Optimal Multiple Stopping Research Articles

Robust Multiple Stopping—A Duality Approach

We develop a method to solve, theoretically and numerically, general optimal stopping problems. Our general setting allows for multiple exercise rights—that is, optimal multiple stopping—for a robust evaluation that accounts for model uncertainty with a dominated family of priors and for general reward processes driven by multidimensional jump-diffusions. Our approach relies on first establishing robust martingale dual representation results for the multiple stopping problem that satisfy appealing almost sure pathwise optimality properties. Next, we exploit these theoretical results to develop upper and lower bounds that, as we formally show, not only converge to the true solution asymptotically, but also constitute genuine prelimiting upper and lower bounds. We illustrate the applicability of our approach in a few examples and analyze the impact of model uncertainty on optimal multiple stopping strategies. Funding: R. J. A. Laeven received financial support from the Netherlands Organization for Scientific Research (NWO) [Grants NWO-Vidi and NWO-Vici]. J. G. M. Schoenmakers received financial support from the Deutsche Forschungsgemeinschaft Excellence Cluster Math+ Berlin [Project AA4-2]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2021.0237 .

Asymptotic Duration for Optimal Multiple Stopping Problems

We study the asymptotic duration of optimal stopping problems involving a sequence of independent random variables that are drawn from a known continuous distribution. These variables are observed as a sequence, where no recall of previous observations is permitted, and the objective is to form an optimal strategy to maximise the expected reward. In our previous work, we presented a methodology, borrowing techniques from applied mathematics, for obtaining asymptotic expressions for the expectation duration of the optimal stopping time where one stop is permitted. In this study, we generalise further to the case where more than one stop is permitted, with an updated objective function of maximising the expected sum of the variables chosen. We formulate a complete generalisation for an exponential family as well as the uniform distribution by utilising an inductive approach in the formulation of the stopping rule. Explicit examples are shown for common probability functions as well as simulations to verify the asymptotic calculations.

Optimal stopping and optimal multiple stopping problem without aggregation of reward family

Optimal stopping and optimal multiple stopping problem without aggregation of reward family

An exit contract optimization problem

We study an exit contract design problem, where one provides a universal exit contract to multiple heterogeneous agents, with which each agent chooses an optimal (exit) stopping time. The problem consists in optimizing the universal exit contract w.r.t. some criterion depending on the contract as well as the agents’ exit times. Under a technical monotonicity condition, and by using Bank-El Karoui’s representation of stochastic processes, we are able to transform the initial contract optimization problem into an optimal control problem. The latter is also equivalent to an optimal multiple stopping problem and the existence of the optimal contract is proved. We next show that the problem in the continuous-time setting can be approximated by a sequence of discrete-time ones, which would induce a natural numerical approximation method. We finally discuss the optimization problem over the class of all Markovian and/or continuous exit contracts.

A new deep neural network algorithm for multiple stopping with applications in options pricing

In this paper, we propose a deep learning method to solve high-dimensional optimal multiple stopping problems. We represent the policies of multiple stopping problems by the composition of functions. Using the new representation, we approximate the optimal stopping policy recursively with simulation samples. We also derive lower and upper bounds and confidence intervals for the values. Finally, we apply the algorithm to the pricing of swing options, and it produces accurate results in high-dimensional problems.

Optimal multiple stopping problem under nonlinear expectation

AbstractIn this paper, we study the optimal multiple stopping problem under the filtration-consistent nonlinear expectations. The reward is given by a set of random variables satisfying some appropriate assumptions, rather than a process that is right-continuous with left limits. We first construct the optimal stopping time for the single stopping problem, which is no longer given by the first hitting time of processes. We then prove by induction that the value function of the multiple stopping problem can be interpreted as the one for the single stopping problem associated with a new reward family, which allows us to construct the optimal multiple stopping times. If the reward family satisfies some strong regularity conditions, we show that the reward family and the value functions can be aggregated by some progressive processes. Hence, the optimal stopping times can be represented as hitting times.

Intrusion Prevention Through Optimal Stopping

We study automated intrusion prevention using reinforcement learning. Following a novel approach, we formulate the problem of intrusion prevention as an (optimal) multiple stopping problem. This formulation gives us insight into the structure of optimal policies, which we show to have threshold properties. For most practical cases, it is not feasible to obtain an optimal defender policy using dynamic programming. We therefore develop a reinforcement learning approach to approximate an optimal threshold policy. We introduce T- SPSA, an efficient reinforcement learning algorithm that learns threshold policies through stochastic approximation. We show that T- SPSA outperforms state-of-the-art algorithms for our use case. Our overall method for learning and validating policies includes two systems: a simulation system where defender policies are incrementally learned and an emulation system where statistics are produced that drive simulation runs and where learned policies are evaluated. We show that this approach can produce effective defender policies for a practical IT infrastructure.

