Optimal Stopping Rule Research Articles

An Energy-Saving Transmit Method Between Internet of Things Device and Base Station Under Fading Channel

In the Internet of things (IoT), the energy-saving of battery-powered IoT terminal is a key problem. To address it, a novel transceiver is proposed, and a transmission scheme is presented by exploiting the fading characteristic of the wireless channel. Based on the novel transceiver design, the IoT terminal can periodically observe the wireless channel gain and determine the appropriate time to upload information to the base station (BS). When the instantaneous wireless channel gain is lower than the wireless channel gain threshold that can be found using optimal stopping rules, the IoT keeps silent and continues to observe the wireless channel gain while when it is higher than the threshold, the IoT terminal uploads its information to the BS. We analyze the energy consumption performance of the proposed transmission scheme and compare it with that of the normal transmission scheme which transmits the information to the BS regardless of the instantaneous wireless channel gain. Simulations results show that the proposed transmission scheme significantly outperforms the normal ones especially under the scenarios featuring long information bits, low received signal-to-noise, and small Rician factor.

Energy efficiency maximization strategy for Sink node in SWIPT-enabled sensor-cloud based on optimal stopping rules

Leveraging energy harvesting abilities in wireless network devices has emerged as an effective way to prolong the lifetime of energy constrained systems. The system gains are usually optimized by designing resource allocation algorithm appropriately. However, few works focus on the interaction that channel's time-vary characters make the energy transfer inefficiently. To address this, we propose a novel system operation sequence for sensor-cloud system where the Sinks provide SWIPT for sensor nodes opportunistically during downlink phase and collect the data transmitted from sensor nodes in uplink phase. Then, the energy-efficiency maximization problem of the Sinks is presented by considering the time costs and energy consumption of channel detection. It is proved that the formulated problem is an optimal stopping process with optimal stopping rules. An optimal energy-efficiency (OEE) algorithm is designed to obtain the optimal stopping rules for SWIPT. Finally, the simulations are performed based on the OEE algorithm compared with the other two strategies to verify the effectiveness and gains in improving the system efficiency.

Costly Multi-Unit Search

We formulate and solve a costly multi-unit search problem for the optimal selling of a stock of goods. Our showcase application is an inventory liquidation problem with fixed holding costs, such as warehousing, salaries or floor planning. A seller faces a stream of buyers periodically arriving with random capped demands. At each decision point, he decides how to price each unit and also whether to stop search or not. We set this as a dynamic programming problem and solve it inductively by characterizing optimal search rules and reservation prices. We show that combining multiple units with a fixed per period search cost might translate into non-monotone selling costs and reservation prices. This lack of monotonicity naturally leads to discontinuities of the pricing strategy. In particular, the seller optimally employs strategies such as bundling, and more sophisticated ones that endogenously combine purchase premiums, when inventory is large, with clearance sales and discounts, when inventory is low. Our model extends search theory by explicitly accounting for the effects of fixed costs on optimal multi-unit pricing strategies, pushing it into a richer class of problems and offering solutions that extend beyond optimal stopping rules.

Multistage Estimation of the Rayleigh Distribution Variance

In this paper we discuss the multistage sequential estimation of the variance of the Rayleigh distribution using the three-stage procedure that was presented by Hall (Ann. Stat. 9(6):1229–1238, 1981). Since the Rayleigh distribution variance is a linear function of the distribution scale parameter’s square, it suffices to estimate the Rayleigh distribution’s scale parameter’s square. We tackle two estimation problems: first, the minimum risk point estimation problem under a squared-error loss function plus linear sampling cost, and the second is a fixed-width confidence interval estimation, using a unified optimal stopping rule. Such an estimation cannot be performed using fixed-width classical procedures due to the non-existence of a fixed sample size that simultaneously achieves both estimation problems. We find all the asymptotic results that enhanced finding the three-stage regret as well as the three-stage fixed-width confidence interval for the desired parameter. The procedure attains asymptotic second-order efficiency and asymptotic consistency. A series of Monte Carlo simulations were conducted to study the procedure’s performance as the optimal sample size increases. We found that the simulation results agree with the asymptotic results.

