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Optimal stopping under model uncertainty: Randomized stopping times approach
ρφ t (X) = sup Q∈Qt (EQ[-X|Ft] - Q∈[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 sup τ∈T ρφ 0 (-Yτ ) = inf x∈R {sup τ∈T E[φ∗ (x + Yτ )- 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. © Institute of Mathematical Statistics, 2016.