Optimal Multiple Stopping Problems Under g-expectation

In this paper, we study the optimal multiple stopping problem under Knightian uncertainty both under discrete-time case and continuous-time case. The Knightian uncertainty is modeled by a single real-valued function g, which is the generator of a kind of backward stochastic differential equations. We show that the value function of the multiple stopping problem coincides with the one corresponding to a new reward sequence or process. For the discrete-time case, this problem can be solved by an induction method which is a straightforward generalization of the single stopping theory. For the continuous-time case, we furthermore need to establish the continuity of the new reward family. This result can be applied to the pricing problem for swing options in financial markets, which gives the holder of this contract at least two times rights to exercise it.

New class of optimal multiple stopping times problems

This paper is devoted to study a new discounted nonlinear optimal multiple stopping times problem with discounted factor β > 0 and infinite horizon. Under the right continuity of the underlying process, we show that the problem can be reduced to a sequence of ordinary optimal stopping problems. Next in the Markovian case, we characterise the value function of the problem in terms of β-excessive functions. Finally, in the special case of a diffusion process, we give explicit expressions for the value function of the problem as well as the optimal stopping strategy. As an explicit example in finance, we apply our theoretical results to manage a new generalised swing contract which gives its holder n rights to mark the price X of a stock, where the payment is only allowed at the final exercise time rather than at each exercise time as in the classical swing contact.

A Study on Optimal Multiple Stopping and Swing Options Pricing

In this paper, we have studied the optimal stopping of random process as well as the costing of Swing options, specially the valuation of electricity market which is considered to an American style option having multiple practicing rights. Since this type of options are widely used in investing, so it requires some methods for valuation and that should be as precise as possible. So, we discuss two numerical methods for getting swing options prices in the field of electricity market, namely Monte Carlo and Finite difference. Finally, we compare our obtained results numerically and graphically with the help of MATLAB. GANIT J. Bangladesh Math. Soc. 40.2 (2020) 145-155

Two-Stage Bayesian Sequential Change Diagnosis

In this paper, we formulate and solve a two-stage Bayesian sequential change diagnosis (SCD) problem. Different from the one-stage sequential change diagnosis problem considered in the existing work, after a change has been detected, we can continue to collect low-cost samples so that the post-change distribution can be identified more accurately. The goal of a two-stage SCD rule is to minimize the total cost including delay, false alarm probability, and misdiagnosis probability. To solve the two-stage SCD problem, we first convert the problem into a two-ordered optimal stopping time problem. Using tools from optimal multiple stopping time theory, we obtain the optimal SCD rule. Moreover, to address the high computational complexity issue of the optimal SCD rule, we further propose a computationally efficient threshold-based two-stage SCD rule. By analyzing the asymptotic behaviors of the delay, false alarm, and misdiagnosis costs, we show that the proposed threshold SCD rule is asymptotically optimal as the per-unit delay costs go to zero.

Modelling and computation of optimal multiple investment timing in multi-stage capacity expansion infrastructure projects

<p style='text-indent:20px;'>So far, the optimal investment timing to maximize the total profit of multi-stage capacity expansion infrastructure projects is not clear. In the case of uncertain demands, the optimal multiple stopping time theory is adopted to model the optimal decision-making of investment timing for multi-stage expansion infrastructure projects in a finite time horizon. In this context, the first-stage of the project involves a dedicated asset investment for later expansion, and the capacity of the project at each stage is constrained, which makes the cash flow of the project exhibit the characteristic of bull call spread. The upwind finite difference method and multi-least squares Monte Carlo simulation are combined to solve the project value and determine the optimal exercise boundaries at all stages described by a sequence of demand thresholds. A multi-stage power plant project is taken as an example to validate the model. Through the example, the optimal investment strategies and the value of the multi-stage project are provided; the effects of the dedicated asset and capacity constraint are illustrated. This study novelly reveals the effect of the capacity constraints on the project value using the bull call spread theory.</p>

Partial liquidation under reference-dependent preferences

We propose a multiple optimal stopping model where an investor can sell a divisible asset position at times of her choosing. Investors have S-shaped reference-dependent preferences, whereby utility is defined over gains and losses relative to a reference level and is concave over gains and convex over losses. For a price process following a time-homogeneous diffusion, we employ the constructive potential-theoretic solution method developed by Dayanik and Karatzas (Stoch. Process. Appl. 107:173–212, 2003). As an example, we revisit the single optimal stopping model of Kyle et al. (J. Econ. Theory 129:273–288, 2006) to allow partial liquidation. In contrast to the extant literature, we find that the investor may partially liquidate the asset at distinct price thresholds above the reference level. Under other parameter combinations, the investor sells the asset in a block, either at or above the reference level.

