Optimal Stopping Research Articles

Wealth maximisation and residential energy-efficiency retrofits: Insights from a real options model

The slow adoption of residential energy-efficiency retrofits continues to hamper the energy transition. We study incentives for adoption by proposing a model of optimal investment under uncertainty where the wealth-maximising agent has the option to delay. Stochastic portfolio returns and energy prices are taken into account. An extension of the model where the energy carrier is switched, e.g. from gas to electricity, is also considered. Exercise boundaries for the optimal stopping problem are estimated numerically for recent case studies of German buildings. Investment is generally not optimal at current energy prices and market conditions. Increasing correlation between gas and electricity prices erodes the value of technology switching. Comparative statics reveal that energy-efficiency investments become optimal at relatively lower energy prices as wealth, income, and savings behaviour increase, and portfolio drift and volatility decrease. Consequently, incentive to invest in retrofits is far more heterogeneous along wealth dimensions than standard discounted cash flow analyses suggest. An examination of retrofit subsidies demonstrates how free-riding by wealthier agents occurs when subsidies are not appropriately targeted. We also show that the pursuit of economic efficiency in subsidy design might have regressive effects on the wealth distribution.

To Bag is to Prune

Abstract It is notoriously difficult to build a bad Random Forest (RF). Concurrently, RF blatantly overfits in-sample without apparent consequences out-of-sample. Arguments like the bias-variance trade-off or double descent cannot rationalize this paradox. I propose a new explanation: bootstrap aggregation and model perturbation as implemented by RF automatically prune a latent “true” tree. More generally, I document that randomized ensembles of greedily optimized learners implicitly perform optimal early stopping out-of-sample. So, letting RF overfit the training data is a dominant tuning strategy against nature’s undisclosed choice of noise level. Additionally, novel ensembles of Boosting and MARS are also eligible. I empirically demonstrate the property, with simulated and real data, by reporting that these new completely overfitting ensembles perform similarly to their tuned counterparts – or better.

Study on Surrounding Rock Control of Withdrawal Space in Fully Mechanized Caving Mining of a 19 m Extra-Thick Coal Seam

The section span of the withdrawal space of fully mechanized top coal caving in an extra-thick coal seam is large, and with the gradual withdrawal of the hydraulic support, a series of strong dynamic pressure disasters occur in the withdrawal space, and the difficulty of surrounding rock support control increases sharply. In order to study the control mechanism of surrounding rock in the final mining withdrawal space in detail and put forward a reasonable support technology scheme, taking the large-section withdrawal space of an 8309 fully mechanized caving face in an extra-thick coal seam of a mine as the research object—through the theoretical investigation of whether the key blocks of the main roof are stably hinged under varied stopping coal caving distances and fracture locations of the main roof—the reasonable and optimal stopping coal caving distances and roadway formation time are determined. Using numerical simulation and similar simulation methods, the vertical stress and the maximum shear stress research indicators were introduced to verify the accuracy of the theoretical analysis results. The results show the following: (1) The reasonable stopping coal caving span is 1~2 times the cycle weighting interval, the best stopping coal caving distance in this geological condition is 30 m, and the best fracture position of the main roof is located above the goaf. (2) The migration of overlying strata in the withdrawal space has obvious zoning characteristics, and the zoning is as follows: a stopping coal caving area, support area of the hydraulic support, withdrawal channel area, and stopping coal pillar area. (3) According to the zoning characteristics of overlying strata movement, the asymmetric zoning support control scheme of the withdrawal space is proposed. The field monitoring results show that the maximum roof subsidence in the withdrawal space is 151 mm, the maximum internal squeezing amount of the stopping coal pillar is 82 mm, and the supporting and anchoring effect of each partition in the withdrawal space is good. The set of partition asymmetric support control schemes has been successfully applied to field practice.

