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Braking distances play an important role in the organization of rail traffic and exploitation of rail vehicles. The braking distances of rail vehicles affect several factors that cannot be defined and determined precisely. In this paper, a mathematical model, for train moving through braking, is presented by taking into account the relevant train braking resistance and a procedure is given for solving a differential equation of movement of the rail vehicle during braking. The procedure allows determining influential factors that have an effect on the length of braking distances. The influence and the changes of specific resistance during the movement of the train, braking force, and adhesion between wheel and rail during the braking process are presented. The results obtained with the presented mathematical model are accurate and match with the results obtained experimentally. [ABSTRACT FROM AUTHOR]

Based on a deterministic and stochastic process hybrid model, we use white noises to account for patient variabilities in treatment outcomes, use a hyperparameter to represent patient heterogeneity in a cohort, and construct a stochastic model in terms of Ito stochastic differential equations for testing the efficacy of three different treatment protocols in CAR T cell therapy. The stochastic model has three ergodic invariant measures which correspond to three unstable equilibrium solutions of the deterministic system, while the ergodic invariant measures are attractors under some conditions for tumor growth. As the stable dynamics of the stochastic system reflects long-term outcomes of the therapy, the transient dynamics provide chances of cure in short-term. Two stopping times, the time to cure and time to progress, allow us to conduct numerical simulations with three different protocols of CAR T cell treatment through the transient dynamics of the stochastic model. The probability distributions of the time to cure and time to progress present outcome details of different protocols, which are significant for current clinical study of CAR T cell therapy., Competing Interests: Declaration of competing interest There is non-financial interest in this research., (Copyright © 2024 Elsevier Inc. All rights reserved.)

In this paper, we present a switching property for squared Bessel process. We prove that the starting point and the time parameter can be in a sense switched. As a consequence, we obtain new distributional dependencies among a squared Bessel process R, a Brownian motion with drift B μ , μ ≥ 0 , B t μ ≔ B t + μt for a Brownian motion B , and an integral functional of geometric Brownian motion A(μ) connected by the Lamperti relation. Finally, we deduce a new explicit form of the joint distribution of the Lamperti triple R T A t μ exp 2 B t μ for fixed t > 0 andT > 0. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Modeling correlated or highly stratified multiple‐response data is a common data analysis task in many applications, such as those in large epidemiological studies or multisite cohort studies. The generalized estimating equations method is a popular statistical method used to analyze these kinds of data, because it can manage many types of unmeasured dependence among outcomes. Collecting large amounts of highly stratified or correlated response data is time‐consuming; thus, the use of a more aggressive sampling strategy that can accelerate this process—such as the active‐learning methods found in the machine‐learning literature—will always be beneficial. In this study, we integrate adaptive sampling and variable selection features into a sequential procedure for modeling correlated response data. Besides reporting the statistical properties of the proposed procedure, we also use both synthesized and real data sets to demonstrate the usefulness of our method. [ABSTRACT FROM AUTHOR]

Background: The Elliot wave principle commonly characterizes the impulsive and corrective wave trends for both financial market trends and electrocardiograms. The impulsive wave trends of electrocardiograms can annotate several wave components of heart-beats including pathological heartbeat waveforms. The stopping time inquires which ordinal element satisfies the assumed mathematical condition within a numerical set. The proposed work constitutes several algorithmic states in reinforcement learning from the stopping time decision, which determines the impulsive wave trends. Each proposed algorithmic state is applicable to any relevant algorithmic state in reinforcement learning with fully numerical explanations. Because commercial electrocardiographs still misinterpret myocardial infarctions from extraordinary electrocardiograms, a novel algorithm needs to be developed to evaluate myocardial infarctions. Moreover, differential diagnosis for right ventricle infarction is required to contraindicate a medication such as nitroglycerin.Methods: The proposed work implements the stopping time theory to impulsive wave trend distribution. The searching process of the stopping time theory is equivalent to the actions toward algorithmic states in reinforcement learning. The state value from each algorithmic state represents the numerically deterministic annotated results from the impulsive wave trend distribution. The shape of the impulsive waveform is evaluated from the interoperable algorithmic states via least-first-power approximation and approximate entropy. The annotated electrocardiograms from the impulsive wave trend distribution utilize a structure of neural networks to approximate the isoelectric baseline amplitude value of the electrocardiograms, and detect the conditions of myocardial infarction. The annotated results from the impulsive wave trend distribution consist of another reinforcement learning environment for the evaluation of impulsive waveform direction.Results: The accuracy to discern myocardial infarction was found to be 99.2754% for the data from the comma-separated value format files, and 99.3579% for those containing representative beats. The clinical dataset included 276 electrocardiograms from the comma-separated value files and 623 representative beats.Conclusions: Our study aims to support clinical interpretation on 12-channel electrocardiograms. The proposed work is suitable for a differential diagnosis under infarction in the right ventricle to avoid contraindicated medication during emergency. An impulsive waveform that is affected by myocardial infarction or the electrical direction of electrocardiography is represented as an inverse waveform. [ABSTRACT FROM AUTHOR]

