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In this paper, we introduce the definitions of stopping time, forward and backward solutions to interval-valued differential equations under generalized Hukuhara differentiability, which could be applied to discuss the evolution of dynamical systems with practical backgrounds. By using these definitions, we study stopping time problems for the Malthusian population model and the logistic model in details. Then, some general conclusions about stopping time problems for interval-valued differential equations are considered and the results are shown to be feasible by providing some examples.

In this paper we will give a simple stopping time proof in the $\mathbb{R}^d$ setting of the matrix weighted Triebel-Lizorkin bounds proved by F. Nazarov/S. Treil and A. Volberg, respectively. Furthermore, we provide explicit matrix A${}_p$ characteristic dependence and also discuss some interesting open questions., 13 pages, no figures, submitted

Many parts or components in a facility are sometime vulnerable to failure causing downtime and production loss. So, in order to decrease the downtime, parts of the same type that follow the same lifespan distribution need to be replaced at least once to minimize the cost of unscheduled labor and the damages to the products. A decision on replacing all such vulnerable parts decreases the total cost including penalty cost, purchasing cost and holding cost. Different preempt times of a machine result in different purchasing quantity, procurement cost and carrying costs. So, a rational decision of allowable stopping time of a machine and order cycle for vulnerable parts jointly minimizes the total cost of supply chain. This study proposes a novel model to minimize the total cost for the vulnerable parts which have joint influence on machine performance. An integrated nonlinear cost model developed here is optimally solved and tested with several different distributions. Strong results are propounded to support the proposed model in terms of manufacturing time and system cost.

In this work we solve in a closed form the problem of an agent who wants to optimise the inter-temporal utility of both his consumption and leisure by choosing: (i) the optimal inter-temporal consumption, (ii) the optimal inter-temporal labour supply, (iii) the optimal share of wealth to invest in a risky asset, and (iv) the optimal retirement age. The wage of the agent is assumed to be stochastic and correlated with the risky asset on the financial market. The problem is split into two sub-problems: the optimal consumption, labour, and portfolio problem is solved first, and then the optimal stopping time is approached. The martingale method is used for the first problem, and it allows to solve it for any value of the stopping time which is just considered as a stochastic variable. The problem of the agent is solved by assuming that after retirement he received a utility that is proportional to the remaining human capital. Finally, a numerical simulation is presented for showing the behaviour over time of the optimal solution.

Establishing protective projects such as breakwaters and flood barriers is the key to preventing marine disasters caused by extreme sea conditions. One of the core technical problems in implementing these large-scale projects is how to determine the fortification criteria reasonably. In the past, research based on random variables studied the determined state of time for the stochastic process. For the first time, this paper studies the statistical characteristics of ocean environment elements from both the time and space with the real perspective of the stochastic process and introduces the concept of stopping time in stochastic processes into the analysis of storm surge. The relationship between the stopping time and threshold selection for the measured data is discussed. The relationship between the wave front displacement and the time exceeding the threshold $$\lambda$$ is given as $$\tau = \inf \left\{ {t \ge 0:\xi \left( t \right) > \lambda } \right\}$$. When the distribution is symmetrical and has Markov characteristics, there is $$P\left( {\tau \le t} \right) = 2P\left( {\xi \left( t \right) > \lambda } \right)$$. Additionally, the relationship between the calculation and the return period of the wave height is given. It is proved that the stopping time $$N = \inf \left\{ {n \ge 1:X_{n} > x_{0} } \right\}$$ obeys the geometric distribution, and the expectation of the stopping time is the return period in the ocean engineering. Through the analysis of the stopping time parameters, the rationality of applying the Gumbel distribution in extreme sea conditions for calculating return period is given. With the relationship between the service life of marine engineering and the commonly used parameters, the average life expectancy of offshore engineering is: $${\text{ET}} = 5$$.