Never stop or never start? Optimal stopping under a mixture of CPT and EUT preferences

We consider the problem of finding the best time to stop a diffusion process for an agent with a preference model that is a mixture of expected utility theory (EUT) and cumulative prospect theory (CPT). In view of time-inconsistency, we consider two types of agents: a naive agent, who is not aware of the time-inconsistency and thus re-plans at every instant, and a sophisticated agent, who is aware of the time-inconsistency and takes a so-called intra-personal equilibrium strategy by correctly anticipating her actions in the future. We show that under a wide range of the CPT preference parameter values, the naive agent will never stop. For a sophisticated agent, we use a different notion of intra-personal equilibrium from the one employed by Ebert and Strack (2018). We show that any two-threshold strategy, which is to stop when the diffusion process reaches either an upper threshold or a lower threshold, cannot be an intra-personal equilibrium if the agent overweights worst, unlikely outcomes disproportionally. We derive a sufficient and necessary condition for the strategy of stopping everywhere to be an intra-personal equilibrium and show that this condition does not hold and thus the sophisticated agent may choose to start the diffusion process for some commonly used probability weighting functions.

Fee structure and equity fund manager’s optimal locking in profits strategy

We study the effects of fee structures on fund managers’ strategies for locking in profits. Utilizing the optimal stopping time method, we identify two critical portfolio value thresholds that signal when a manager will choose to lock in profits. Fee components such as management fees, self-investment ratios, and high-water marks significantly influence these decisions. Specifically, higher management fees are associated with increased risk aversion, leading to a narrower continuation region, indicating a preference for lower risk. Conversely, performance fees encourage greater risk-taking. We use the S&P 500 Index and NASDAQ Composite index as representatives of managers’ portfolios and apply our model to illustrate how managers adjust their profit-locking strategies in response to their desired rewards.

A note on reflected BSDEs in infinite horizon with stochastic Lipschitz coefficients

We consider an infinite horizon, obliquely reflected backward stochastic differential equation (RBSDE). The main contribution of the present work is that we generalize previous results on infinite horizon RBSDEs to the setting where the driver has a stochastic Lipschitz coefficient. As an application, we consider robust optimal stopping problems for functional stochastic differential equations (FSDEs) where the driver has linear growth.

Tailoring optimal KERMA, projected range, and mass stopping power, and gamma-ray shielding capabilities through BaO/ZnO/CdO/SrO incorporation into Na2O–SiO2 glasses

This study explores the radiation shielding potential of Na2O–SiO2 glass matrices doped with BaO, ZnO, CdO, and SrO, analysed across a broad energy spectrum. The research employed various computational tools such as Phy-X/PSD and PAGEX codes to calculate various shielding parameters, including Kinetic Energy Released in Material (KERMA), mass stopping power, fast neutron, and gamma-ray attenuation properties. Among the glass samples, S3, with its BaO content, exhibited the most advantageous properties in terms of radiation attenuation. The results showed that S3 has the lowest Half Value Layer (HVL) of 0.032 cm at 0.015 MeV and the highest KERMA value, peaking at 0.3 MeV with 0.18 MeV/g. The mass stopping power values for alpha particles in the S3 sample showed a pronounced peak at 0.7 MeV, reaching 150 MeV cm2/g, indicating superior energy absorption. The S3 sample, composed of 20% Na2O, 20% BaO, and 60% SiO2, demonstrated the highest density (i.e., 3.225 g/cm³) and effective atomic number (i.e., 17.5), which contributed to its superior shielding performance. The study's findings show that the careful manipulation of glass composition, particularly through the incorporation of high-atomic-number oxides, can significantly enhance shielding effectiveness. It can be concluded that the S3 glass sample would be an optimal configuration for practical applications in radiation protection, where the Na2O–SiO2 glass matrices doped with BaO in terms of optimal material properties.

Is Reinforcement Learning Good at American Option Valuation?

