Search

What is the number of rolls of fair six-sided dice until the first time the total sum of all rolls is a prime? We compute the expectation and the variance of this random variable up to an additive error of less than 10 − 4 . This is a solution to a puzzle suggested by DasGupta in the Bulletin of the Institute of Mathematical Statistics, where the published solution is incomplete. The proof is simple, combining a basic dynamic programming algorithm with a quick Matlab computation and basic facts about the distribution of primes. [ABSTRACT FROM AUTHOR]

This paper studies the issue of coordinating equipment maintenance operations with capital investment strategy in the presence of random equipment failures. The traditional approach, developed by Kamien and Schwartz (KS) in their celebrated paper published in 1971, is to formulate the problem as a deterministic optimal control problem with the probability of machine failure as the state variable. With this approach, the optimal policy is deterministic. As a major departure from the KS approach, we explicitly model the underlying stochastic process of machine failures. Our analysis of the stochastic dynamic programming model offers new insights into the problem. Under a long planning horizon with a limited replacement opportunity, each individual machine serves as a revenue generator and contributes a significant amount to the profit of the system. In contrast, when the replacement budget is quite generous over a relatively short planning horizon, adding one extra machine only helps as a backup for unexpected failures of the machines purchased before it. An interesting result derived from this comparison is that a deterministic policy turns out to be optimal for the former, while a state-contingent policy must be applied to the latter. In other words, the deterministic KS approach does not work in general when a chain of machine replacement is considered. We further characterize the effects of the discount rate, productivity deterioration, learning, decision delay, and technology advancement on the optimal policy. [ABSTRACT FROM AUTHOR]

We study a class of non-linear Backward stochastic differential equations (BSDE) with a superlinear driver process f adapted to a filtration F and over a random time interval [[0, S]] where S is a stopping time of F. The terminal condition ξ is allowed to take the value +∞, i.e., singular. We call a stopping time S solvable with respect to a given BSDE and filtration if the BSDE has a minimal supersolution with terminal value ∞ at terminal time S. Our goal is to show existence of solutions to the BSDE for a range of singular terminal values under the assumption that S is solvable. We will do so by proving that the minimal supersolution to the BSDE is a solution, i.e., it is continuous at time S and attains the terminal value with probability 1. We consider three types of terminal values: 1) Markovian: i.e., ξ is of the form ξ = g(ΞS) where Ξ is a continuous Markovian diffusion process, S is a hitting time of Ξ and g is a deterministic function 2) terminal conditions of the form ξ1 = ∞· 1(τ≥) and 3) ξ2 = ∞· 1(τ≥) where τ is another stopping time. For general ξ we prove that minimal supersolution has a limit at time S provided that F is left continuous at time S. Finally, we discuss the implications of our results about Markovian terminal conditions to the solution of non-linear elliptic PDE with singular boundary conditions. [ABSTRACT FROM AUTHOR]

All the available modes of travel and their respective travel parameters must be known to the commuters before their trip. Otherwise they may either spend more money or more time for the trip. In addition to this, recent pandemic, rapidly spreading novel corona virus is demanding a smart solution for contactless commuting. This paper suggests a practical solution to make both the above possible and it emphasizes the applicability of two developed android applications, one for travel data collection and another to predict travel time for a multimodal trip within the study area. If the whole trip is by a single mode, the user can get the corresponding travel time estimate from "Google maps". But, if the trip is by multiple modes, it is not possible to get the total travel time estimate for the whole trip at a time from "Google maps". A separate travel mode for "auto" is unavailable in "Google maps" alongside drive, two-wheeler, train or bus and walk alternatives. It is also observed that the travel time estimate of "Google maps" for the city buses is inaccurate. Hence, the two modes (Buses and Autos) were chosen for the study. Unless and until the travel times and stopping times of the two modes are known, it is not possible to predict their trip times. Hence, the mobility analysis was performed for the two modes in the study area to find their respective average travel rate at peak hours, across 15 corridors and the results were presented. [ABSTRACT FROM AUTHOR]

