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.