This paper investigates algorithms for identifying the optimal policy for pricing American Options. The American Option pricing is reformulated as a Sequential Decision-Making problem with two binary actions (Exercise or Continue), transforming it into an optimal stopping time problem. Both the least square Monte Carlo simulation method (LSM) and Reinforcement Learning (RL)-based methods were utilized to find the optimal policy and, hence, the fair value of the American Put Option. Both Classical Geometric Brownian Motion (GBM) and calibrated Stochastic Volatility models served as the underlying uncertain assets. The novelty of this work lies in two aspects: (1) Applying LSM- and RL-based methods to determine option prices, with a specific focus on analyzing the dynamics of “Decisions” made by each method and comparing final decisions chosen by the LSM and RL methods. (2) Assess how the RL method updates “Decisions” at each batch, revealing the evolution of the decisions during the learning process to achieve optimal policy.

On the Monotonicity of the Stopping Boundary for Time-Inhomogeneous Optimal Stopping Problems

We consider a class of time-inhomogeneous optimal stopping problems and we provide sufficient conditions on the data of the problem that guarantee monotonicity of the optimal stopping boundary. In our setting, time-inhomogeneity stems not only from the reward function but, in particular, from the time dependence of the drift coefficient of the one-dimensional stochastic differential equation (SDE) which drives the stopping problem. In order to obtain our results, we mostly employ probabilistic arguments: we use a comparison principle between solutions of the SDE computed at different starting times, and martingale methods of optimal stopping theory. We also show a variant of the main theorem, which weakens one of the assumptions and additionally relies on the renowned connection between optimal stopping and free-boundary problems.

Irreversible reinsurance: minimization of capital injections in presence of a fixed cost

AbstractWe propose a model in which, in exchange to the payment of a fixed transaction cost, an insurance company can choose the retention level as well as the time at which subscribing a perpetual reinsurance contract. The surplus process of the insurance company evolves according to the diffusive approximation of the Cramér-Lundberg model, claims arrive at a fixed constant rate, and the distribution of their sizes is general. Furthermore, we do not specify any particular functional form of the retention level. The aim of the company is to take actions in order to minimize the sum of the expected value of the total discounted flow of capital injections needed to avoid bankruptcy and of the fixed activation cost of the reinsurance contract. We provide an explicit solution to this problem, which involves the resolution of a static nonlinear optimization problem and of an optimal stopping problem for a reflected diffusion. We then illustrate the theoretical results in the case of proportional and excess-of-loss reinsurance, by providing a numerical study of the dependency of the optimal solution with respect to the model’s parameters.

Nonzero-Sum Optimal Stopping Game with Continuous vs. Periodic Exercise Opportunities

Nonzero-Sum Optimal Stopping Game with Continuous vs. Periodic Exercise Opportunities

Comparison of real data and simulated data analysis of a stopping rule based on the standard error of measurement in computerized adaptive testing for medical examinations in Korea: a psychometric study.

This study aimed to compare and evaluate the efficiency and accuracy of computerized adaptive testing (CAT) under 2 stopping rules (standard error of measurement [SEM]=0.3 and 0.25) using both real and simulated data in medical examinations in Korea. This study employed post-hoc simulation and real data analysis to explore the optimal stopping rule for CAT in medical examinations. The real data were obtained from the responses of 3rd-year medical students during examinations in 2020 at Hallym University College of Medicine. Simulated data were generated using estimated parameters from a real item bank in R. Outcome variables included the number of examinees’ passing or failing with SEM values of 0.25 and 0.30, the number of items administered, and the correlation. The consistency of real CAT result was evaluated by examining consistency of pass or fail based on a cut score of 0.0. The efficiency of all CAT designs was assessed by comparing the average number of items administered under both stopping rules. Both SEM 0.25 and SEM 0.30 provided a good balance between accuracy and efficiency in CAT. The real data showed minimal differences in pass/ fail outcomes between the 2 SEM conditions, with a high correlation (r=0.99) between ability estimates. The simulation results confirmed these findings, indicating similar average item numbers between real and simulated data. The findings suggest that both SEM 0.25 and 0.30 are effective termination criteria in the context of the Rasch model, balancing accuracy and efficiency in CAT.

