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 \[ρΦₜ(X)=_{Qₜ}(EQ[-X|Fₜ]-E[Φ(dQ/dP)|Fₜ]),\] 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 E[Φ (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 (ρₜΦ), we derive a novel representation {eqnarray*}τρΦ₀(-Yτ)=infₓ\{τE[Φ*(x+Yτ)-x]\},{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.