The paper considers the problem of optimal control in models with disorder and provides the computational scheme of its solution. The proposed method is based on replacing the disorder with its estimate by the stopping time. The calculations take place on the binary tree with nodes divided into two classes. The division of tree nodes into two classes is carried out using a decision rule associated either with testing two hypothesis, or with the optimal stopping of a Markov process developing on a binary tree. The estimate of the disorder is the moment at which the walk on the tree is placed in a node of the first class for the first time. The example of the optimal control problem is related to the problem of quantile hedging of a payoff. [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]

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 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]

This paper analyzes the asymptotic performances of fully distributed sequential hypothesis testing procedures as the type-I and type-II error rates approach zero, in the context of a sensor network without a fusion center. In particular, the sensor network is defined by an undirected graph, where each sensor can observe samples over time, access the information from the adjacent sensors, and perform the sequential test based on its own decision statistic. Different from most literature, the sampling process and the information exchange process in our framework take place simultaneously (or, at least in comparable time-scales), thus cannot be decoupled from one another. Our goal is to achieve second-order asymptotically optimal performance at every sensor, i.e., the average detection delay is within a constant gap from the centralized optimal sequential test as the error rates approach zero for the fixed number of sensors. To that end, a type of test procedure that resembles the well-known sequential probability ratio test (SPRT), termed as distributed SPRT (DSPRT) in this paper, is studied based on two message-passing schemes, respectively. The first scheme features the dissemination of the raw samples. In specific, every sample propagates over the network by being relayed from one sensor to another until it reaches all the sensors in the network. Although the sample propagation-based DSPRT is shown to yield the asymptotically optimal performance at each sensor, it incurs excessive inter-sensor communication overhead due to the exchange of raw samples with index information. The second scheme adopts the consensus algorithm, where the local decision statistic is exchanged between sensors instead of the raw samples, thus significantly lowering the communication requirement compared with the first scheme. In particular, the decision statistic for DSPRT at each sensor is updated by the weighted average of the decision statistics in the neighborhood at every message-passing step. We show that, under certain regularity conditions, the consensus algorithm-based DSPRT also yields the second-order asymptotically optimal performance at all sensors given a fixed number of sensors. Our asymptotic analyses of the two message-passing-based DSPRTs are then corroborated by simulations using the Gaussian and Laplacian samples. [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]

This paper deals with the problem of sequentially detecting statistical changes. In particular, the focus is on transient change detection, in which a probability minimizing optimal criterion is desirable. This is in contrast with the traditional minimization of the detection delay, proposed in quickest change detection problems.Afinite moving average stopping time is proposed for the general transient change detection problem. The statistical performance of this stopping time is investigated and compared to other methods available in the literature. The proposed stopping time and theoretical findings are applied to quality monitoring, including reliabilitymonitoring in industrial processes and signal quality monitoring in global navigation satellite systems. Numerical simulations are presented to assess the goodness of the presented theoretical results, and the performance of the considered stopping times. This will show the superiority of the proposed scheme. [ABSTRACT FROM AUTHOR]

We start proceeding with the stopping time theory in discrete time with the help of the Mizar system [1], [4]. We prove, that the expression for two stopping times k1 and k2 not always implies a stopping time (k1 + k2) (see Theorem 6 in this paper). If you want to get a stopping time, you have to cut the function e.g. (k1 + k2) ∪ T (see [2, p. 283 Remark 6.14]). Next we introduce the stopping time in continuous time. We are focused on the intervals [0, r] where r ∈ R. We prove, that for I = [0, r] or I = [0,+∞[ the set {A ⋂ I : A ∈ Borel-Sets} is a σ-algebra of I (see Definition 6 in this paper, and more general given in [3, p.12 1.8e]). The interval I can be considered as a timeline from now to some point in the future. This set is necessary to define our next lemma. We prove the existence of the σ-algebra of the τ -past, where τ is a stopping time (see Definition 11 in this paper and [6, p.187, Definition 9.19]). If τ1 and τ2 are stopping times with τ1 is smaller or equal than τ2 we can prove, that the σ-algebra of the τ1-past is a subset of the σ-algebra of the τ2-past (see Theorem 9 in this paper and [6, p.187 Lemma 9.21]). Suppose, that you want to use Lemma 9.21 with some events, that never occur, see as a comparison the paper [5] and the example for ST(1)={+∞} in the Summary. We don't have the element +1 in our above-mentioned time intervals [0, r[ and [0,+1[. This is only possible if we construct a new σ-algebra on R ∪ {−∞,+∞}. This construction is similar to the Borel-Sets and we call this σ-algebra extended Borel sets (see Definition 13 in this paper and [3, p. 21]). It can be proved, that {+∞} is an Element of extended Borel sets (see Theorem 21 in this paper). Now we use the interval [0,+∞] as a basis. We construct a σ-algebra on [0,+∞] similar to the book ([3, p. 12 18e]), see Definition 18 in this paper, and call it extended Borel subsets. We prove for stopping times with this given σ-algebra, that for τ1 and τ2 are stopping times with τ1 is smaller or equal than τ2 we have the σ-algebra of the τ1-past is a subset of the σ-algebra of the τ2-past, see Theorem 25 in this paper. It is obvious, that {+∞} 2 extended Borel subsets. In general, Lemma 9.21 is important for the proof of the Optional Sampling Theorem, see 10.11 Proof of (i) in [6, p. 203]. [ABSTRACT FROM AUTHOR]