History Chronicles: A Tribute to Peter Armitage

Peter Armitage CBE recently passed away (February 2024). He was a pio¬neer in medical statistics and was best known for his innovative developments in clinical trial design and analysis, especially the use of sequential analysis and optimal stopping decision criteria. Sequen¬tial analysis was originally devised as a tool for industrial quality control, and it was Armitage’s genius to recognize its importance for medical research. This column includes tributes and memories of Armitage’s life and accomplishments.

Frontiers in Operations: Optimal Genetic Testing of Families

Problem definition: Through the laws of inheritance, knowing an individual’s genetic status informs disease risk for family members, but current protocols for deciding whom to genetically test only consider one person at a time rather than design an optimal testing plan for the entire family. Methodology/results: We develop a Markov decision process framework for maximizing the net benefits of genetic testing that integrates a Bayesian network of genetic statuses, with a functional representation of cost-effectiveness. Our model provides a contingent sequence of family members to test one at a time, that is, a plan that dynamically incorporates new test results, revealed sequentially at random, to decide who next to test. In the general case, we show that optimal stopping follows a structure with two-sided thresholds, previously known only for individual testing. Although the optimal testing sequence, in general, is contingent on the family test results, in the special case of sibling-only tests we can identify this sequence a priori. Our numerical case study, which was conducted in a realistic BRCA1/2 testing setting, demonstrates that an optimal policy significantly improves cost-effectiveness over existing policies. Thus, our framework offers a promising and powerful new approach to genetic testing. Managerial implications: In an optimal policy, prioritizing testing family members who might otherwise not have been tested can lead to an overall improvement in familial health value, surpassing even the most cost-effective existing protocols. From a management perspective, healthcare organizations and insurance companies can potentially save costs by implementing this approach for such families. History: This paper has been accepted in the Manufacturing & Service Operations Management Frontiers in Operations Initiative. Funding: D. Adelman is grateful for financial support from Booth School of Business. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2023.0057 .

Nash equilibria in greenhouse gas offset credit markets

One approach to reducing greenhouse gas (GHG) emissions is to incentivise carbon capturing and carbon reducing projects while simultaneously penalising excess GHG output. In this work, we present a novel market framework and characterise the optimal behaviour of GHG offset credit (OC) market participants in both single-player and two-player settings. The single player setting is posed as an optimal stopping and control problem, while the two-player setting is posed as optimal stopping and mixed-Nash equilibria problem. We demonstrate the importance of acting optimally using numerical solutions and Monte Carlo simulations and explore the differences between the homogeneous and heterogeneous players. In both settings, we find that market participants benefit from optimal OC trading and OC generation.

The Secretary Problem with Independent Sampling

The secretary problem is probably the most well-studied optimal stopping problem with many applications in economics and management. In the secretary problem, a decision maker faces an unknown sequence of values, revealed successively, and has to make irrevocable take-it-or-leave-it decisions. Her goal is to select the maximum value in the sequence. Although in the classic secretary problem the values of upcoming elements are entirely unknown, in many realistic situations, the decision maker has access to some information, for example, from past data. In this paper, we take a sampling approach and assume that before starting the sequence, each element is sampled independently with probability p. We study both the adversarial and the random arrival models. Our main result is to obtain the best possible algorithms for both settings and all values of p. As p grows to one, the obtained guarantees converge to the optimal guarantees in the full information case. Notably, we establish that the best possible algorithm in the adversarial order setting is a fixed threshold algorithm. In the random order setting, we characterize the best possible algorithm by a sequence of thresholds, dictating at which point in time we should accept a value. Surprisingly, this sequence is independent of p. We complement our theoretical results with numerical experiments on data of people playing the secretary problem repeatedly. Our results help explain some behavioral issues they raised and indicate that people play a strategy similar to our optimal algorithms from the start onwards, albeit slightly suboptimally. This paper was accepted by Chung Piaw Teo, optimization. Funding: This work was partially funded by the Agencia Nacional de Investigación y Desarrollo Chile [Grants FB210005 (Centro de Modelamiento Matamático), AIM23_0004 (Millenium Institute for Research in Market Imperfections and Public Policy), AFB230002 (Instituto Sistemas Complejos de Ingeniería), ACT210005, and Fondo Nacional de Desarrollo Científico y Tecnológico 1220054], the Alexander von Humboldt Foundation with funds from the German Federal Ministry of Education and Research, and the German Research Foundation within the Research Training Group Advanced Optimization in a Networked Economy [Grant Graduiertenkolleg 2201]. The views expressed in this paper do not necessarily reflect those of the European Central Bank or the Eurosystem. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2021.01580 .