OEDDBOS: An Efficient Data Distributing Strategy with Energy Saving in Sensor-Cloud Systems

Sensor-cloud is a developing technology and popular paradigm for various applications. It integrates wireless sensor into a cloud computing environment. On the one hand, the cloud offers extensive data storage and analytical and processing capabilities not available in sensor nodes. On the other hand, data distribution (such as time synchronization and configuration files) is always an important topic in such sensor-cloud systems, which leads to a rapid increase in energy consumption by sensors. In this paper, we aim to reduce the energy consumption of data dissemination in sensor-cloud systems and study the optimization of energy consumption with time-varying channel quality when multiple nodes use the same channel to transmit data. Suppose that there is a certain probability that the nodes send data for competing channel. And then, they decide to distribute data in terms of channel quality for saving energy after getting the channel successfully whether or not. Firstly, we construct the maximization problem of average energy efficiency for distributing data with delay demand. Then, this maximization problem transferred an optimal stopping problem which generates the optimal stopping rule. At last, the thresholds of the optimal transmission rate in each period are solved by using the optimal stopping theory, and the optimal energy efficiency for data distribution is achieved. Simulation results indicate that the strategy proposed in this paper can to some extent improve average energy efficiency and delivery ratio and enhance energy optimization effect and network performance compared with other strategies.

Optimal Stopping Time for Geometric Random Walks with Power Payoff Function

Two optimal stopping problems for geometric random walks with the observer’s power payoff function, on the finite and infinite horizons, are solved. For these problems, an explicit form of the cut value and also optimal stopping rules are established. It is proved that the optimal stopping rules are nonrandomized thresholds and describe the corresponding free boundary. An explicit form of the free boundary is presented.

The Dynkin game with regime switching and applications to pricing game options

This paper is concerned with the Dynkin game (a zero-sum optimal stopping game). The dynamic of the system is modeled by a regime switching diffusion, in which the regime switching mechanism provides the structural changes of the random environment. The goal is to find a saddle point for the payoff functional up to one of the players exiting the game. Taking advantage of the method of penalization and the dynamic programming principle, the value function of the game problem is shown to be the unique viscosity solution to the associated variational inequalities. We also consider a financial example of pricing game option under a regime switching market. Both optimal stopping rules for the buyer and the seller and fair price of the option are numerically demonstrated in this example. All the results are markedly different from the traditional cases without regime switching.

Optimal rules for the sequential selection of uniform spacings

ABSTRACT The unit interval is broken at random into n spacings with the breaking points given by a random sample of size n−1 from the uniform distribution on . A decision-maker observes the lengths of these spacings sequentially and must decide either to select the present spacing or to reject it and continue to observe the next one. No recall of preceding observations is permitted. In this paper, we first find an optimal stopping rule to maximize the probability of selecting the largest spacing. Furthermore, we establish a connection to the classical secretary problem, from which we derive a lower bound for the maximum probability. Moreover, we conjecture that the limiting optimal probability is the same as in the full-information best-choice problem. Next, we find an optimal stopping rule to maximize the expected length of the selected spacing. It is shown that the maximum obtainable expected length lies between the expected value of the second-largest and the largest spacing.

Asymptotically optimal prior-free clock auctions

Clock auctions have a number of properties that make them attractive for practical purposes. They are weakly group strategy-proof, make bidding truthfully an obviously dominant strategy, and preserve trading agents' privacy. However, optimal reserve prices and stopping rules depend on the details of underlying distributions, and so clock auctions have proved challenging to implement in a prior-free, asymptotically optimal way. In this paper, we develop a prior-free clock auction that is asymptotically optimal by exploiting a relationship between hazard rates and the spacings between order statistics. Extensions permit price discrimination among heterogeneous groups, minimum revenue thresholds, and quantity caps.

Optimal stopping for the exponential of a Brownian bridge

AbstractWe study the problem of stopping a Brownian bridgeXin order to maximise the expected value of an exponential gain function. The problem was posed by Ernst and Shepp (2015), and was motivated by bond selling with non-negative prices.Due to the non-linear structure of the exponential gain, we cannot rely on methods used in the literature to find closed-form solutions to other problems involving the Brownian bridge. Instead, we must deal directly with a stopping problem for a time-inhomogeneous diffusion. We develop techniques based on pathwise properties of the Brownian bridge and martingale methods of optimal stopping theory, which allow us to find the optimal stopping rule and to show the regularity of the value function.

