Finite Time Horizon Research Articles

Approximations of random periodic solutions for path‐dependent stochastic differential equations with finite/infinite delay

The aim of this work is to show the numerical approximation of random periodic solutions for path‐dependent stochastic differential equations with a finite and an infinite delay, in which the drifts are only dissipative on average with respect to the time parameter. The key findings are as follows: (i) by using a combination of synchronous coupling approaches and continuous‐time Euler–Maruyama schemes, we study the existence and uniqueness of numerical random periodic solutions to path‐dependent stochastic differential equations on an infinite time horizon; (ii) by introducing a new technical method, we investigate the strong convergence in the mean‐square sense of a numerical approximation to path‐dependent stochastic differential equations in the case of an infinite time horizon; (iii) to the best of our knowledge, our result is the first one upon numerical approximations of random periodic solutions for path‐dependent stochastic differential equations (SDEs) with infinite delay. We expect that our approximated method can be extended to a more general structure.

Dynamic Contract Design in the Presence of Double Moral Hazard

We consider a stylized incentive management problem over an infinite time horizon, where the principal hires an agent to provide services to customers. Customers request service in one of two ways: either via an online or a traditional offline channel. The principal does not observe the offline customers’ arrivals, nor does she observe whether the agent exerts (costly) effort that can increase the arrival rate of customers. This creates an opportunity for the agent (i) to divert cash (that is, to under-report the number of offline customers and pocket respective revenues) and also (ii) to shirk (that is, not to exert effort), thus leading to a novel and thus far unexplored double moral hazard problem. To address this problem, we formulate a constrained, continuous-time, stochastic optimal control problem and derive an optimal contract with a simple intuitive structure that includes a payment scheme and a potential termination time of the agent. We enrich the model to allow the principal to either (i) dynamically adjust the prices for the services in both channels or (ii) monitor the agent. Both tools help the principal to alleviate the double moral hazard problem. We derive respective optimal strategies for using those tools that guarantee the highest profits. We show that the worse the agent’s past performance is, the lower the prices should be set and the more the principal should monitor the agent. This paper was accepted by Ilia Tsetlin, behavioral economics and decision analysis. Funding: The work of F. Tian is financially supported by the Hong Kong RGC Funding [RGC General Research Fund 2023/24 17502023], HKU Seed Fund for Basic Research for New Staff (2022), and the HKU Business School Shenzhen Research Institute [SZRI2023-BRF-04]. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2022.01169 .

Residence time density (RTd) identification through deconvolution of finite time horizon input output data and its Application to a novel analytical dispersion process RTd model

This work presents first an approach to the identification of Residence Time density (RTd) and Residence Time Distribution (RTD) functions, through deconvolution of finite time horizon, input/output data generated by a zero-order hold input, using the novel concept of zero-order hold, average, impulse response coefficients. Subsequently, a novel RTd analytical expression is derived for an open-open axial dispersion process with a finite experimental section in a tube of infinite length, and an explanation is provided for this RTd model’s differences from the literature. This new RTd expression is then employed in two case studies to demonstrate the effectiveness of the proposed identification through a deconvolution conceptual framework in identifying RTd model parameters, and in assisting in the selection of RTd models which are compatible with experimental data.

On the infinite time horizon approximation for Lévy-driven McKean-Vlasov SDEs with non-globally Lipschitz continuous and super-linearly growth drift and diffusion coefficients

This paper studies the numerical approximation for McKean-Vlasov stochastic differential equations driven by Lévy processes. We propose a tamed-adaptive Euler-Maruyama scheme and consider its strong convergence in both finite and infinite time horizons when applying for some classes of Lévy-driven McKean-Vlasov stochastic differential equations with non-globally Lipschitz continuous and super-linearly growth drift and diffusion coefficients.

Lp-solutions of Multi-dimensional Oblique Reflected BSDEs and Optimal Switching Problem on Finite or Infinite Time Horizon

Lp-solutions of Multi-dimensional Oblique Reflected BSDEs and Optimal Switching Problem on Finite or Infinite Time Horizon

Joint optimization of production, inspection, and maintenance under finite time for smart manufacturing systems

