OPTIMAL STOPPING UNDER MODEL UNCERTAINTY: RANDOMIZED STOPPING TIMES APPROACH

In this work, we consider optimal stopping problems with conditional convex risk measures of the form $\rho _t^\phi (X) = \mathop {\sup }\limits_{Q \in {Q_t}} ({\mathbb{E}_Q}[ - X|{F_t}] - \mathbb{E}[\phi (\frac{{dQ}}{{dP}})|Ft])$ where Φ : [0, ∞[→ [0, ∞] is a lower semicontinuous convex mapping and Qt stands for the set of all probability measures Q which are absolutely continuous w.r.t. a given measure P and Q = P on Ft. Here, the model uncertainty risk depends on a (random) divergence $\mathbb{E}[\phi (\frac{{dQ}}{{dP}})|Ft]$ measuring the distance between a hypothetical probability measure we are uncertain about and a reference one at time t. Let (Yt)t∈[0, T] be an adapted nonnegative, right-continuous stochastic process fulfilling some proper integrability condition and let 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 $\mathop {\sup }\limits_{\tau \in T} \rho _0^\phi ( - {Y_\tau }) = \mathop {\inf }\limits_{x \in \mathbb{R}} \{ \mathop {\sup }\limits_{\tau \in T} \,\mathbb{E}[\phi * (x + {Y_\tau }) - x]\} $, 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.