We consider determining the optimal stopping time for the glue curing of wood panels in an automatic process environment. Using the near-infrared spectroscopy technology to monitor the manufacturing process ensures substantial savings in energy and time. We collect a time-series of curves from a near-infrared spectrum probe consisting of 72 spectra and aim to detect an optimal stopping time. We propose an estimation procedure to determine the optimal stopping time of wood panel compression and the estimation uncertainty associated with the estimated stopping time. Our method first divides the entire data set into a training sample and a testing sample, then iteratively computes integrated squared forecast errors based on the testing sample. We then apply a structural break detection method with one breakpoint to determine an estimated optimal stopping time from a univariate time-series of the integrated squared forecast errors. We also investigate the finite sample performance of the proposed method via a series of simulation studies.

A modification of a classical Bachelier model by letting a stock price absorb at zero is revisited. Alternative proofs are given to derive option pricing formulas under the modified Bachelier model and numerical comparison with the Black–Scholes formula is provided. Quantile hedging methodology is developed for both classical and modified Bachelier models and application to pricing the pure endowment with fixed guarantee life insurance contracts is demonstrated, both theoretically and by means of a numerical example.

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.

We study the problem of tracking stopping time in a distributed circumstance. More specially, a Gaussian random walk with positive drift is observed simultaneously by multiple detectors in a noisy fashion. The goal is to estimate the first-passage time when the random walk reaches a predefined level. After characterizing the optimal estimator with mean absolute p-moment, a distributed tracking stopping time estimation algorithm is proposed, where each detector only exchanges observation with their neighbors based on the average consensus scheme. Then, we provide theoretical performance bound and testify the asymptotic optimality of proposed distributed stopping time estimator. Finally, performance analysis is depicted by the numerical simulation.

It is proved that the law of a possibly killed Levy process X , seen up to and including (resp. up to strictly before) a stopping time, determines already the law of X (resp. up to a compound Poisson component and killing).

This paper attempts to study the optimal stopping time for semi-Markov processes (SMPs) under the discount optimization criteria with unbounded cost rates. In our work, we introduce an explicit construction of the equivalent semi-Markov decision processes (SMDPs). The equivalence is embodied in the value functions of SMPs and SMDPs, that is, every stopping time of SMPs can induce a policy of SMDPs such that the value functions are equal, and vice versa. The existence of the optimal stopping time of SMPs is proved by this equivalence relation. Next, we give the optimality equation of the value function and develop an effective iterative algorithm for computing it. Moreover, we show that the optimal and {\epsilon}-optimal stopping time can be characterized by the hitting time of the special sets. Finally, to illustrate the validity of our results, an example of a maintenance system is presented in the end., Comment: 24 pages, 0 figure

What are the optimal times for an Internet of Things (IoT) device to act as a blockchain miner? The aim is to minimize the energy consumed by low-power IoT devices that log their data into a secure (tamper-proof) distributed ledger. We formulate the energy-efficient blockchain mining for IoT devices as a multiple-stopping time partially observed Markov decision process (POMDP) to maximize the probability of adding a block in the blockchain; we also present a model to optimize the number of stops (mining instants). In general, POMDPs are computationally intractable to solve, but we show mathematically using submodularity that the optimal mining policy has a useful structure: 1) it is monotone in belief space, and 2) it exhibits a threshold structure, which divides the belief space into two connected sets. Exploiting the structural results, we formulate a computationally-efficient linear mining policy for the blockchain-enabled IoT device. We present a policy gradient technique to optimize the parameters of the linear mining policy. Finally, we use synthetic and real Bitcoin datasets to study the performance of our proposed mining policy. We demonstrate the energy efficiency achieved by the optimal linear mining policy in contrast to other heuristic strategies.