Given the flexible and configurable characteristics of smart manufacturing systems with a limited time per manufacturing task, the assumption of infinite time for prostems is no longer applicable to the joint-control strategy. Consequently, a joint-control model that considers production, inspection, and maintenance within a finite-time scenario for smart manufacturing systems is proposed in this paper. The objective is to optimize overall production and maintenance functions to minimize the total system cost. Comparing the joint strategy under infinite time with the proposed finite-time approach reveals significant differences in unit costs between the two scenarios. To enhance the effectiveness of the model, a discrete iterative algorithm with multiple loops was developed. Through a case study, it was observed that 1) joint strategies implemented within a finite time horizon were more cost-effective than those under infinite time, thus emphasizing the need for business managers to develop strategies within a finite time frame; 2) different production planning and efficiency levels had varying effects on the final joint strategy, necessitating customized strategies based on different production durations. Overall, the research gap regarding joint strategies within a finite-time context was addressed in this research, serving as a methodological foundation for practitioners to develop various strategies that minimize total costs across diverse real-world scenarios.

Infinite-Horizon Probability of Ruin for a Variable-Memory Counting Process (Hawkes Process)

The reserve R(t) of an insurance company denotes the accumulated capital up to time ttt throughout its operational span. While this reserve can be modeled using stochastic processes, the endeavor remains notably intricate. The mathematical complexity presents a considerable obstacle for researchers within this domain. Prior research has predominantly focused on reserve models constructed from Markovian processes. In this manuscript, our attention shifts to risk models derived from non-Markovian processes, particularly those with claim arrivals governed by Hawkes processes. Before delving into the core subject matter, we provide an extensive review of the literature on counting processes with variable memory. This includes a thorough exploration of Hawkes processes, Gerber-Shiu functions, integro-differential equations, and Laplace transforms. These elements are crucial for computing the probability of ruin over an infinite time horizon, especially in scenarios where interdependence exists between inter-claim times. Additionally, we present the requisite mathematical tools essential for comprehending the contents of this article, specifically tailored for individuals with a foundational understanding of actuarial science. Furthermore, we have successfully determined the probability of ruin, marking a significant milestone in our investigation.

Finite-horizon energy allocation scheme in energy harvesting-based linear wireless sensor network

Linear wireless sensor networks (LWSNs) are a specialized topology of wireless sensor networks (WSNs) widely used for environmental monitoring. Traditional WSNs rely on batteries for energy supply, limiting their performance due to battery capacity constraints, while renewable energy harvesting technology is an effective approach to alleviating the battery capacity bottleneck. However, the stochastic nature of renewable energy makes designing an efficient energy management scheme for network performance improvement a compelling research problem. In this paper, we investigate the problem of maximizing throughput over a finite-horizon time period for an energy harvesting-based linear wireless sensor network (EH-LWSN). The solution to the original problem is very complex, and this complexity mainly arises from two factors. First, the optimal energy allocation scheme has temporal coupling, i.e., the current optimal strategy relies on the energy harvested in the future. Second, the optimal energy allocation scheme has spatial coupling, i.e., the current optimal strategy of any node relies on the available energy of other nodes in the network. To address these challenges, we propose an iterative energy allocation algorithm for EH-LWSN. Firstly, we theoretically prove the optimality of the algorithm and analyze the time complexity of the algorithm. Next, we design the corresponding distributed version and consider the case of estimating the energy harvest. Finally, through experiments using a real-world renewable energy dataset, the results show that the proposed algorithm outperforms the other two heuristics energy allocation schemes in terms of network throughput.

Group machinery intelligent maintenance: Adaptive health prediction and global dynamic maintenance decision-making

Intelligent preventive maintenance powered by health data analytics is essential to ensure operation safety and performance of diverse industrial equipment. Despite the rapid advancement of preventive maintenance methodologies in recent several decades, their implementation to global dynamic maintenance of complex systems across infinite time horizon, in particular leveraging real-time prognosis information to realize adaptive group partition and ordering, has been largely an under-explored domain. To this end, this study devises a generic global-dynamic group maintenance policy for multi-component degrading systems. As opposed to existing models, the policy automatically renews the entire system health information instantly upon the completion of each group maintenance task; therefore, it realizes the dynamic union of (a) predictive group maintenance and (b) unplanned opportunistic maintenance over an infinite time horizon for the first time, such that to promote decision precision and efficiency. A dynamic grouping algorithm is designed to explore the structure of optimal maintenance solutions. The applicability is exemplified by comparative experiments that show substantial advantages over heuristic policies.

