## The Annals of Applied Probability

### Optimal stopping under model uncertainty: Randomized stopping times approach

#### Abstract

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

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

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

https://projecteuclid.org/euclid.aoap/1458651832

Digital Object Identifier
doi:10.1214/15-AAP1116

Mathematical Reviews number (MathSciNet)
MR3476637

Zentralblatt MATH identifier
1339.60043

#### Citation

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. https://projecteuclid.org/euclid.aoap/1458651832

#### References

• [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.