The article deals with an arbitrary elastic 3D-system (body) that performs a controlled finite rotation with respect to some fixed axis and small nonstationary oscillations. The system oscillations occur due to external load (power control) or inertial load of the rotational transportation of the carrying body (kinematic control). The linear equations of oscillations are used in normal coordinates, in which motion is represented by eigenmodes of vibrations for system that is free in the rotation angle (including system rotation as a solid body in the case of power control) and for system fixed in rotation angle in the case of kinematic control. It is assumed that the (power or inertial)load acting on the system is proportional to some controlling finite time function from a certain class. The purpose of this article is to solve the problem of system rotation for a certain time from one rest position to another at a given finite angle using the given control function and to eliminate the elastic oscillations on the several lowest eigenmodes at the stopping time. The relations between the time of the system rotation under the action of a given control function and the eigenmodes frequencies for oscillations being eliminated are obtained on the basis of the exact solutions of the equations in normal coordinates. These relations satisfy the zero initial and final conditions. They are “tuned” by minimizing the positive definite quadratic form written for them by varying the system parameters to fulfill these relations simultaneously for several eigenfrequencies. As an example, the calculations for a model of a symmetrical spacecraft with two identical elastic solar cell panels consisting of four planar non-deformable sections connected by elastic hinges are carried out for comparison and analysis of the results accuracy. The finite rolling motion of the system with the damping at the stopping time of rotation for several (from one to three) lowest eigenmodes of antisymmetric vibrations is considered. The comparisons of the initial equations of motion for the system in generalized coordinates using several simple control functions and the found parameters of the “tuned” system with numerical solutions are accomplished.

Evaluation measures are more or less explicitly based on user models which abstract how users interact with a ranked result list and how they accumulate utility from it. However, traditional measures typically come with a hard-coded user model which can be, at best, parametrized. Moreover, they take a deterministic approach which leads to assign a precise score to a system run.In this paper, we take a different angle and, by relying on Markov chains and random walks, we propose a new family of evaluation measures which are able to accommodate for different and flexible user models, allow for simulating the interaction of different users, and turn the score into a random variable which more richly describes the performance of a system. We also show how the proposed framework allows for instantiating and better explaining some state-of-the-art measures, like AP, RBP, DCG, and ERR.

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.

The velocity distribution in the vicinity of the surface of an axisymmetric body rotating in a viscous medium at frequency ω directed along its axis is determined. The dissipative function has been calculated and used for deriving the equation of motion, from which an analytic expression for the stopping time of the body (until its complete stoppage) is obtained. The time of entrainment of an external stationary cylinder coaxial with the body is calculated by solving the time-dependent Navier–Stokes equation.

In this paper we obtain a recursive formula for the density of the double-barrier Parisian stopping time. We present a probabilistic proof of the formula for the first few steps of the recursion, and then a formal proof using explicit Laplace inversions. These results provide an efficient computational method for pricing double-barrier Parisian options.

This paper concerns stopping-time management of smart sensing nodes, a kind of very large scale integration system, based on the tradeoffs between sensing accuracy and power consumption, which are foundations of Internet of Things systems and cyber-physical systems. In practice, smart sensing nodes work periodically, and more samples leads to higher accuracy but more energy cost, so when to stop sampling in a periodic cycle to achieve the optimal tradeoff is an interesting issue. This paper formulates this issue as a partially observable Markov decision process (POMDP) and develops a POMDP-based Optimal Stopping-time Algorithm to make the above tradeoff. Field experiments demonstrate its performance.

In this work, we consider a stochastic differential equation that generalizes the well known Paris' equation from fracture of materials. The model describes the propagation of cracks on solids, and it includes a deterministic summand and a stochastic component in terms of a Brownian motion. The use of Ito's stochastic integral gives an equivalent stochastic integral equation that is further generalized here. We note that the probability distribution of the stopping time of the general model satisfies a deterministic diffusion-advection partial differential equation for which the solution is known only in a reduced number of particular cases. Motivated by these analytical results, we develop a fast finite-difference method to approximate the distribution of the stopping time. The method is an explicit exponential-like technique that preserves the main features of a probability distribution, namely, the non-negativity, the boundedness from above by 1 as well as the spatial monotonicity. Moreover, the method is a monotone technique that is also capable of preserving the temporal monotonicity of the approximations. These properties of the proposed methodology are thoroughly established in the present manuscript. A continuity condition of the numerical solutions in terms of the initial conditions and the temporal computational parameter is established also, together with a limiting property of the methodology when the free parameter tends to infinity. For comparison purposes, we are providing an implicit and stable discretization of the mathematical model which has a second order of convergence but for which conditions that guarantee the positivity, the boundedness and the monotonicity of approximations are not available. The numerical simulations obtained with implementations of our techniques show that the explicit method is an efficient scheme that preserves the characteristics of interest (non-negativity, boundedness from above by 1 and monotonicity), and that the numerical approximations are in good agreement with the known exact solutions.