We consider an optimal switching problem where the terminal reward depends on the entire control trajectory. We show existence of an optimal control by applying a probabilistic technique based on the concept of Snell envelopes. We then apply this result to solve an impulse control problem for stochastic delay differential equations driven by a Brownian motion and an independent compound Poisson process. Furthermore, we show that the studied problem arises naturally when maximizing the revenue from operation of a group of hydro-power plants with hydrological coupling. [ABSTRACT FROM AUTHOR]

A single jump filtration generated by a random variable γ with values in on a probability space (F, P) is defined as follows: a set A F belongs to Ft if A∩{γ > t} is either or {γ > t}. A processM is proved to be a local martingale with respect to this filtration if and only if it has a representationMt = F(t)1{t<γ }+L1{γ }, where F is a deterministic function and L is a random variable such that EMt < ∞ and E(Mt ) = E(M0) for every t {t R+ : P(γ t) > 0}. This result seems to be new even in a special case that has been studied in the literature, namely, where F is the smallest σ-field with respect to which γ is measurable (and then the filtration is the smallest one with respect to which γ is a stopping time). As a consequence, a full description of all local martingales is given and they are classified according to their global behaviour. [ABSTRACT FROM AUTHOR]

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 λ is given as τ = inf t ≥ 0 : ξ t > λ . When the distribution is symmetrical and has Markov characteristics, there is P τ ≤ t = 2 P ξ t > λ . 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 n ≥ 1 : X n > x 0 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: ET = 5 . [ABSTRACT FROM AUTHOR]

We consider sequential detection based on quantized data in the presence of eavesdropper. Stochastic encryption is employed as a counter measure that flips the quantization bits at each sensor according to certain probabilities, and the flipping probabilities are only known to the legitimate fusion center (LFC) but not the eavesdropping fusion center (EFC). As a result, the LFC employs the optimal sequential probability ratio test (SPRT) for sequential detection whereas the EFC employs a mismatched SPRT (MSPRT). We characterize the asymptotic performance of the MSPRT in terms of the expected sample size as a function of the vanishing error probabilities. We show that when the detection error probabilities are set to be the same at the LFC and EFC, every symmetric stochastic encryption is ineffective in the sense that it leads to the same expected sample size at the LFC and EFC. Next, in the asymptotic regime of small detection error probabilities, we show that every stochastic encryption degrades the performance of the quantized sequential detection at the LFC by increasing the expected sample size, and the expected sample size required at the EFC is no fewer than that is required at the LFC. Then the optimal stochastic encryption is investigated in the sense of maximizing the difference between the expected sample sizes required at the EFC and LFC. Although this optimization problem is nonconvex, we show that if the acceptable tolerance of the increase in the expected sample size at the LFC induced by the stochastic encryption is small enough, then the globally optimal stochastic encryption can be analytically obtained; and moreover, the optimal scheme only flips one type of quantized bits (i.e., 1 or 0) and keeps the other type unchanged. [ABSTRACT FROM AUTHOR]

We study a finite horizon balance sheet optimal multi-modes switching problem related to trade-off strategies between expected profit and cost cash flows. The problem is formulated in terms of Snell envelopes for the profit and the cost yields which act as obstacles to each other, moreover we fully characterize the optimal strategies. Then using the link between the Snell envelope of processes and reflected backward stochastic differential equations (RBSDEs for short), solving the problem turns out actually to solving the related system of RBSDEs, for which we prove the existence of a continuous minimal solution using an approximation scheme. [ABSTRACT FROM AUTHOR]

Abstract 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.

