The Annals of Applied Probability

Optimal stopping under model uncertainty: Randomized stopping times approach

Denis Belomestny and Volker Krätschmer

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In this work, we consider optimal stopping problems with conditional convex risk measures of the form

\[\rho^{\Phi}_{t}(X)=\sup_{\mathrm{Q}\in\mathcal{Q}_{t}}(\mathbb{E}_{\mathrm{Q}}[-X|\mathcal{F}_{t}]-\mathbb{E}[\Phi(\frac{d\mathrm{Q}}{d\mathrm{P}})\Big|\mathcal{F}_{t}]),\] where $\Phi:[0,\infty[\,\rightarrow[0,\infty]$ is a lower semicontinuous convex mapping and $\mathcal{Q}_{t}$ stands for the set of all probability measures $\mathrm{Q}$ which are absolutely continuous w.r.t. a given measure $\mathrm{P}$ and $\mathrm{Q}=\mathrm{P}$ on $\mathcal{F}_{t}$. Here, the model uncertainty risk depends on a (random) divergence $\mathbb{E}[\Phi (\frac{d\mathrm{Q}}{d\mathrm{P}})|\mathcal{F}_{t}]$ measuring the distance between a hypothetical probability measure we are uncertain about and a reference one at time $t$. Let $(Y_{t})_{t\in[0,T]}$ be an adapted nonnegative, right-continuous stochastic process fulfilling some proper integrability condition and let $\mathcal{T}$ be the set of stopping times on $[0,T]$; then without assuming any kind of time-consistency for the family $(\rho_{t}^{\Phi})$, we derive a novel representation \begin{eqnarray*}\sup_{\tau\in\mathcal{T}}\rho^{\Phi}_{0}(-Y_{\tau})=\inf_{x\in\mathbb{R}}\{\sup_{\tau\in\mathcal{T}}\mathbb{E}[\Phi^{*}(x+Y_{\tau})-x]\},\end{eqnarray*} which makes the application of the standard dynamic programming based approaches possible. In particular, we generalize the additive dual representation of Rogers [Math. Finance 12 (2002) 271–286] to the case of optimal stopping under uncertainty. Finally, we develop several Monte Carlo algorithms and illustrate their power for optimal stopping under Average Value at Risk.

Article information

Ann. Appl. Probab., Volume 26, Number 2 (2016), 1260-1295.

Received: March 2014
Revised: April 2015
First available in Project Euclid: 22 March 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Optimized certainty equivalents optimal stopping primal representation additive dual representation randomized stopping times thin sets


Belomestny, Denis; Krätschmer, Volker. Optimal stopping under model uncertainty: Randomized stopping times approach. Ann. Appl. Probab. 26 (2016), no. 2, 1260--1295. doi:10.1214/15-AAP1116.