A common approach to retirement planning focuses on building sufficient funds to retire at a fixed, predefined retirement time horizon. Optimal portfolio strategies are applied in an effort to achieve this goal. These strategies typically lead to the wealth at retirement having substantial uncertainty, making it difficult for investors to understand how much to save and how the savings rate impacts their wealth when they retire. A cash investment overcomes these problems, but is generally too expensive for most savers. Instead, we optimize on statistics of the stopping time at which one has achieved a target wealth. This is related to the notion of labor optionality introduced in "Labor supply flexibility and portfolio choice in a life cycle model", 1992, by Bodie, Merton, and Samuelson, and avoids some of the shortcomings of predefined retirement time approaches. We solve this retirement stopping time problem using stochastic optimization with a Markov control. The objective function is relatively flat near the optimum, so, for example, reducing both variance and trading frequency have little impact on the optimum. We also prove that shorting strategies are stochastically dominated by ones which do not short, which is of general interest for financial optimization problems.

As the first part in the present paper, we study a class of backward stochastic differential equation (BSDE, for short) driven by Teugels martingales associated with some Levy processes having moment of all orders and an independent Brownian motion. We obtain an existence and uniqueness result for this type of BSDEs when the final time is allowed to be random. As the second part, we prove, under a monotonicity condition, an existence and uniqueness result for fully coupled forward–backward stochastic differential equation (FBSDE, for short) driven by Teugels martingales in stopping time duration. As an illustration of our theoretical results, we deal with a portfolio selection in Levy-type market.

Departing from a general stochastic differential equation with Brownian diffusion, we establish that the distribution of probability of the stopping time is governed by a parabolic partial differential equation. A particular form of the problem under investigation may be associated to a stochastic generalization of the well-known Paris’ law from structural mechanics, in which case, the solution of the boundary-value problem represents the probability distribution of the hitting time. An implicit, convergent and probability-based discretization to approximate the solution of the boundary-value problem is proposed in this work. Using a convenient vector representation of our scheme, we prove that the method preserves the most relevant properties of a probability distribution function, namely, the non-negativity, the boundedness from above by 1, and the monotonicity. In addition, we establish that our method is a convergent technique, and provide some illustrative comparisons against known exact solu...

Unexpected events are introduced to the asset replacement algorithm, and modeled by a Poisson process for all cash-flows. Resale price result is expected to decrease randomly with jumps. A risk-neutral stopping time, evaluating the probability of a gain from the first minimum in the valuation sequence in comparison to a longer cycle, is illustrated with a stochastic logistic process. The solution to this option reveals a Paradox – the probabilities for a longer cycle and the risk-free rate are mutually exclusive.

I propose a new methodology to assess the effect of foreign exchange (FX) intervention, based on the probability of an FX rate reaching. The variable is the probability of an FX rate reaching a particular threshold before reaching another. Importantly, the probability depends on not only the level, but also the trend and volatility of a current FX rate. When an intervention changes the probability in a desired direction, the intervention is effective. The notable feature of the probability is that it considers both the level and volatility of an FX rate comprehensively, while previous studies have examined these effects of FX intervention separately. Empirical results based on regression and nearest-neighbor analyses applied to Japanese data indicate that publicity and size are significant in the effectiveness of intervention.