We give under weak assumptions a complete combinatorial characterization of identifiability for linear mixtures of finite alphabet sources, with unknown mixing weights and unknown source signals, but known alphabet. This is based on a detailed treatment of the case of a single linear mixture. Notably, our identifiability analysis applies also to the case of unknown number of sources. We provide sufficient and necessary conditions for identifiability and give a simple sufficient criterion together with an explicit construction to determine the weights and the source signals for deterministic data by taking advantage of the hierarchical structure within the possible mixture values. We show that the probability of identifiability is related to the distribution of a hitting time and converges exponentially fast to one when the underlying sources come from a discrete Markov process. Finally, we explore our theoretical results in a simulation study. This paper extends and clarifies the scope of scenarios for which blind source separation becomes meaningful. [ABSTRACT FROM AUTHOR]

. We start with the definition of stopping time according to [4], p.283. We prove, that different definitions for stopping time can coincide. We give examples of stopping time using constant-functions or functions defined with the operator max or min (defined in [6], pp.37-38). Finally we give an example with some given filtration. Stopping time is very important for stochastic finance. A stopping time is the moment, where a certain event occurs ([7], p.372) and can be used together with stochastic processes ([4], p.283). Look at the following example: we install a function ST: {1,2,3,4} → {0, 1, 2} ∪ {+∞}, we define:. a. ST(1)=1, ST(2)=1, ST(3)=2, ST(4)=2. b. The set {0,1,2} consists of time points: 0=now, 1=tomorrow, 2=the day after tomorrow. We can prove: c. {w, where w is Element of Ω: ST.w=0}=ø & {w, where w is Element of Ω: ST.w=1}={1,2} & {w, where w is Element of Ω: ST.w=2}={3,4} and ST is a stopping time. We use a function Filt as Filtration of {0,1,2}, Σ where Filt(0)=Ωnow, Filt(1)=Ωfut1 and Filt(2)=Ωfut2. From a.,b. and c. we know that: d. {w, where w is Element of Ω: ST.w=0} in Ωnow and {w, where w is Element of Ω: ST.w=1} in Ωfut1 and {w, where w is Element of Ω: ST.w=2} in Ωfut2. The sets in d. are events, which occur at the time points 0(=now), 1(=tomorrow) or 2(=the day after tomorrow), see also [7], p.371. Suppose we have ST(1)=+∞, then this means that for 1 the corresponding event never occurs. As an interpretation for our installed functions consider the given adapted stochastic process in the article [5]. ST(1)=1 means, that the given element 1 in {1,2,3,4} is stopped in 1 (=tomorrow). That tells us, that we have to look at the value f2(1) which is equal to 80. The same argumentation can be applied for the element 2 in {1,2,3,4}. ST(3)=2 means, that the given element 3 in {1,2,3,4} is stopped in 2 (=the day after tomorrow). That tells us, that we have to look at the value f3(3) which is equal to 100. ST(4)=2 means, that the given element 4 in {1,2,3,4} is stopped in 2 (=the day after tomorrow). That tells us, that we have to look at the value f3(4) which is equal to 120. In the real world, these functions can be used for questions like: when does the share price exceed a certain limit? (see [7], p.372). [ABSTRACT FROM AUTHOR]