Event-triggered quasi-time-varying [formula omitted] filtering for switched systems via multiple trigger-dependent Lyapunov functionals

This paper investigates the event-triggered filtering problem for switched systems from the perspective of switched impulsive systems (SISs) by exploiting the characteristics of trigger and impulse behavior. By constructing multiple trigger-dependent Lyapunov functionals (MTDLFs), an event-triggered quasi-time-varying (ET-QTV) H∞ filtering scheme is developed. Since the MTDLFs properly capture the event-induced impulse property of SISs, they release less conservatism. The filtering stability in the scenario of frequent plant switching during an interevent interval is also guaranteed by incorporating the average dwell time switching. The filtering synthesis condition is formulated as a set of QTV matrix inequalities defined over a finite time horizon, based on which a sum of squares-based filter design algorithm is presented further. Additionally, the derivation of a lower bound on interevent intervals is clarified for the event-triggering mechanism (ETM) to exclude the Zeno phenomenon. Finally, the effectiveness and potential practicability of the proposed filtering scheme are validated using a morphing aircraft model.

Discrete-time optimal control problems with time delay argument: New discrete-time Euler–Lagrange equations with delay

Discrete-time optimal control problems are a crucial type of control problems that deal with a dynamic system evolving in discrete time-steps. This paper introduces a new technique for solving linear discrete-time optimal control problems with state delays, applicable to both finite and infinite time horizons. Our method employs a Riccati matrix equation, optimizing control strategies and ensuring system stability through bounded control inputs. We adopt a Bolza problem for the performance index, which guides the classification of control issues. The technique simplifies problems into manageable Riccati matrix equations using the Euler–Lagrange equations and Pontryagin maximum principle, ensuring stability and necessary condition compliance. The paper validates the approach with numerical examples from quantum mechanics and classical physics, demonstrating its practicality and potential for broader application.

Pricing for perpetual American strangle options under stochastic volatility with fast mean reversion

A perpetual American strangle option refers to an investment strategy combining the features of both call and put options on a single underlying asset, with an infinite time horizon. Investors are known to use this trading strategy when they expect the stock price to fluctuate significantly but cannot predict whether it will rise or fall. In this study, we consider the perpetual American strangle options under a stochastic volatility model and investigate the corrected option values and free boundaries using an asymptotic analysis technique. Further, we examine the pricing accuracy of the approximated formulas for perpetual American strangle options under stochastic volatility by comparing our solutions with the prices that are obtained from Monte Carlo simulations. We also investigate the sensitivities of the option values and free boundaries with respect to several model parameters.

Risk-Averse Markov Decision Processes Through a Distributional Lens

By adopting a distributional viewpoint on law-invariant convex risk measures, we construct dynamic risk measures (DRMs) at the distributional level. We then apply these DRMs to investigate Markov decision processes, incorporating latent costs, random actions, and weakly continuous transition kernels. Furthermore, the proposed DRMs allow risk aversion to change dynamically. Under mild assumptions, we derive a dynamic programming principle and show the existence of an optimal policy in both finite and infinite time horizons. Moreover, we provide a sufficient condition for the optimality of deterministic actions. For illustration, we conclude the paper with examples from optimal liquidation with limit order books and autonomous driving. Funding: This work was supported by Natural Sciences and Engineering Research Council of Canada [Grants RGPAS-2018-522715 and RGPIN-2018-05705].

Risk-sensitive benchmarked portfolio optimization under non-linear market dynamics

We discuss a continuous-time portfolio optimization problem to beat a stochastic benchmark. The model considers non-linear stochastic differential equations (SDEs) to model the dynamics of assets and economic factors. Unlike existing literature on risk-sensitive criteria, the proposed framework allows the model to capture the non-linearity in assets and factors dynamics. This article contributes to the two essential aspects of the problem; first, the existence and uniqueness of optimal investment strategy, which we prove using stochastic control theory and shows optimal strategies remain unchanged for the finite and infinite time horizon problems. Second, we use forecasting to compare investment performance under different economic factors. We analyze returns for the proposed model for three years in two important financial markets; S&P 100 and Dow 30. Risk versus return analysis indicates the importance of choosing relevant economic factors and their model for better portfolio performance. The results suggest an excellent bet to select the non-linear model over the linear one.

