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Author: Schoenmakers, John G. M. × Subject: randomization ×

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    From optimal martingales to randomized dual optimal stopping
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Belomestny, Denis; Schoenmakers, John G. M.
    In this article we study and classify optimal martingales in the dual formulation of optimal stopping problems. In this respect we distinguish between weakly optimal and surely optimal martingales. It is shown that the family of weakly optimal and surely optimal martingales may be quite large. On the other hand it is shown that the Doob-martingale, that is, the martingale part of the Snell envelope, is in a certain sense the most robust surely optimal martingale under random perturbations. This new insight leads to a novel randomized dual martingale minimization algorithm that does`nt require nested simulation. As a main feature, in a possibly large family of optimal martingales the algorithm efficiently selects a martingale that is as close as possible to the Doob martingale. As a result, one obtains the dual upper bound for the optimal stopping problem with low variance.
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    Solving optimal stopping problems via randomization and empirical dual optimization
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Belomestny, Denis; Bender, Christian; Schoenmakers, John G. M.
    In this paper we consider optimal stopping problems in their dual form. In this way we reformulate the optimal stopping problem as a problem of stochastic average approximation (SAA) which can be solved via linear programming. By randomizing the initial value of the underlying process, we enforce solutions with zero variance while preserving the linear programming structure of the problem. A careful analysis of the randomized SAA algorithm shows that it enjoys favorable properties such as faster convergence rates and reduced complexity as compared to the non randomized procedure. We illustrate the performance of our algorithm on several benchmark examples.