We define and study order continuity, topological continuity, γ -Hölder-continuity and Kolmogorov–Čentsov-continuity of continuous-time stochastic processes in vector lattices and show that every such kind of continuous submartingale has a continuous compensator of the same kind. The notion of variation is introduced for continuous time stochastic processes and for a γ -Hölder-continuous martingale with finite variation, we prove that it is a constant martingale. The localization technique for not necessarily bounded martingales is introduced and used to prove our main result which states that the quadratic variation of a continuous-time γ -Hölder continuous martingale X is equal to its compensator 〈 X 〉 . [ABSTRACT FROM AUTHOR]

This paper considers the sequential composite hypothesis test with multiple sensors. The sensors observe random samples in parallel and communicate with a fusion center, who makes the global decision based on the sensor inputs. On the one hand, in the centralized scenario, where local samples are precisely transmitted to the fusion center, the generalized sequential likelihood ratio test (GSPRT) is shown to be asymptotically optimal in terms of the expected sample size as error rates tend to zero. On the other hand, for systems with limited power and bandwidth resources, decentralized solutions that only send a summary of local samples (we particularly focus on a one-bit communication protocol) to the fusion center is of great importance. To this end, we first consider a decentralized scheme where sensors send their one-bit quantized statistics every fixed period of time to the fusion center. We show that such a uniform sampling and quantization scheme is strictly suboptimal and its suboptimality can be quantified by the KL divergence of the distributions of the quantized statistics under both the hypotheses. We then propose a decentralized GSPRT based on level-triggered sampling. That is, each sensor runs its own GSPRT repeatedly and reports its local decision to the fusion center asynchronously. We show that this scheme is asymptotically optimal as the local thresholds and global thresholds grow large at different rates. Finally, two particular models and their associated applications are studied to compare the centralized and decentralized approaches. Numerical results are provided to demonstrate that the proposed level-triggered sampling based decentralized scheme aligns closely with the centralized scheme with substantially lower communication overhead, and significantly outperforms the uniform sampling and quantization-based decentralized scheme. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper provides well-posedness results and stochastic representations for the solutions to equations involving both the right- and the left-sided generalized operators of Caputo type. As a special case, these results show the interplay between two-sided fractional differential equations and two-sided exit problems for certain Lévy processes. [ABSTRACT FROM AUTHOR]

This paper studies the order-fulfillment process of a supplier producing multiple customized capital goods. The times when orders are confirmed by customers are random. The supplier can only work on one product at any time due to capacity constraints. The supplier must determine the optimal time to start the process for each order so that the total expected cost of having the goods ready before or after their orders are confirmed is minimized. We formulate this problem as a discrete time Markov decision process. The optimal policy is complex in general. It has a threshold-type structure and can be fully characterized only for some special cases. Based on our formulation, we compute the optimal policy and quantify the value of jointly managing the order fulfillment processes of multiple orders and the value of taking into account demand arrival time uncertainty. [ABSTRACT FROM AUTHOR]

While many efforts have been devoted to optimizing the power output for a finite-time thermodynamic process, thermodynamic optimization under realistic situations is not necessarily concerned with power alone; rather, it may be of great relevance to optimize generic objective functions that are combinations of power, entropy production, and/or efficiency. One can optimize the objective function for a given model; generally the obtained results are strongly model dependent. However, if the thermodynamic process in question is operated in the linear response regime, then we show in this work that it is possible to adopt a unified approach to optimizing the objective function, thanks to Onstage's theory of linear irreversible thermodynamics. A dissipation bound is derived, and based on it, the efficiency associated with the optimization problem, which is universal in the linear response regime and irrespective of model details, can be obtained in a unified way. Our results are in good agreement with previous findings. Moreover, we unveil that the ratio between the stopping time of a finite-time process and the optimized duration time plays a pivotal role in determining the corresponding efficiency in the case of linear response. [ABSTRACT FROM AUTHOR]