We study a finite horizon optimal multi-modes switching problem with many nodes. The switching is based on the optimal expected profit and cost yields, moreover both sides of the balance sheet are considered. The profit and cost yields per unit time are respectively assumed to be coupled through a coupling term which is the average of profit and cost yields. The corresponding system of Snell envelopes is highly complex, so we consider the aggregated yields where a mean-field approximation is used for the coupling term. First, the problem is formulated by the mean of the Snell envelope of processes. Then, in terms of backward SDEs, the problem is equivalent to a system of mean-field reflected backward SDEs with interconnected and nonlinear obstacles. More precisely, the driver function depends also on the mean of the unknown process (expected profit or cost yields) which makes the mean-field interaction in the driver nonlinear. The first main result of this paper, is to show the existence of a continuous minimal solution of the system of mean-field reflected backward SDEs, which is done by using the Picard iteration method. The second main result concerns the optimality of the switching strategies which we fully characterize. [ABSTRACT FROM AUTHOR]

In this work we discuss almost sure convergence for sums of arbitrarily dependent stochastic sequence under different conditions of Chung's type. Our approach is based on the stopping time technique and the theorem of convergence for martingale difference sequence. Meanwhile, the results here include some relevant classical conclusions. [ABSTRACT FROM AUTHOR]

In this paper we study continuous-time two-player zero-sum optimal switching games on a finite horizon. Using the theory of doubly reflected backward stochastic differential equations with interconnected barriers, we show that this game has a value and an equilibrium in the players' switching controls. [ABSTRACT FROM AUTHOR]

We propose a two-stage sequential method for obtaining tandem-width confidence intervals for a Bernoulli proportion p. The term "tandem-width" refers to the fact that the half-width of the confidence interval is not fixed beforehand; it is instead required to satisfy two different half-width upper bounds, h0 and h1, depending on the (unknown) values of p. To tackle this problem, we first propose a simple but useful sequential method for obtaining fixed-width confidence intervals for p, whose stopping rule is based on the minimax estimator of p. We observe Bernoulli(p) trials sequentially, and for some fixed half-width h = h0 or h1, we develop a stopping time T such that the resulting confidence interval for p, [ ], covers the parameter with confidence at least where is the maximum likelihood estimator of p at time T. Furthermore, we derive theoretical properties of our proposed fixed-width and tandem-width methods and compare their performances with existing alternative sequential schemes. The proposed minimax-based fixed-width method performs similarly to alternative fixed-width methods, while being easier to implement in practice. In addition, the proposed tandem-width method produces effective savings in sample size compared to the fixed-width counterpart and provides excellent results for scientists to use when no prior knowledge of p is available. [ABSTRACT FROM AUTHOR]

Let X1, X2,... be independent random variables observed sequentially and such that X1,..., Xθ−1 have a common probability density p0, while Xθ, Xθ+1,... are all distributed according to p1 ≠ p0. It is assumed that p0 and p1 are known, but the time change θ ∈ ℤ+ is unknown and the goal is to construct a stopping time τ that detects the change-point θ as soon as possible. The standard approaches to this problem rely essentially on some prior information about θ. For instance, in the Bayes approach, it is assumed that θ is a random variable with a known probability distribution. In the methods related to hypothesis testing, this a priori information is hidden in the so-called average run length. The main goal in this paper is to construct stopping times that are free from a priori information about θ. More formally, we propose an approach to solving approximately the following minimization problem: Δ (θ ; τ α) → min τ α subject to α (θ ; τ α) ≤ α for any θ ≥ 1 , where α(θ; τ) = Pθ{τ < θ} is the false alarm probability and Δ(θ; τ) = Eθ(τ − θ)+ is the average detection delay computed for a given stopping time τ. In contrast to the standard CUSUM algorithm based on the sequential maximum likelihood test, our approach is related to a multiple hypothesis testing methods and permits, in particular, to construct universal stopping times with nearly Bayes detection delays. [ABSTRACT FROM AUTHOR]

Let and denote Markov processes on being the results of the following transformations of a symmetric -stable process on : the first one coincides with the process killed at the stopping time and the second one is determined by its transition probability density defined for , , and by the equality , where g is transition probability density of the process . If those transformations lead us to the same process. In the case of they are different. Some properties of these processes are investigated. We restrict ourselves, in this paper, to the case of . [ABSTRACT FROM AUTHOR]

