Skip to main content
SpringerLink
Log in

Optimal stopping under ambiguity in continuous time

  • Published:
Mathematics and Financial Economics Aims and scope Submit manuscript

Abstract

We develop a theory of optimal stopping problems under ambiguity in continuous time. Using results from (backward) stochastic calculus, we characterize the value function as the smallest (nonlinear) supermartingale dominating the payoff process. For Markovian models, we derive an adjusted Hamilton–Jacobi–Bellman equation involving a nonlinear drift term that stems from the agent’s ambiguity aversion. We show how to use these general results for search problems and American options.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bachelier, L.: Théorie de la Spéculation. Annales Scientifiques de l’École Normale Supérieure, pp. 21–86 (1900)

  2. Bayraktar E., Yao S.: Optimal stopping for nonlinear expectations—part I. Stoch. Process. Appl. 121, 185–211 (2011a)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bayraktar E., Yao S.: Optimal stopping for nonlinear expectations—part II. Stoch. Process. Appl. 121, 212–264 (2011b)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bayraktar, E., Karatzas, I., Yao, S.: Optimal stopping for dynamic convex risk measures. Illinois J. Math. 54(3), 1025–1067 (Fall 2010)

  5. Bertola G.: Irreversible investment. Working Paper. Princeton University, Princeton (1988)

    Google Scholar 

  6. Chen Z., Peng S.: A general downcrossing inequality for g-martingales. Stat. Probab. Lett. 46, 169–175 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen Z., Epstein L.: Ambiguity, risk and asset returns in continuous time. Econometrica 70, 1403–1443 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chernoff H.: Sequential Analysis and Optimal Design. SIAM, Philadelphia (1972)

    Book  MATH  Google Scholar 

  9. Coquet F., Hu Y., Mémin J., Peng S.: Filtration-consistent nonlinear expectations and related g-expectations. Probab. Theory Relat. Fields 123, 1–27 (2002)

    Article  MATH  Google Scholar 

  10. Courtault, J.-M., Kabanov, Y.: Louis Bachelier: Aux Origines de la Finance Mathématique. Presses universitaires de Franche-Comté (2002)

  11. Crandall M., Ishii H., Lions P.: User’s guide to viscosity solutions of second-order partial differential equations. Bull. Am. Math. Soc. 27(1), 1–67 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  12. DeGroot M.H.: Optimal Statistical Decisions. Wiley, New York (2004)

    Book  MATH  Google Scholar 

  13. Delbaen, F., Peng, S., Rosazza Gianin, E.: Representation of the penalty term of dynamic concave utilities. Working Paper (2009)

  14. Dixit , Pindyck R.: Investment Under Uncertainty. Princeton University Press, Princeton (1994)

    Google Scholar 

  15. Duffie D., Epstein L.: Stochastic differential utility. Econometrica 60, 353–394 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. El Karoui N.: Les aspects probabilistes du controle stochastique. In: Dellacherie, C., Meyer, P. (eds) Ecole dEte de Probabilites de Saint-Flour, pp. 74–238. Springer, New York (1979)

    Google Scholar 

  17. El Karoui N., Peng S., Quenez M.: Backward stochastic differential equations in finance. Math. Financ. 7(1), 1–71 (1997a)

    Article  MathSciNet  MATH  Google Scholar 

  18. El Karoui N., Kapoudjian C., Pardoux E., Peng S., Quenez M.: Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s. Ann. Appl. Probab. 25(2), 702–737 (1997b)

    Article  MathSciNet  MATH  Google Scholar 

  19. Fleming W., Soner H.: Controlled Markov Processes and Viscosity Solutions. Springer, New York (2006)

    MATH  Google Scholar 

  20. Föllmer H., Protter P., Shiryaev A.: Quadratic covariation and an extension of itô’s formula. Bernoulli 1, 149–169 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jacka S.: Optimal stopping and the American put. Math. Financ. 1, 1–14 (1991)

    Article  MATH  Google Scholar 

  22. Jacka S., Lynn J.: Finite-horizon optimal stopping, obstacle problems and the shape of the continuation region. Stoch. Stoch. Rep. 39, 25–42 (1992)

    MathSciNet  MATH  Google Scholar 

  23. Karatzas I.: On the pricing of American options. Appl. Math. Optim. 17, 37–60 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  24. Karatzas I., Shreve S.E.: Brownian Motion and Stochastic Calculus. Springer, New York (1991)