We study a singular one-dimensional parabolic problem with initial data in the BV space, i.e. the energy space, for various boundary data. We pay special attention to Dirichlet conditions, which need not be satisfied in a pointwise manner. The equation we study has two singular slopes, so that in principle solutions to the value problems may have an infinite number of oscillations, which seems surprising for a parabolic problem. We investigate this issue. We also study the facet creation process and the stopping of solutions caused by the evolution of facets. Our estimate of stopping time is based on the comparison principle for viscosity solutions. For this purpose we show that our solutions are viscosity solutions in the sense of [10]. [ABSTRACT FROM AUTHOR]

We obtain tight super- and sub-solutions, or hard bounding functions for the probabilistic representation of the solution, to the boundary value problem for second-order elliptic partial differential equations in an explicit polynomial form. Unlike accurate approximate solutions by the existing discretization-based methods, our hard bounding polynomial functions act as pointwise 100% confidence intervals within which the solution is guaranteed to exist. Our methodology can deal with a wide range of boundary conditions even of fully mixed type with Dirichlet, Neumann and Robin components, without increasing the complexity. The quality of hard bounds can be improved by introducing piecewise polynomial functions and domain decomposition. Convergence of hard bounding functions is proved under technical conditions. Numerical results are presented throughout the paper to support our theoretical developments and to illustrate the effectiveness of our methodology. [ABSTRACT FROM AUTHOR]

Let X be a progressively measurable, almost surely right-continuous stochastic process such that X τ ∈ L 1 and E [ X τ ] = E [ X 0 ] for each finite stopping time τ . In 2006, Cherny showed that X is then a uniformly integrable martingale provided that X is additionally nonnegative. Cherny then posed the question whether this implication also holds even if X is not necessarily nonnegative. We provide an example that illustrates that this implication is wrong, in general. If, however, an additional integrability assumption is made on the limit inferior of | X | then the implication holds. Finally, we argue that this integrability assumption holds if the stopping times are allowed to be randomized in a suitable sense. [ABSTRACT FROM AUTHOR]

The study is objected to investigate the response of a magnetorheological brake (MRB) system under thefree move inertial mass. The disk-type MRB comprises of a rotating disk immersed in magnetorheological fluids (MRFs) and surrounded by an electromagnet coil. The magnetized coil causes a solidification of the MR fluid so that the shear stress between the moving part and static part increases resulting in the decrement speed of the moving parts. The shear stress can be varied by applying different electric current to the coil. The study began with the part design using the 3D modeling software, following by the magnetostatic analysis. The flux density across the magnetorheological fluid could be predicted through this finite element magnetic simulation. The quantity of magnetic flux was then used to predict the shear stress between static and moving parts. The fabricated MRB was integrated onto a test rig which employs load cell and speed sensor as well as completely instrumented with data acquisition. Since the MRB test rig performed a simple free rotation system, a linear second order differential equation was derived to model the stopping time and braking torque behaviors. The equation of motion was built in a Simulink model, and the simulation results were compared to the real measurement. The achievable braking torque was also presented based on the average value from the load cell. [ABSTRACT FROM AUTHOR]

Abstract: We prove that if a pair of weights satisfies a sharp -bump condition in the scale of all log bumps or certain loglog bumps, then Haar shifts map into with a constant quadratic in the complexity of the shift. This in turn implies the two weight boundedness for all Calderón–Zygmund operators. This gives a partial answer to a long-standing conjecture. We also give a partial result for a related conjecture for weak-type inequalities. To prove our main results we combine several different approaches to these problems; in particular we use many of the ideas developed to prove the conjecture. As a byproduct of our work we also disprove a conjecture by Muckenhoupt and Wheeden on weak-type inequalities for the Hilbert transform. This is closely related to the recent counterexamples of Reguera, Scurry and Thiele. [Copyright &y& Elsevier]

This paper deals with minimization of the variances of the total discounted costs for constrained Continuous-Time Markov Decision Processes (CTMDPs). The costs consist of cumulative costs incurred between jumps and instant costs incurred at jump epochs. We interpret discounting as an exponentially distributed stopping time. According to existing theory, for the expected total discounted costs optimal policies exist in the forms of randomized stationary and switching stationary policies. While the former is typically unique, the latter forms a finite set whose number of elements grows exponentially with the number of constraints. This paper investigates the problem when the process stops immediately after the first jump. For costs up to the first jump we provide an index for selection of actions by switching stationary policies and show that the indexed switching policy achieves a smaller variance than the randomized stationary policy. For problems without instant costs, the indexed switching policy achieves the minimum variance of costs up to the first jump among all the equivalent switching policies. [ABSTRACT FROM AUTHOR]

