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1. Optimal stopping with expectation constraints

   https://doi.org/10.1214/23-AAP1980  

   Bayraktar, Erhan; Yao, Song (February 2024, The Annals of Applied Probability)

   We analyze an optimal stopping problem with a series of inequality-type and equality-type expectation constraints in a general non-Markovian framework. We show that the optimal stopping problem with expectation constraints (OSEC) in an arbitrary probability setting is equivalent to the constrained problem in weak formulation (an optimization over joint laws of stopping rules with Brownian motion and state dynamics on an enlarged canonical space), and thus the OSEC value is independent of a specific probabilistic setup. Using a martingale-problem formulation, we make an equivalent characterization of the probability classes in weak formulation, which implies that the OSEC value function is upper semianalytic. Then we exploit a measurable selection argument to establish a dynamic programming principle in weak formulation for the OSEC value function, in which the conditional expected costs act as additional states for constraint levels at the intermediate horizon. 

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2. Unbiased Optimal Stopping via the MUSE

   https://doi.org/10.1016/j.spa.2022.12.007  

   Zhou, Zhengqing; Wang, Guanyang; Blanchet, Jose H.; Glynn, Peter W. (December 2022, Stochastic Processes and their Applications)
3. Optimal Stopping with a Probabilistic Constraint

   https://doi.org/10.1007/s10957-017-1183-3  

   Palmer, Aaron Zeff; Vladimirsky, Alexander (December 2017, Journal of Optimization Theory and Applications)
4. Optimal Stopping of Adaptive Dose-Finding Trials

   https://doi.org/10.1287/serv.2020.0261  

   Nasrollahzadeh, Amir Ali; Khademi, Amin (June 2020, Service Science)

   null (Ed.)

   The primary objective of this paper is to develop computationally efficient methods for optimal stopping of an adaptive Phase II dose-finding clinical trial, where the decision maker may terminate the trial for efficacy or abandon it as a result of futility. We develop two solution methods and compare them in terms of computational time and several performance metrics such as the probability of correct stopping decision. One proposed method is an application of the one-step look-ahead policy to this problem. The second proposal builds a diffusion approximation to the state variable in the continuous regime and approximates the trial’s stopping time by optimal stopping of a diffusion process. The secondary objective of the paper is to compare these methods on different dose-response curves, particularly when the true dose-response curve has no significant advantage over a placebo. Our results, which include a real clinical trial case study, show that look-ahead policies perform poorly in terms of the probability of correct decision in this setting, whereas our diffusion approximation method provides robust solutions. 

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5. Conditional optimal stopping: A time-inconsistent optimization

   https://doi.org/10.1214/19-AAP1540  

   Nutz, Marcel; Zhang, Yuchong (August 2020, The Annals of Applied Probability)

   null (Ed.)