    MATH  Google Scholar 

  25. Karatzas I., Wang H.: A barrier option of American type. Appl. Math. Optim. 42, 259–279 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. Koopmans T.: Stationary ordinal utility and impatience. Econometrica 28, 287–309 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  27. Krylov N.: Controlled Diffusion Processes. Springer, New York (1980)

    Book  MATH  Google Scholar 

  28. Maccheroni, F., Marinacci, M., Rustichini, A.: dynamic Variational Preferences. J. Econ. Theory. 128, 4–44 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. McDonald R., Siegel D.: The value of waiting to invest. Q. J. Econ. 101, 707–727 (1986)

    Article  Google Scholar 

  30. Myneni R.: The pricing of the American option. Ann. Appl. Probab. 2, 1–23 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  31. Nishimura K.G., Ozaki H.: Search and Knightian uncertainty. J. Econ. Theory 119, 299–333 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  32. Nishimura K.G., Ozaki H.: Irreversible investment and Knightian uncertainty. J. Econ. Theory 136, 668–694 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  33. Peng S.: BSDE and related g-expectation. In: El Karoui, N., Mazliak, L. (eds) Backward Stochastic Differential Equation, no 364 in Pitman Research Notes in Mathematics Series, Addison Wesley Longman, London (1997)

    Google Scholar 

  34. Peng S.: Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob–Meyer’s type. Probab. Theory Relat. Fields 113, 473–499 (1999)

    Article  MATH  Google Scholar 

  35. Peskir G., Shiryaev A.: Optimal Stopping and Free Boundary Problems Lectures in Mathematics. ETH Zurich, Birkhauser (2006)

    Google Scholar 

  36. Pham H.: Optimal stopping, free boundary, and American option in a jump-diffusion model. Appl. Math. Optim. 35, 145–167 (1997)

    MathSciNet  MATH  Google Scholar 

  37. Pindyck R.S.: Irreversible investment, capacity choice, and the value of the firm. Am. Econ. Rev. 78, 969–985 (1988)

    Google Scholar 

  38. Riedel F.: Optimal stopping with multiple priors. Econometrica 77, 857–908 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Riedel F., Su X.: On Irreversible Investment. Financ. Stoch. 4, 223–250 (2009)

    Google Scholar 

  40. Rogers L., Williams D.: Diffusions, Markov Processes, and Martingales vol. 2 of Series in Probability and Mathematical Statistics. Wiley, Chichester (1987)

    Google Scholar 

  41. Sargent T.: Dynamic Macroeconomic Theory. Harvard University Press, Cambridge (1987)

    Google Scholar 

  42. Skiadas C.: Robust control and recursive utility. Financ. Stoch. 7, 745–789 (2003)

    Article  MathSciNet  Google Scholar 

  43. Smit H., Trigeorgis L.: Strategic Investment: Real Options and Games. Princeton University Press, Princeton (2004)

    Google Scholar 

  44. Trevino, E.: American options in incomplete markets: upper and lower snell envelopes and robust partial hedging. Ph.D. thesis, Humboldt University at Berlin (2008)

  45. Trevino, E.: Optimal stopping under model uncertainty and the regularity of lower snell envelopes. Quant. Financ. 12(6) (2012). doi:10.1080/14697688.2010.488653

  46. Trigeorgis L.: Real Options: Managerial Flexibility and Strategy in Resource Allocation. MIT Press, Cambridge (1996)

    Google Scholar 

  47. Trojanowska M., Kort P.: The worst case for real options. J. Optim. Theory Appl. 146, 709–734 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  48. Weitzman M.L.: Optimal search for the best alternative. Econometrica 47, 641–654 (1979)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Financial Mathematics, Peking University, Beijing, China

    Xue Cheng

  2. Institute of Mathematical Economics, Bielefeld University, Bielefeld, Germany

    Frank Riedel

Authors
  1. Xue Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Frank Riedel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Frank Riedel.

Additional information

Tese results have been presented at seminars in Freiburg (December 2008), CENOS Workshop, Münster (May 2009), ZiF Bielefeld (September 2009), IDEISCOR Conference Risk Sharing and Finance, Toulouse (September 2009), the Workshop on Risk Measures and Robust Optimization in Finance, NUS Singapore, November 2009, Oxford University (February 2010), and the Third Chinese-German Workshop on Stochastic Analysis and Related Fields, Beijing, 2010.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cheng, X., Riedel, F. Optimal stopping under ambiguity in continuous time. Math Finan Econ 7, 29–68 (2013). https://doi.org/10.1007/s11579-012-0081-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11579-012-0081-6

Keywords

JEL Classification

Search

Navigation