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  • [1] Anantharaman, R. (2012). Thin subspaces of $L^{1}(\lambda)$. Quaest. Math. 35 133–143.
  • [2] Andersen, L. and Broadie, M. (2004). A primal-dual simulation algorithm for pricing multidimensional American options. Management Sciences 50 1222–1234.
  • [3] Ankirchner, S. and Strack, P. (2011). Skorokhod embeddings in bounded time. Stoch. Dyn. 11 215–226.
  • [4] Baxter, J. R. and Chacon, R. V. (1977). Compactness of stopping times. Z. Wahrsch. Verw. Gebiete 40 169–181.
  • [5] Bayraktar, E., Karatzas, I. and Yao, S. (2010). Optimal stopping for dynamic convex risk measures. Illinois J. Math. 54 1025–1067.
  • [6] Bayraktar, E. and Yao, S. (2011). Optimal stopping for non-linear expectations. Stochastic Process. Appl. 121 185–211.
  • [7] Bayraktar, E. and Yao, S. (2011). Optimal stopping for non-linear expectations. Stochastic Process. Appl. 121 212–264.
  • [8] Belomestny, D. (2013). Solving optimal stopping problems by empirical dual optimization. Ann. Appl. Probab. 23 1988–2019.
  • [9] Ben-Tal, A. and Teboulle, M. (1987). Penalty functions and duality in stochastic programming via $\phi$-divergence functionals. Math. Oper. Res. 12 224–240.
  • [10] Ben-Tal, A. and Teboulle, M. (2007). An old-new concept of convex risk measures: The optimized certainty equivalent. Math. Finance 17 449–476.
  • [11] Bion-Nadal, J. (2008). Dynamic risk measures: Time consistency and risk measures from BMO martingales. Finance Stoch. 12 219–244.
  • [12] Borwein, J. M. and Zhuang, D. (1986). On Fan’s minimax theorem. Math. Program. 34 232–234.
  • [13] Cheng, X. and Riedel, F. (2013). Optimal stopping under ambiguity in continuous time. Math. Financ. Econ. 7 29–68.
  • [14] Cheridito, P., Delbaen, F. and Kupper, M. (2004). Coherent and convex monetary risk measures for bounded càdlàg processes. Stochastic Process. Appl. 112 1–22.
  • [15] Cheridito, P., Delbaen, F. and Kupper, M. (2006). Dynamic monetary risk measures for bounded discrete-time processes. Electron. J. Probab. 11 57–106.
  • [16] Cheridito, P. and Li, T. (2009). Risk measures on Orlicz hearts. Math. Finance 19 189–214.
  • [17] Delbaen, F., Peng, S. and Rosazza Gianin, E. (2010). Representation of the penalty term of dynamic concave utilities. Finance Stoch. 14 449–472.
  • [18] Detlefsen, K. and Scandolo, G. (2005). Conditional and dynamic convex risk measures. Finance & Stochastics 9 539–561.
  • [19] Edgar, G. A., Millet, A. and Sucheston, L. (1981). On Compactness and Optimality of Stopping Times. Lecture Notes in Math. 939 36–61. Springer, Berlin.
  • [20] Edgar, G. A. and Sucheston, L. (1992). Stopping Times and Directed Processes. Cambridge Univ. Press, Cambridge.
  • [21] Edwards, D. A. (1987). On a theorem of Dvoretsky, Wald and Wolfowitz concerning Liapounov measures. Glasg. Math. J. 29 205–220.
  • [22] Fan, K. (1953). Minimax theorems. Proc. Natl. Acad. Sci. USA 39 42–47.
  • [23] Floret, K. (1980). Weakly Compact Sets. Lecture Notes in Math. 801. Springer, Berlin.
  • [24] Föllmer, H. and Penner, I. (2006). Convex risk measures and the dynamics of their penalty functions. Statist. Decisions 24 61–96.
  • [25] Föllmer, H. and Schied, A. (2010). Stochastic Finance, 3rd. ed. de Gruyter, Berlin.
  • [26] Frittelli, M. and Rosazza Gianin, E. (2002). Putting order in risk measures. J. Bank. Financ. 26 1473–1486.
  • [27] Frittelli, M. and Rosazza Gianin, E. (2004). Dynamic convex risk measures. In Risk Measures for the 21st Century (G. Szegö, ed.) 227–248. Wiley, New York.
  • [28] Kaina, M. and Rüschendorf, L. (2009). On convex risk measures on $L^{p}$-spaces. Math. Methods Oper. Res. 69 475–495.
  • [29] Kingman, J. F. C. and Robertson, A. P. (1968). On a theorem of Lyapunov. J. Lond. Math. Soc. 43 347–351.
  • [30] Krätschmer, V. and Schoenmakers, J. (2010). Representations for optimal stopping under dynamic monetary utility functionals. SIAM J. Financial Math. 1 811–832.
  • [31] Kühn, Z. and Rösler, U. (1998). A generalization of Lyapunov’s convexity theorem with applications in optimals stopping. Proc. Amer. Math. Soc. 126 769–777.
  • [32] Kupper, M. and Schachermayer, W. (2009). Representation results for law invariant time consistent functions. Math. Financ. Econ. 2 189–210.
  • [33] Riedel, F. (2009). Optimal stopping with multiple priors. Econometrica 77 857–908.
  • [34] Rockafellar, R. T. and Wets, J.-B. (1998). Variational Analysis. Springer, Berlin/Heidelberg.
  • [35] Rogers, L. C. G. (2002). Monte Carlo valuation of American options. Math. Finance 12 271–286.
  • [36] Witting, H. and Müller-Funk, U. (1995). Mathematische Statistik II. Teubner, Stuttgart.
  • [37] Xu, Z. Q. and Zhou, X. Y. (2013). Optimal stopping under probability distortion. Ann. Appl. Probab. 23 251–282.