastic calculus and applications, Applications of Mathematics (New York), vol. 18, Springer-Verlag, New York, 1982.
  • H. F öllmer and A. Schied, Stochastic finance: An introduction in discrete time, extended ed., de Gruyter Studies in Mathematics, vol. 27, Walter de Gruyter & Co., Berlin, 2004.
  • H. F öllmer and I. Penner, Convex risk measures and the dynamics of their penalty functions, Statist. Decisions 24 (2006), no. 1, 61–96.
  • I. Karatzas and S. E. Shreve, Brownian motion and stochastic calculus, 2nd ed., Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1991.
  • I. Karatzas and I. M. Zamfirescu, Martingale approach to stochastic control with discretionary stopping, Appl. Math. Optim. 53 (2006), no. 2, 163–184.
  • I. Karatzas and I. M. Zamfirescu, Martingale approach to stochastic differential games of control and stopping, Ann. Probab. 36 (2008), no. 4, 1495–1527.
  • I. Karatzas and S. E. Shreve, Methods of Mathematical Finance, Applications of Mathematics (New York), vol. 39, Springer-Verlag, New York, 1998.
  • N. Kazamaki, Continuous exponential martingales and BMO, Lecture Notes in Math., vol. 1579, Springer-Verlag, Berlin, 1994.
  • S. Kl öppel and M. Schweizer, Dynamic indifference valuation via convex risk measures, Math. Finance 17 (2007), no. 4, 599–627.
  • M. Kobylanski, J. P. Lepeltier, M. C. Quenez and S. Torres, Reflected BSDE with superlinear quadratic coefficient, Probab. Math. Statist. 22 (2002), no. 1, Acta Univ. Wratislav. No. 2409, 51–83.
  • J.-P. Lepeltier, On a general zero-sum stochastic game with stopping strategy for one player and continuous strategy for the other, Probab. Math. Statist. 6 (1985), no. 1, 43–50.
  • J. Neveu, Discrete-parameter martingales, revised ed., North-Holland Mathematical Library, vol. 10. North-Holland Publishing Co., Amsterdam, 1975, Translated from the French by T. P. Speed,
  • R. T. Rockafellar, Convex analysis, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997, Reprint of the 1970 original, Princeton Paperbacks.
  • D. Williams, Probability with martingales, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1991.