Joint detection and estimation refers to deciding between two or more hypotheses and, depending on the test outcome, simultaneously estimating the unknown parameters of the underlying distribution. This problem is investigated in a sequential framework under mild assumptions on the underlying random process. We formulate an unconstrained sequential decision problem, whose cost function is the weighted sum of the expected run-length and the detection/estimation errors. Then, a strong connection between the derivatives of the cost function with respect to the weights, which can be interpreted as Lagrange multipliers, and the detection/estimation errors of the underlying scheme is shown. This property is used to characterize the solution of a closely related sequential decision problem, whose objective function is the expected run-length under constraints on the average detection/estimation errors. We show that the solution of the constrained problem coincides with the solution of the unconstrained problem with suitably chosen weights. These weights are characterized as the solution of a linear program, which can be solved using efficient off-the-shelf solvers. The theoretical results are illustrated with two example problems, for which optimal sequential schemes are designed numerically and whose performance is validated via Monte Carlo simulations. [ABSTRACT FROM AUTHOR]

The buffer behavior of routers has an important influence on the quality of service of network services. Exploring theory and methods to analyze the characteristics of the buffer is an interesting job, especially when aggregate traffic is transmitted. This letter investigates the buffer behavior of a single server queuing system with multiple arrival flows, which models the operations of a router. A martingale analytical framework is proposed, which could evaluate buffer performance for aggregate traffic arrivals with different distributions. We use the processes of arrival and service to construct martingales, and analyze the queuing system with heterogeneous arrivals in martingale domain. Applying the stopping time theory of martingale, the transient performance of queue buffer is presented. The probability of buffer overflow and the expectation of buffer stopping time are derived together in this model. The simulations are given to validate the analytical results. This investigation provides a valuable reference for router buffer design. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying stopping time problem of interval-valued differential equation. Based on length constraint, we discuss the stopping time without Lipschitz condition, monotonicity and nonnegativity hypotheses. Then, stopping times with respects to upper and lower boundednesses are studied without monotonicity hypothesis, which is necessary in [27]. The results can be applied to discuss stopping time problem of more general interval-valued differential equations. We also provide some examples to verify our results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of local smooth solution to stochastic magneto-hydrodynamic equations without diffusion forced by additive noise in R 3 . We first transform the system into a random system via a simple change of variable and borrow the result obtained for classical magneto-hydrodynamic equations, then we show that this random transformed system is measurable with respect to the stochastic element. Finally we extend the solution to the maximality solution. Due to the coupled construction of this system, we need more elaborate and complicated estimates with respect to stochastic Euler equation.

We investigate the convergence of hitting times for jump-diffusion processes. Specifically, we study a sequence of stochastic differential equations with jumps. Under reasonable assumptions, we establish the convergence of solutions to the equations and of the moments when the solutions hit certain sets.

In sequential analysis, an experimenter may gather information regarding an unknown parameter by observing random samples in successive steps. We emphasize a number of specific parametric models under a variety of loss functions. The total number of observations collected at termination is a positive integer-valued random variable, customarily denoted by . The exact probability distribution of N is often hard to obtain. However, under a set of regulatory conditions, our standardized version of the stopping variable from Definition 2.1 in Section 2.2 would follow a normal distribution in the asymptotic sense. In this article, we first show how these regulatory conditions build upon one another in order to conclude the asymptotic normality of such standardized stopping variable, . We provide exploratory data analysis (EDA) via a number of interesting illustrations obtained through large-scale simulation studies. We demonstrate the broad applicability of purely sequential methodologies included in this article along with the appropriateness of our conclusions regarding the asymptotic normality of the standardized stopping variable as a practical guideline. [ABSTRACT FROM AUTHOR]

The invariance formulas for stopping times of squared Bessel processes with positive index are presented. For a stopping time τ satisfying some relatively mild assumptions and every positive Borel function f, we have the invariance formula where and γμ is a gamma random variable independent of σ(τ, R(τ)). The second invariance formula is where K is a constant and Δμ is an explicitly given function. The applications of invariance formulas are presented. In particular, new results on the distribution of a killed squared Bessel process, a conditional inverse local time, and the first hitting time of a graph by a squared Bessel process are delivered. [ABSTRACT FROM AUTHOR]

We consider a type of optimal switching problems with non-uniform execution delays and ramping. Such problems frequently occur in the operation of economical and engineering systems. We first provide a solution to the problem by applying a probabilistic method. The main contribution is, however, a scheme for approximating the optimal control by limiting the information in the state-feedback. In a numerical example the approximation routine gives a considerable computational performance enhancement when compared to a conventional algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we obtain fixed-width confidence interval for covariate-adjusted response-adaptive designs. Specifically, we consider logistic regression model and the normal regression model for binary and continuous responses, respectively, both in the situations for presence and absence of treatment–covariate interactions. Simulation study and real-data analysis are carried out. [ABSTRACT FROM AUTHOR]