A stochastic Poincaré-Bendixson theorem and its generalized theorem are given in this paper. The aim of these theorems is to show the existence of a crater-like stationary distribution for a stochastic planar autonomous system. The main theorems are applied on a stochastic predator-prey system as an instruction. For example, stochastic Hopf bifurcation phenomenon is observed when the qualitative change of the shape of the stationary distribution is taken as an indicator of bifurcation. [ABSTRACT FROM AUTHOR]

With a finite number of samples drawn from one of two possible distributions sequentially revealed, an opportunistic detection rule is proposed, which possibly makes an early decision in favor of the alternative hypothesis, while always deferring the decision of the null hypothesis until collecting all the samples. Properties of this opportunistic detection rule are discussed and its key asymptotic behavior in the large sample size limit is established. Specifically, a Chernoff–Stein lemma type of characterization of the exponential decay rate of the miss probability under the Neyman–Pearson criterion is established, and consequently, a performance metric of asymptotic exponential efficiency loss is proposed and discussed, which is exactly the ratio between the Kullback–Leibler distance and the Chernoff information of the two hypotheses. Analytical results are corroborated by numerical experiments. [ABSTRACT FROM AUTHOR]

A Gaussian random walk (or a Wiener process), possibly with drift, is observed in a noisy or delayed fashion. The problem considered in this paper is to estimate the first time \tau the random walk reaches a given level. Specifically, the average p-moment (p\geq 1) optimization problem \inf \eta {\BBE} \vert \eta -\tau \vert ^{p} is investigated where the infimum is taken over the set of stopping times that are defined on the observation process. When there is no drift, optimal stopping rules are characterized for both types of observations. When there is a drift, upper and lower bounds on \inf \eta {\BBE} \vert \eta -\tau \vert ^{p} are established for both types of observations. The bounds are tight in the large-level regime for noisy observations and in the large-level-large-delay regime for delayed observations. Noteworthy, for noisy observations there exists an asymptotically optimal stopping rule that is a function of a single observation. Simulation results are provided that corroborate the validity of the results for non-asymptotic settings. [ABSTRACT FROM PUBLISHER]

For a homogeneous diffusion process ( X), we consider problems related to the distribution of the stopping times . The results obtained are used to construct an inductive procedure allowing us to find the distribution of the increments of the process X between two adjacent kagi and renko instants of time. [ABSTRACT FROM AUTHOR]

Abstract: In this article, we consider the properties of hitting times for -martingales and the stopped processes. We prove that the stopped processes for -martingales are still -martingales and that the hitting times for a class of -martingales including one-dimensional -Brownian motion are quasi-continuous. As an application, we improve the -martingale representation theorems of . [Copyright &y& Elsevier]

Abstract: Under the framework of -expectation and -Brownian motion, we introduce Itô’s integral for stochastic processes without assuming quasi-continuity. Then we can obtain Itô’s integral on stopping time interval. This new formulation permits us to obtain Itô’s formula for a general -function, which essentially generalizes the previous results of Peng (2006, 2008, 2009, 2010, 2010)  as well as those of Gao (2009)  and Zhang et al. (2010) . [Copyright &y& Elsevier]

Abstract: Users in Service-oriented architecture (SOA) seek the best Quality of service (QoS) by service selection from the candidates responding in succession. In case the QoS changes dynamically, choosing one service and stop the searching is problematic for a service user who makes the choice online. Lack of accurate knowledge of service distribution, the user is unable to make a good decision. The Local Martingale Difference (LMD) approach is developed in this paper to help users to achieve optimal results, in the sense of probability. The stopping time is proved to be bounded to ensure the existence of an optimal solution first. Then, a global estimation over the time horizon is transformed to a local determination based on current martingale difference to make the algorithm feasible. Independent of any predetermined threshold or manual intervention, LMD enables users to stop around the optimal time, based on the information collected during the stochastic process. Verified to be efficient by comparison with three traditional methods, LMD is adaptable in vast applications with dynamic QoS. [Copyright &y& Elsevier]

