OPTIMAL STOPPING UNDER MODEL UNCERTAINTY: RANDOMIZED STOPPING TIMES APPROACH

被引:17
作者
Belomestny, Denis [1 ,2 ]
Kraetschmer, Volker [1 ]
机构
[1] Duisburg Essen Univ, Fac Math, Thea Leymann Str 9, D-45127 Essen, Germany
[2] Natl Res Univ, Higher Sch Econ, Moscow, Russia
关键词
Optimized certainty equivalents; optimal stopping; primal representation; additive dual representation; randomized stopping times; thin sets; CONVEX RISK MEASURES; NONLINEAR EXPECTATIONS; THEOREM; PART;
D O I
10.1214/15-AAP1116
中图分类号
O21 [概率论与数理统计]; C8 [统计学];
学科分类号
020208 ; 070103 ; 0714 ;
摘要
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.
引用
收藏
页码:1260 / 1295
页数:36
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