Enabling Space-Based Computed Cloud Tomography with a Mixed Integer Linear Programming Scheduler

Cumuliform clouds in Earth’s lower atmosphere play important roles in the radiative balance and hydrological cycle of the climate system. Yet key processes unfolding in these clouds, such as aerosol interactions and precipitation initiation, are still poorly understood. Current space-based cloud observation methods are inadequate for inferring the internal structure of clouds outside of a narrow transect that is probed actively by radar and lidar. Cloud tomography is an emerging technique that uses passive imaging of a cloud target from multiple locations with a large angular range to infer internal cloud structure. Many low convective clouds only have a lifetime of 15–25 min, necessitating autonomous scheduling of observation targets as they appear. Onboard autonomous scheduling is formulated as a mixed-integer optimization problem (MILP) with a finite time horizon and a reward scheme designed to maximize the angular range of the observations. A finite time horizon MILP scheduler is well suited to this mission because the short lifetime of low convective clouds creates a natural time horizon. This MILP can solve for an optimal observation schedule in a maximum time of 15 ms on a desktop CPU. The MILP scheduler is able to observe nearly 60% more targets than a conventional push-broom camera configuration. This initial result is promising and demonstrates the need for continued research in this area.

Optimal pair trading: Consumption-investment problem with finite and infinite horizon

Abstract We present a simple solution of the consumption-investment problem pair trading on a finite time horizon. The proof is based on the remark that the HJB equation can be reduced to a linear parabolic equation solvable explicitly. As a further development we obtain also a solution of the problem for the infinite time horizon.

Scalable Distributed State Estimation over Binary Sensor Networks with Energy Harvester

This paper deals with distributed state estimation problem for discrete time-varying systems over binary sensor networks, where every binary sensor is equipped with an energy harvester. The input of every binary sensor considers the randomly occurring missing measurements. The differences between the real and estimated inputs of binary sensor are employed to derive useful information in order to address the insufficient information for estimation purpose. The information from neighboring nodes is transmitted only if its energy level is positive, where a random variable is introduced to formulate the energy level. By means of the available information, distributed estimator is constructed for each binary sensor and the desirable performance constraints is given for the dynamic characteristics of estimation errors within a finite time horizon. Sufficient conditions are established for the existence of desired distribution estimation quantities through local performance analysis methods. Also, the desired distributed estimator gains are calculated recursively, which means the desirable scalability. Ultimately, the viability and efficiency of the distributed scheme are exhibited through a practical illustration.

Optimal profit in two-level trade credit eoq model with default risk and reminder cost under finite time horizon having time-dependent demand and deterioration

Trade credit is a type of promotional activity that generally increases demand and revenue but also invites default risk due to dishonest customers. Due to default risk, revenue is lost, and to overcome this, an arrangement is made to remind the defaulters. A retailer dealing with a perishable item wants to exhaust the stock quickly after a certain deterioration level. Demand and deterioration of the products are normally dynamic. The business period of seasonal products is uncertain. Considering these facts, we formulate and analyze a two-level trade credit inventory model, where the wholesaler and retailer give credit periods to their corresponding downstream customers. After a certain level of deterioration, the retailer increases the credit period for the customers for early stock exhaustion, and to reduce default risk, a reminder cost is introduced. These activities increase the profit. The mathematical models under different circumstances are formulated for different time horizons. Some existing results are deduced. The models are numerically solved using a parametric study and the Generalised Reduced Gradient method through LINGO 19.0 software. Some lemmas and theorems are deduced to establish the analytical outcomes. Trade-offs between the number of the business cycle, trade credit, and reminder cost against optimum profit are separately demonstrated. The results with and without reminder cost are compared, and it is shown that the model with reminder cost fetches more profit. Profit under different uncertain environments are evaluated, and they differ marginally. Some beneficial impacts are discussed.

The one-station bike repositioning problem

In bike sharing systems the quality of the service to the users strongly depends on the strategy adopted to reposition the bikes. The bike repositioning problem is in general very complex as it involves different interrelated decisions: the routing of the repositioning vehicles, the scheduling of their visits to the stations, the number of bikes to load or unload for each station and for each vehicle that visits the station. In this paper we study the problem of optimally loading/unloading vehicles that visit the same station at given time instants of a finite time horizon. The goal is to minimize the total lost demand of bikes and free stands in the station. We model the problem as a mixed integer linear programming problem and present an optimal algorithm that runs in linear time in the size of the time horizon.

Expected Value of the Occupation Times of Brownian Motion

Occupation times of a stochastic process describes the amount of time the process spends inside a spatial interval during a certain finite time horizon. It appears in the fiber lay-down process in nonwoven production industry. The occupation time can be interpreted as the mass of fiber material deposited inside some region. From application point of view, it is important to know the average mass per unit area of the final fleece. In this paper derive an expression for the expected value of the occupation times in terms of Gaussian error functions.