Bayesian sequential testing with expectation constraints

We study a stopping problem arising from a sequential testing of two simple hypotheses H0 and H1 on the drift rate of a Brownian motion. We impose an expectation constraint on the stopping rules allowed and show that an optimal stopping rule satisfying the constraint can be found among the rules of the following type: stop if the posterior probability for H1 attains a given lower or upper barrier; or stop if the posterior probability comes back to a fixed intermediate point after a sufficiently large excursion. Stopping at the intermediate point means that the testing is abandoned without accepting H0 or H1. In contrast to the unconstrained case, optimal stopping rules, in general, cannot be found among interval exit times. Thus, optimal stopping rules in the constrained case qualitatively differ from optimal rules in the unconstrained case.

A random walk version of Robbins' problem: small horizon

In Robbins' problem of minimizing the expected rank, a finite sequence of $n$ independent, identically distributed random variables are observed sequentially and the objective is to stop at such a time that the expected rank of the selected variable (among the sequence of all $n$ variables) is as small as possible. In this paper we consider an analogous problem in which the observed random variables are the steps of a symmetric random walk. Assuming continuously distributed step sizes, we describe the optimal stopping rules for the cases $n=2$ and $n=3$ in two versions of the problem: a version in which the actual steps of the random walk are disclosed to the decision maker; and a version in which only the relative ranks of the positions taken by the random walk are observed. When $n=3$, the optimal rule and expected rank depend on the distribution of the step sizes. We give sharp bounds for the optimal expected rank in the partial information version, and fairly sharp bounds in the full information version.

Early Stopping for Kernel Boosting Algorithms: A General Analysis With Localized Complexities

Early stopping of iterative algorithms is a widely-used form of regularization in statistics, commonly used in conjunction with boosting and related gradient-type algorithms. Although consistency results have been established in some settings, such estimators are less well-understood than their analogues based on penalized regularization. In this paper, for a relatively broad class of loss functions and boosting algorithms (including L2-boost, LogitBoost and AdaBoost, among others), we exhibit a direct connection between the performance of a stopped iterate and the localized Gaussian complexity of the associated function class. This connection allows us to show that local fixed point analysis of Gaussian or Rademacher complexities, now standard in the analysis of penalized estimators, can be used to derive optimal stopping rules. We derive such stopping rules in detail for various kernel classes, and illustrate the correspondence of our theory with practice for Sobolev kernel classes.

Search Among Queues Under Quality Differentiation

Customers looking for service providers often face search frictions and have to trade off quality and availability. To understand customers’ search behavior when they are confronted with a large collection of vertically differentiated, congested service providers, we build a model in which arriving customers conduct a costly sequential search to resolve uncertainty about service providers’ quality and queue length and select one to join by optimal stopping rules. Customers search, in part, because of variations in waiting time across service providers, which, in turn, is determined by the search behavior of customers. Thus, an equilibrium emerges. We characterize customers’ equilibrium search/join behavior in a mean field model as the number of service providers grows large. We find that reducing either the search cost or customer arrival rate may increase the average waiting time in the system as customers substitute toward high-quality service providers. Moreover, with lower search costs, the improved quality obtained by customers may not make up for the prolonged wait, therefore degrading the average search reward and, more importantly, decreasing customer welfare; when customers search, their welfare can even be lower than if they are not allowed to search at all. This paper was accepted by Gad Allon, operations management.

A direct solution method for pricing options in regime‐switching models

AbstractPricing financial or real options with arbitrary payoffs in regime‐switching models is an important problem in finance. Mathematically, it is to solve, under certain standard assumptions, a general form of optimal stopping problems in regime‐switching models. In this article, we reduce an optimal stopping problem with an arbitrary value function in a two‐regime environment to a pair of optimal stopping problems without regime switching. We then propose a method for finding optimal stopping rules using the techniques available for nonswitching problems. In contrast to other methods, our systematic solution procedure is more direct as we first obtain the explicit form of the value functions. In the end, we discuss an option pricing problem, which may not be dealt with by the conventional methods, demonstrating the simplicity of our approach.

