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We consider the statistical concept of causality in continuous time within filtered probability spaces based on Granger's definition of causality. The given causality concept is connected to the optional and predictable processes important in stochastic integration. More precisely, we establish that the preservation of predictability with respect to larger filtrations is implied by the considered notion of (self-)causality. We also consider the connections between the given causality concept and the optional and predictable projections of a stochastic process, which play an important role in the general theory of stochastic processes, semimartingale theory, and stochastic calculus. Some results show that the (self-)causality implies indistinguishability of the optional (or predictable) projections with respect to considered filtrations from those with respect to larger filtrations. [ABSTRACT FROM AUTHOR]

We consider the problem of jointly testing multiple hypotheses and estimating a random parameter of the underlying distribution. This problem is investigated in a sequential setup under mild assumptions on the underlying random process. The optimal method minimizes the expected number of samples while ensuring that the average detection/estimation errors do not exceed a certain level. After converting the constrained problem to an unconstrained one, we characterize the general solution by a nonlinear Bellman equation, which is parameterized by a set of cost coefficients. A strong connection between the derivatives of the cost function with respect to the coefficients and the detection/estimation errors of the sequential procedure is derived. Based on this fundamental property, we further show that for suitably chosen cost coefficients the solutions of the constrained and the unconstrained problem coincide. We present two approaches to finding the optimal coefficients. For the first approach, the final optimization problem is converted into a linear program, whereas the second approach solves it with a projected gradient ascent. To illustrate the theoretical results, we consider two problems for which the optimal schemes are designed numerically. Using Monte Carlo simulations, it is validated that the numerical results agree with the theory. [ABSTRACT FROM AUTHOR]

We derive subexponential tail asymptotics for the distribution of the maximum of a compound renewal process with linear component and of a Lévy process, both with negative drift, over random time horizon τ that does not depend on the future increments of the process. Our asymptotic results are uniform over the whole class of such random times. Particular examples are given by stopping times and by τ independent of the processes. We link our results with random walk theory. [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]

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]

. 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]

In this paper we study a discrete-time optimal switching problem on a finite horizon. The underlying model has a running reward, terminal reward, and signed (positive and negative) switching costs. Using optimal stopping theory for discrete-parameter stochastic processes, we extend a well-known explicit dynamic programming method for computing the value function and the optimal strategy to the case of signed switching costs. [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]

In this article, we address stochastic differential games of mixed type with both control and stopping times. Under standard assumptions, we show that the value of the game can be characterized as the unique viscosity solution of corresponding Hamilton-Jacobi-Isaacs (HJI) variational inequalities. [ABSTRACT FROM PUBLISHER]

Let (Xt)0≤t≤T be a one-dimensional stochastic process with independent and stationary increments, either in discrete or continuous time. In this paper we consider the problem of stopping the process (Xt) 'as close as possible' to its eventual supremum MT := sup0≤t≤T Xt when the reward for stopping at time τ ≤ T is a nonincreasing convex function of MT -- Xtau. Under fairly general conditions on the process (Xt), it is shown that the optimal stopping time τ takes a trivial form: it is either optimal to stop at time 0 or at time T. For the case of a random walk, the rule τ ≡ T is optimal if the steps of the walk stochastically dominate their opposites, and the rule τ ≡ 0 is optimal if the reverse relationship holds. An analogous result is proved for Lévy processes with finite Lévy measure. The result is then extended to some processes with nonfinite Lévy measure, including stable processes, CGMY processes, and processes whose jump component is of finite variation. [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]

The notions of stopping times and stopped processes for continuous stochastic processes are defined and studied in the framework of Riesz spaces. This leads to a formulation and proof of Doob's optional sampling theorem. [ABSTRACT FROM AUTHOR]

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]

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]

This paper presents a generic framework that is common to many stochastic models. For the virtual wait process in an M/G/1 queueing system, we obtain the expected number of events during a sojourn time above or below a state-space level. Similar quantities are derived for the “extended age” process in a standard G/M/1 queue. These results generalize the expected number of events in a busy period. We also apply the framework to a renewal model with a barrier. We obtain the expected number of renewals required to exceed the barrier. [ABSTRACT FROM AUTHOR]

We deduce an analog of the classical Wald’s identity E S τ = E τE ξ in the case of the infinite mean of summands. We find the conditions on τ under which Emin( S τ, x) ∼ E τE min( ξ, x) as x→∞. [ABSTRACT FROM AUTHOR]

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]