Deep neural network expressivity for optimal stopping problems

This article studies deep neural network expression rates for optimal stopping problems of discrete-time Markov processes on high-dimensional state spaces. A general framework is established in which the value function and continuation value of an optimal stopping problem can be approximated with error at most ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\varepsilon $\\end{document} by a deep ReLU neural network of size at most κdqε−r\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\kappa d^{\\mathfrak{q}} \\varepsilon ^{-\\mathfrak{r}}$\\end{document}. The constants κ,q,r≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\kappa ,\\mathfrak{q},\\mathfrak{r} \\geq 0$\\end{document} do not depend on the dimension d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$d$\\end{document} of the state space or the approximation accuracy ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\varepsilon $\\end{document}. This proves that deep neural networks do not suffer from the curse of dimensionality when employed to approximate solutions of optimal stopping problems. The framework covers for example exponential Lévy models, discrete diffusion processes and their running minima and maxima. These results mathematically justify the use of deep neural networks for numerically solving optimal stopping problems and pricing American options in high dimensions.

Strategies people use buying airline tickets: a cognitive modeling analysis of optimal stopping in a changing environment

Strategies people use buying airline tickets: a cognitive modeling analysis of optimal stopping in a changing environment

Sampling for Remote Estimation of the Wiener Process over an Unreliable Channel

In this paper, we study a sampling problem where a source takes samples from a Wiener process and transmits them through a wireless channel to a remote estimator. Due to channel fading, interference, and potential collisions, the packet transmissions are unreliable and could take random time durations. Our objective is to devise an optimal causal sampling policy that minimizes the long-term average mean square estimation error. This optimal sampling problem is a recursive optimal stopping problem, which is generally quite difficult to solve. However, we prove that the optimal sampling strategy is, in fact, a simple threshold policy where a new sample is taken whenever the instantaneous estimation error exceeds a threshold. This threshold remains a constant value that does not vary over time. By exploring the structure properties of the recursive optimal stopping problem, a low-complexity iterative algorithm is developed to compute the optimal threshold. This work generalizes previous research by incorporating both transmission errors and random transmission times into remote estimation. Numerical simulations are provided to compare our optimal policy with the zero-wait and age-optimal policies.

Discounted nonzero-sum optimal stopping games under Poisson random intervention times

We present solutions to some discounted nonzero-sum optimal stopping games of two players related to the perpetual game-type contingent claims with payoffs representing linear functions of the running values of a geometric Brownian motion. It is assumed that the underlying process can be stopped by the both players only at certain random intervention times which coincide with the jump times of the two appropriate independent Poisson processes. The optimal stopping times forming a Nash equilibrium are shown to be the first times at which the underlying process is either below or above certain lower or upper constant boundaries at the jump times of the appropriate Poisson processes. The proof is based on the reductions of the original games to the associated coupled free-boundary problems and the solutions to the latter problems by means of the smooth-fit conditions at the optimal boundaries for every player. We show that the optimal stopping constant lower and upper boundaries are determined as (possibly multiple) solutions to the equivalent coupled systems of arithmetic equations. The obtained results can be interpreted as the rational valuation of some perpetual randomized Bermudian game-type contingent claims in the Black-Merton-Scholes model.