EURO 2024 Copenhagen
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1917. Guaranteed bounds for optimal stopping problems using kernel-based non-asymptotic uniform confidence bands

Invited abstract in session TA-35: Optimization under uncertainty: theory and solution algorithms, stream Stochastic, Robust and Distributionally Robust Optimization.

Tuesday, 8:30-10:00
Room: 44 (building: 303A)

Authors (first author is the speaker)

1. Martin Glanzer
University of Mannheim Business School
2. Sebastian Maier
Imperial College London
3. Georg Pflug
Department of Statistics and Decision Support Systems, University of Vienna

Abstract

In this talk, a new approach is presented for obtaining guaranteed upper and lower bounds on the true optimal value of stopping problems by nonparametric regression. Unlike existing simulation-and-regression approaches, such as those based on least squares Monte Carlo and information relaxation, whose estimates depend on the sampled paths of the underlying stochastic process and are therefore stochastic in nature, our guaranteed bounds allow replacing the sampling error with a pre-specified confidence level. The key to our approach is the use of non-asymptotic uniform confidence bands based on a kernel ridge regression (KRR) estimator to over- and underestimate the conditional expected value from continuation. Since our high- and low-biased KRR estimators are guaranteed to over- and underestimate, respectively, the true but unknown conditional expectation at every step of the backward recursive procedure with a pre-specified confidence level, we obtain probabilistically guaranteed bounds on the problem's true optimal value. We derive closed-form formulas for both the fixed design and the random design case, which makes our approach readily applicable in both simulation and data-driven situations. We illustrate the applicability of the proposed bounding procedure by valuing a Bermudan-style put option.

Keywords

Status: accepted

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