We consider the optimal stopping problem with a possible compensated refusal of reward. We discuss functionals of exponential Brownian motion. The optimal stopping time is defined on the set of all finite stopping times. The functionals under consideration correspond to payments for standard American options. [ABSTRACT FROM AUTHOR]

The nested case-control design is a relatively new type of observational study whereby a case-control approach is employed within an established cohort. In this design, we observe cases and controls longitudinally by sampling all cases whenever they occur but controls at certain time points. Controls can be obtained at time points randomly scheduled or prefixed for operational convenience. This design with longitudinal observations is efficient in terms of cost and duration, especially when the disease is rare and the assessment of exposure levels is difficult. In our design, we propose sequential sampling methods and study both (group) sequential testing and estimation methods so that the study can be stopped as soon as the stopping rule is satisfied. To make such a longitudinal sampling more efficient in terms of both numbers of subjects and replications, we propose applying sequential sampling methods to subjects and replications, simultaneously, until the information criterion is fulfilled. This simultaneous sequential sampling on subjects and replicates is more flexible for practitioners designing their sampling schemes, and is different from the classical approaches used in longitudinal studies. We newly define the σ-field to accommodate our proposed sampling scheme, which contains mixtures of independent and correlated observations, and prove the asymptotic optimality of sequential estimation based on the martingale theories. We also prove that the independent increment structure is retained so that the group sequential method is applicable. Finally, we present results by employing sequential estimation and group sequential testing on both simulated data and real data on children's diarrhea. [ABSTRACT FROM AUTHOR]

A variable-to-fixed length encoder partitions the source string into variable-length phrases that belong to a given and fixed dictionary. Tunstall, and independently Khodak, designed variable-to-fixed length codes for memoryless sources that are optimal under certain constraints. In this paper, we study the Tunstall and Khodak codes using variety of techniques ranging from stopping times for sums of independent random variables to Tauberian theorems and Mellin transform. After proposing an algebraic characterization of the Tunstall and Khodak codes, we present new results on the variance and a central limit theorem for dictionary phrase lengths. This analysis also provides a new argument for obtaining asymptotic results about the mean dictionary phrase length and average redundancy rates. [ABSTRACT FROM AUTHOR]

Let {Xn}n∞=0 be a stationary real-valued Gaussian time series. We estimate the conditional expectation E(Xn+1|X0,...,Xn) from a growing number of observations X0,...,Xn in a pointwise consistent way along a sequence of stopping times. [ABSTRACT FROM AUTHOR]

The paper analyzes the implications of extreme events on the proper choice of discounting. Any discounting with constant or declining rates can be linked to random “stopping time” events, which define the internal discount-related horizons of evaluations. Conversely, any stopping time induces a discounting, in particular, with the standard discount rates. The expected duration of the stopping time horizon for discount rates obtained from capital markets does not exceed a few decades and, as such, these rates may significantly underestimate the net benefits of long-term decisions. The alternative undiscounted stopping time criterion allows to induce social discounting focusing on arrival times of potential extreme events rather then horizons of market interests. Induced discount rates are conditional on the degree of social commitment to mitigate risk. In general, extreme events affect these rates, which alter the optimal mitigation efforts that, in turn, change events. The use of undiscounted stopping time criteria requires stochastic optimisation methods. [ABSTRACT FROM AUTHOR]

Abstract: Under the conditions of coefficients being non-Lipschitz and the diffusion coefficient being elliptic, we study the strong Feller property and irreducibility for the transition probability of solutions to general multivalued stochastic differential equations by using the coupling method, Girsanov''s theorem and a stopping argument. Thus we can establish the exponential ergodicity and the spectral gap. [Copyright &y& Elsevier]

Abstract: This paper studies drawdown and drawup processes in a general diffusion model. The main result is a formula for the joint distribution of the running minimum and the running maximum of the process stopped at the time of the first drop of size . As a consequence, we obtain the probabilities that a drawdown of size precedes a drawup of size and vice versa. The results are applied to several examples of diffusion processes, such as drifted Brownian motion, Ornstein–Uhlenbeck process, and Cox–Ingersoll–Ross process. [Copyright &y& Elsevier]