Optimal multiple stopping under catastrophic event

Optimal multiple stopping under catastrophic event

Optimal multiple stopping under catastrophic event

In this paper, we introduce a new optimal multiple stopping times problem, where we assume each exercise right happens before the date of release of a catastrophic event modelled by a random variable and this catastrophe can be natural (e.g., earthquake, tsunami) or technological (e.g., nuclear event). Since a sudden catastrophe will have a direct influence on prices variation, especially those of underlying as well as option's prime, eventual catastrophic event will be modelled by the first time the underlying's price exceeds some large barrier. The originality of this paper comes from a mathematical model taking account of a nonlinear criteria of sum of the underlying stopped at stopping times of the holder's filtration information as well as a random number of exercise rights at sopping times involving prior to a catastrophic event. This will generalises the concept of swing contracts, where the exercise rights number is only deterministic and finite.

New class of optimal multiple stopping times problems

New class of optimal multiple stopping times problems

Non-Nested Monte Carlo Dual Bounds for Multi-Exercisable Options

We study the optimal marginal value of discrete-time optimal multiple stopping problems and find that it can be formulated as a single optimal stopping optimization as well. Based on this result propose a marginal-value-based lower bound method to achieve a small bound on the iterative error. We further introduce a non-nested upper bound method. The convergence of both methods is analysed. The implementation details and enhancement techniques are discussed as well. Overall, our methods make a good trade-off between the time-efficiency and the tightness in dual bounds.

Optimal expansion timing decisions in multi-stage PPP projects involving dedicated asset and government subsidies

The topic of investment timing in multi-stage public-private partnership (PPP) projects has not been received much attention so far. This study investigates optimal expansion timing decisions in multi-stage PPP projects under an uncertain demand and where the first-stage greenfield project involving a dedicated asset is immediately implemented as the PPP contract is closed, whereas the timing of the later expansion is flexibly decided according to the demand. In this setting, the optimal multiple stopping timing theory is adopted to model the expansion framework. Furthermore, we integrate a government subsidy, including an investment subsidy and revenue subsidy, into the expansion timing decisions. Through a hypothetical three-stage investment plan for a sanitary sewerage project, the optimal expansion strategies and the value of the multi-stage project before and after the subsidy are provided using a least squares Monte Carlo simulation. Also, the influences of a dedicated asset on the expansion strategies and project value are illustrated. In addition, we compare the incremental value before and after the subsidy and earlier expansion derived from two types of subsidies. The comparisons show that there is more incremental value for the revenue subsidy, and that the investment subsidy brings an earlier expansion.

The viscosity solutions approach to swing options pricing under a regime-switching mean-reverting model

In this paper, we study the valuation of swing options on electricity markets with local volume and refraction time constraints, under the setting that the dynamic of the underlying spot price is a 2-state regime-switching mean-reverting process. We derive the corresponding optimal multiple stopping problem, reduce it to a sequence of optimal single stopping problems, and further find that those value functions satisfy HJB variational inequalities subject to suitable conditions. Then after a prior estimation for the value functions, the viscosity solutions approach is adopted to get existence and uniqueness results in viscosity sense.

Multiple stopping time POMDPs: Structural results & application in interactive advertising on social media

This paper considers a multiple stopping time problem for a Markov chain observed in noise, where a decision maker chooses at most L stopping times to maximize a cumulative objective. We formulate the problem as a Partially Observed Markov Decision Process (POMDP) and derive structural results for the optimal multiple stopping policy. The main results are as follows: (i) The optimal multiple stopping policy is shown to be characterized by threshold curves Γl, for l=1,…,L, in the unit simplex of Bayesian Posteriors. (ii) The stopping sets Sl (defined by the threshold curves Γl) are shown to exhibit the following nested structure Sl−1⊂Sl. (iii) The optimal cumulative reward is shown to be monotone with respect to the copositive ordering of the transition matrix. (iv) A stochastic gradient algorithm is provided for estimating linear threshold policies by exploiting the structural results. These linear threshold policies approximate the threshold curves Γl, and share the monotone structure of the optimal multiple stopping policy. (v) Application of the multiple stopping framework to interactively schedule advertisements in live online social media. It is shown that advertisement scheduling using multiple stopping performs significantly better than currently used methods.