The Annals of Applied Probability

Optimal stopping under model uncertainty: Randomized stopping times approach

Denis Belomestny and Volker Krätschmer

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

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1458651832

Digital Object Identifier
doi:10.1214/15-AAP1116

Mathematical Reviews number (MathSciNet)
MR3476637

Zentralblatt MATH identifier
1339.60043

Subjects
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)

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

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


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