This paper gives an insight to the mathematical theory of the discrete-time martingale, including basic definitions and results. Still, the paper is focused on prooving a variant of the Convergence Theorem based on Doob's Inequality for martingales. The second part of this paper presents a proof of The Law of Large Numbers for discrete-times martingales, as an application of the Convergence Theorem. [ABSTRACT FROM AUTHOR]

Abstract: Shepp’s urn model is a useful tool for analyzing the stopping-rule problems in economics and finance. In [R.W. Chen, A. Zame, C.T. Lin, H. Wu, A random version of Shepp’s urn scheme, SIAM J. Discrete Math. 19 (1) (2005) 149–164], Chen et al. considered a random version of Shepp’s urn scheme and showed that a simple drawing policy (called “the in the hole policy”) can asymptotically maximize the expected value of the game. By extending the work done by Chen et al., this note considers a more general urn scheme that is better suited to real-life price models in which the short-term value might not fluctuate. Further, “the in the hole policy” is shown to be asymptotically optimal for this new urn scheme. [Copyright &y& Elsevier]

A general scale-space algorithm is presented for denoising signals and images with spatially varying dominant scales. The process is formulated as a partial differential equation with spatially varying time. The proposed adaptivity is semilocal and is in conjunction with the classical gradient-based diffusion coefficient, designed to preserve edges. The new algorithm aims at maximizing a local SNR measure of the denoised image. It is based on a generalization of a global stopping time criterion presented recently by the author and his colleagues. Most notably, the method also works well for partially textured images and outperforms any selection of a global stopping time. Given an estimate of the noise variance, the procedure is automatic and can be applied well to most natural images. [ABSTRACT FROM AUTHOR]

Abstract: We extend Troitsky''s study of martingales in Banach lattices to include stopping times. Results from the theory of unconditional Schauder decompositions and multipliers are used to derive an optional stopping theorem for unbounded stopping times. We also apply these techniques to convergent nets of stopped processes, as well as to unconditional Schauder decompositions in vector-valued -spaces (). [Copyright &y& Elsevier]

Formulas for the expected ruin time and the individual ruin probabilities are found for the asymmetric n-player gambler''s ruin problem with equal initial fortunes of n+r dollars, for r a fixed non-negative integer, and n⩾max(r,2); these are evaluated in several cases. Asymptotic results are obtained for the expected time to ruin for the asymmetric game, and for the individual ruin probabilities for the symmetric game. We also prove that the ruin time and which player is ruined are dependent events in the case where the players each start with n+2 dollars and n⩾3. This differs from the asymmetric 2-player game with equal initial fortunes, as well as from the n-player game with n⩾3 and equal initial fortunes less than or equal to n+1 dollars, in which the analogous events have been shown to be independent. [Copyright &y& Elsevier]

We consider the rate of convergence of the paths of some self-attracting diffusions. We prove that these diffusions are attracted by their mean-process. Moreover, under some assumptions on the interaction function f, this attraction becomes strong enough to imply the almost sure convergence of the paths. In this case, we provide an estimate of the rate of convergence which depends on the growth of f near the origin. [Copyright &y& Elsevier]

Given a Brownian motion (Bt)t⩾0 and a general target law μ (not necessarily centered or even in L1) we show how to construct an embedding of μ in B. This embedding is an extension of an embedding due to Perkins, and is optimal in the sense that it simultaneously minimises the distribution of the maximum and maximises the distribution of the minimum among all embeddings of μ. The embedding is then applied to regular diffusions, and used to characterise the target laws for which a Hp-embedding may be found. [Copyright &y& Elsevier]

We develop a novel time-selection strategy for iterative image restoration techniques: the stopping time is chosen so that the correlation of signal and noise in the filtered image is minimized. The new method is applicable to any images where the noise to be removed is uncorrelated with the signal, under the assumptions that the filter used is suitable for the given type of data, and that neither the additive noise nor the filtering procedure alter the average gray value; no other knowledge (e.g. the noise variance, training data etc.) is needed. We analyse the theoretical properties of the method, then test the performance of our time estimation procedure experimentally, and demonstrate that it yields near-optimal results for a wide range of noise levels and for various filtering methods. [ABSTRACT FROM AUTHOR]