Monitoring a Poisson process subject to gradual changes in the arrival rates

We look at a Poisson process where the arrival rates change from a known λ1 to a known λ2. Whereas in most of the literature the change-point is abrupt, we model the more realistic assumption that states that the change happens gradually over a period of time η where η is known. We calculate the probability that the change has started and completed. We also look at optimal stopping rules assuming that there is a cost for a false alarm and a cost per time unit to stop early. We conclude with some numerical results.

OPTIMAL SELECTION OF THE k-TH BEST CANDIDATE

In the subject of optimal stopping, the classical secretary problem is concerned with optimally selecting the best of n candidates when their relative ranks are observed sequentially. This problem has been extended to optimally selecting the kth best candidate for k ≥ 2. While the optimal stopping rule for k=1,2 (and all n ≥ 2) is known to be of threshold type (involving one threshold), we solve the case k=3 (and all n ≥ 3) by deriving an explicit optimal stopping rule that involves two thresholds. We also prove several inequalities for p(k, n), the maximum probability of selecting the k-th best of n candidates. It is shown that (i) p(1, n) = p(n, n) > p(k, n) for 1<k<n, (ii) p(k, n) ≥ p(k, n + 1), (iii) p(k, n) ≥ p(k + 1, n + 1) and (iv) p(k, ∞): = lim n→∞p(k, n) is decreasing in k.

Deep Optimal Stopping

In this paper we develop a deep learning method for optimal stopping problems which directly learns the optimal stopping rule from Monte Carlo samples. As such, it is broadly applicable in situations where the underlying randomness can efficiently be simulated. We test the approach on three problems: the pricing of a Bermudan max-call option, the pricing of a callable multi barrier reverse convertible and the problem of optimally stopping a fractional Brownian motion. In all three cases it produces very accurate results in high-dimensional situations with short computing times.

Intelligent Multi-Agent Based Multilayered Control System for Opportunistic Load Scheduling in Smart Buildings

Today’s power systems are subject to the high penetration of dynamic load. Volatility and intermittency of the dynamic load demand need to be compensated through optimization and scheduling without compromising user comfort. This paper proposes a multi-agent-based multi-layered hierarchical control system for residential load management under real-time pricing environment. The major objectives are to reduce peak load demand, electricity cost, and user discomfort. In doing so, different types of agents, i.e., price agent $p_{a}$ , sensor agent $s_{a}$ , decision agent $d_{a}$ , load agent $l_{a}$ , and action agent $a_{a}$ , are developed to control residential loads, such as normal load ( nl ) and heavy load ( hl ). To handle price uncertainty, dynamically, optimal stopping rule (OSR) theory has been used. Two variants of OSR are proposed: 1) priority inversion logic-based OSR to subsidize the responsive consumers and 2) maximum energy consumption limit $Q$ -based OSR-Q to maximize the profit of energy retailers. Finally, the proposed mechanism is validated on a set of loads to show the applicability and proficiency under a dynamic environment.

An Optimal Dividend Problem with Capital Injections over a Finite Horizon

In this paper, we propose and solve an optimal dividend problem with capital injections over a finite time horizon. The surplus dynamics obeys a linearly controlled drifted Brownian motion that is reflected at the origin: dividends give rise to time-dependent instantaneous marginal profits, whereas capital injections are subject to time-dependent instantaneous marginal costs. The aim is to maximize the sum of a liquidation value at terminal time and of the total expected profits from dividends, net of the total expected costs for capital injections. Inspired by the study of El Karoui and Karatzas [Integration of the optimal risk in a stopping problem with absorption, in Séminaire de Probabilités XXIII, Springer, Berlin, 1989, pp. 405--420] on reflected follower problems, we relate the optimal dividend problem with capital injections to an optimal stopping problem for a drifted Brownian motion that is absorbed at the origin. We show that whenever the optimal stopping rule is triggered by a time-dependent boundary, the value function of the optimal stopping problem gives the derivative of the value function of the optimal dividend problem. Moreover, the optimal dividend strategy is also triggered by the moving boundary of the associated stopping problem. The properties of this boundary are then investigated in a case study in which instantaneous marginal profits and costs from dividends and capital injections are constants discounted at a constant rate.