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(Phi)(t)(X) = sup(Q is an element of Qt) (E-Q[-X vertical bar F-t] - E[Phi(dQ/dP)vertical bar F-t]), where Phi : [0, infinity[->[0, infinity] is a lower semicontinuous convex mapping and Q(t) stands for the set of all probability measures Q which are absolutely continuous w.r.t. a given measure P and Q = P on F-t. Here, the model uncertainty risk depends on a (random) divergence E[Phi(dQ/dP)vertical bar 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 is an element of[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(Phi)(t)), we derive a novel representation sup(tau is an element of T)rho(Phi)(0) (-Y-tau) = inf(x is an element of R){sup(tau is an element of T) 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.