Public transport priority is the development trend in public transport, and signal priority is its main means. In order to further improve the accuracy of delay calculation and realize the priority of bus signals, this paper proposes the idea of multiple conversion criteria and consideration of stop time for the coordination and control of bus and car mixed traffic flow trunk roads. First of all, on the basis of in-depth analysis of the differences in the characteristics of bus and car models, a multi-conversion standard delay calculation method is proposed, and its effectiveness is verified by simulation. The results show that compared with the single conversion standard delay calculation method, the average delay error of cars and buses calculated by this method is reduced by 22.54% and 82.21%, respectively. Then, the influence of bus stops on bus speed and delay is further analyzed, and the coordinated control model of bus priority trunk roads considering bus stops is constructed with the passenger capacity of each bus line and the per capita delay as the goal, and the solution is given. Finally, 178 randomly generated examples are used to verify and analyze the effectiveness and sensitivity of this model.

We give an analytic description for the infinitesimal generator constructed by Applebaum-Estrade for Lévy flights on a broad class of closed Riemannian manifolds including all negatively-curved manifolds, the flat torus and the sphere. Various properties of the associated semigroup and the asymptotics of the expected stopping time for Lévy flight based random searches for small targets, also known as the narrow capture problem, are then obtained using our newfound understanding of the infinitesimal generator. Our study also relates to the Lévy flight foraging hypothesis in the field of biology as we compute the expected time for finding a small target by using the Lévy flight random search. A similar calculation for Brownian motion on surfaces was done in [arXiv:2209.12425]., 40 pages

This research studies the impact of maximum allowable stopping time for machining tools on the production-inventory policy under a variable tools lifespan and demand. A mathematical model is developed to determine the optimal maximal allowing stopping time, the inventory level, and the replenishment interval. A numerical example was conducted using an exhaustive searching method to show the implementation of our model. The results indicate that, compared to the traditional policy which only repairs and replace the broken tools, adopting a maximum allowable stopping time dramatically reduced the total production cost.

This paper deals with the portfolio selection problem of the exponential utility with stopping time under the O-U processes model. By using the dynamic programming and the Feynman-Kac approach, the optimal strategy and the optimal stopping time are obtained.

We consider a class of stopping time problems in which the state of the process evolves according to a discrete-time Markov chain and the action which terminates the process is available only in a certain subset of the state space. For the objective of maximizing total expected discounted reward we develop an infinite-horizon Markov decision process formulation and derive two families of sufficient conditions under which the optimal policy exhibits a control-limit structure. We present numerical examples to illustrate the two families of conditions we derive are satisfiable and do not overrule each other. We also demonstrate that a family of our conditions is weaker than those in the literature which guarantee the optimality of control-limit policies for a special case of the problem we study.

Stopping-time Banach spaces is a collective term for the class of spaces of eventually null integrable processes that are defined in terms of the behaviour of the stopping times with respect to some fixed filtration. From the point of view of Banach space theory, these spaces in many regards resemble the classical spaces such as $L^1$ or $C(\Delta)$, but unlike these, they do have unconditional bases. In the present paper we study the canonical bases in the stopping-time spaces in relation to factorising the identity operator thereon. Since we work exclusively with the dyadic-tree filtration, this set-up enables us to work with tree-indexed bases rather than directly with stochastic processes. \emph{En route} to the factorisation results, we develop general criteria that allow one to deduce the uniqueness of the maximal ideal in the algebra of operators on a Banach space. These criteria are applicable to many classical Banach spaces such as (mixed-norm) $L^p$-spaces, BMO, $\mathrm{SL^\infty}$ and others., Comment: 25 pages, accepted by Advances in Mathematics

In this paper, we consider the optimal stopping problem on semi-Markov processes (SMPs) with finite horizon, and aim to establish the existence and computation of optimal stopping times. To achieve the goal, we first develop the main results of finite horizon semi-Markov decision processes (SMDPs) to the case with additional terminal costs, introduce an explicit construction of SMDPs, and prove the equivalence between the optimal stopping problems on SMPs and SMDPs. Then, using the equivalence and the results on SMDPs developed here, we not only show the existence of optimal stopping time of SMPs, but also provide an algorithm for computing optimal stopping time on SMPs. Moreover, we show that the optimal and "-optimal stopping time can be characterized by the hitting time of some special sets, respectively., Comment: 27 pages, 0 figures

In this paper, we consider a size-dependent renewal risk model with stopping time claim-number process. In this model, we do not make any assumption on the dependence structure of claim sizes and inter-arrival times. We study large deviations of the aggregate amount of claims. For the subexponential heavy-tailed case, we obtain a precise large-deviation formula; our method substantially relies on a martingale for the structure of our models.

In the paper two approaches of estimation of the first boundary crossing time moment are considered. As the first estimate (for the process with the change-point on the interval of observation [0,t]) the conditional mean value expectation of the stopping-time is considered. The second considered estimate is the measurable and observable stopping-time moment for which the conditional mean value of the appropriate process is equal to the value of the boundary.

In protoplanetary disks, zones of dense particle configuration promote planet formation. Solid particles in dense clouds alter their motion through collective effects and back reaction to the gas. The effect of particle-gas feedback with ambient solid-to-gas ratios $\epsilon > 1$ on the stopping time of particles is investigated. In experiments on board the International Space Station we studied the evolution of a dense granular gas while interacting with air. We observed diffusion of clusters released at the onset of an experiment but also the formation of new dynamical clusters. The solid-to-gas mass ratio outside the cluster varied in the range of about $\epsilon_{\rm avg} \sim 2.5 - 60$. We find that the concept of gas drag in a viscous medium still holds, even if the medium is strongly dominated in mass by solids. However, a collective factor has to be used, depending on $\epsilon_{\rm avg} $, i.e. the drag force is reduced by a factor 18 at the highest mass ratios. Therefore, flocks of grains in protoplanetary disks move faster and collide faster than their constituents might suggest.

In Longstaff and Schwartz (2001) a method for American option pricing using simulation and regression is suggested, and since then the method has rapidly gained importance. However, the idea of using regression and simulation for American option pricing was used at least as early as in Carriere (1996). In the present paper we provide a thorough comparison of these two methods and relate them to the work of Tsitsiklis and Van Roy (2001). Although the methods are often considered to be similar this analysis allows us to point out an important but often overlooked difference between the methods. The paper further shows that due to this difference it is possible to provide arguments favoring the method suggested in Longstaff and Schwartz (2001). Finally, the paper compares the methods in a realistic numerical setting and shows that the practitioner does well in choosing the method of Longstaff and Schwartz (2001) instead of the methods of Carriere (1996) or Tsitsiklis and Van Roy (2001) for American option pricing.

This paper demonstrates how to convert a path-dependent optimal stopping time problem into a path-independent problem using a transformation analysis method. We test this method to deal with several problems, especially those in stochastic volatility environments. We introduce stochastic state variables into volatility dynamics and analyse the influence of state-variable volatile characters on investment stopping boundaries. For arbitrary coefficient circumstances, we set up a Riccati equation that satisfies the transformation. For circumstances involving Heston stochastic-volatility, we propose an analytical solution. This paper extends research on the optimal investment stopping issue to a stochastic investment opportunity environment. Our proposed method can enhance the ability of optimal investment stopping theory to describe the real capital market.

Opportunistic multicast schemes for time-sensitive stream traffic are studied for cellular networks where erasure coded packets are transmitted over discrete-time quasi-static forward-link fading channels. Assume that to successfully decode a transmitted stream fragment, it suffices that one receives k packets from the fragment. Two important issues are thus raised on when to stop transmitting a fragment and how to minimize the stopping time (ST) through transmission rate scheduling. Based on available channel state information and scheduled history, we tackle them particularly for small k required for short latency. If the distribution of IID channel states is available, the scheme is to compute and use the optimal constant transmission rate and minimum fixed ST subject to a reliability constraint. We show that the minimum ST grows with the logarithm of multicast group size. If channel state information is available, we propose to minimize random ST through selecting each optimal instantaneous transmission rate for a utility function. The utility function is specifically designed to exploit system transient states and dynamics. Results show that the scheme with an exponential weighted residual work achieves the least mean ST but the scheme maximizing instantaneous effective sum throughput has an edge of low complexity.