Markov and semi-Maskov processes with block tridiagonal transition matrices for their embedded discrete-time Markov chains are underlying stochastic models in many applied probability problems. In particular, identity-by-descent (IBD) problems for uncle-type and cousin-type relationships fall into this class. More specifically, the exact distributions of relevant IBD statistics for two individuals in either an uncle-type or cousin-type relationship are of interest. Such statistics are the amount of genome shared IBD by the two related individuals on a chromosomal segment and the number of IBD pieces on such a segment. These lead to special reward distributions associated with block tridiagonal transition matrices for continuous-time Markov chains. A method is provided for calculating explicit, closed-form expressions for Laplace transforms of general reward functions for such Markov chains. Some calculation results on the cumulative probabilities of relevant IBD statistics via a numerical inversion of the Laplace transforms are also provided for uncle/nephew and first-cousin relationships. [ABSTRACT FROM AUTHOR]

Abstract: A new importance sampling method is presented for computing the first passage probability of elasto-plastic systems under white noise excitations. The importance sampling distribution corresponds to shifting the mean of the excitation to an ‘adapted’ (‘predictable’) stochastic process whose future is determined based on information only up to the present. Choosing the adapted process involves designing an adaptive control force algorithm in a stochastic environment that targets to drive the response to first passage failure based on information up to the present. Algorithms for single-degree-of-freedom linear and elasto-plastic systems are proposed and their resulting computational efficiency investigated. Numerical results show that the use of adapted process is particularly useful for nonlinear hysteretic systems where hysteretic effects undermine the effectiveness of conventional importance sampling method based on fixed design points. [Copyright &y& Elsevier]

We discuss the performance of “closed-end” fluctuation tests, used to detect changes in the structural stability of a model, in the case where the time horizon is large and the observations do not possess high enough moments. It is demonstrated via a simulation study that, in the latter case, the choice of the boundary function should take into account the number of moments existing; otherwise the actual size of the test may exceed the nominal level. We suggest an alternative “open-end” fluctuation procedure and study its asymptotic behavior. It turns out that if an improper boundary function is chosen, then the null hypothesis may be rejected even if it is true. This means that an appropriate choice of the boundary function requires some prior information about the tail heaviness of the observations. This is different from the cumulative sum-based monitoring schemes suggested by Chu et al. (1996) and further studied by Horvath et al. (2004). [ABSTRACT FROM AUTHOR]

In this paper, we consider a vendor realizing a sequence of random order arrivals in random sizes. The vendor has the autonomy to hold/consolidate small orders until an economical dispatch quantity accumulates. Consequently, the actual inventory requirements at the vendor are in part determined by the parameters of the shipment release policy in use. In this context, we investigate the impact of shipment consolidation on the expected long-run average cost by simultaneously computing the optimal order quantity for inventory replenishment at the vendor and the optimal dispatch quantity for outbound shipments. Since we consider the case where demand follows a general stochastic bulk arrival process, obtaining exact analytical expressions for some key operating characteristics of the cost function is intractable. Hence, we provide easy-to-compute approximations which enable efficient numerical solutions for the problem. We also investigate: (i) the cases where consolidated shipments are preferred over immediate deliveries; (ii) the sensitivity of optimal integrated policy variables to demand/cost parameters; (iii) the potential savings that can be obtained by shipment consolidation; and (iv) the tradeoffs between the waiting time induced by shipment consolidation and costs saved. Our results provide insights into the impact of outbound transportation operations on inventory replenishment decisions and outbound distribution system design. Moreover, numerical testing suggests that significant cost savings (up to 57%) are possible with shipment consolidation. [ABSTRACT FROM AUTHOR]

Common software release procedures based on statistical techniques try to optimize the trade-off between further testing costs and costs due to remaining errors. We propose new software release procedures where the aim is to certify with a certain confidence level that the software does not contain errors. The underlying model is a discrete time model similar to the geometric Moranda model. The decisions are based on a mix of classical and Bayesian approaches to sequential testing and do not require any assumption on the initial number of errors. [ABSTRACT FROM AUTHOR]

Abstract: Consider a Galton–Watson process with immigration. This paper studies the limits of the sequential estimator, proposed by [Sriram, T.N., Basawa, I.V., and Huggins, R.M., (1991). Sequential estimation for branching processes with immigration. Ann. Statist. 19, 2232–2243.] and the modified sequential estimator, proposed by [Shete, S., Sriram, T.N., 1998. Fixed precision estimator of the offspring mean in branching processes. Stochastic Process. Appl. 77, 17–33.]. [Sriram, T.N., Basawa, I.V., Huggins, R.M., 1991. Sequential estimation for branching processes with immigration. Ann. Statist. 19, 2232–2243.] proved that the sequential estimators are asymptotically normal in the subcritical and critical cases. In this paper it is proved that the sequential estimators are asymptotically normal in the supercritical case and that the limiting distributions of the modified estimators, after being properly standardized, are normal as well. [Copyright &y& Elsevier]

In this work, we solve completely the starting and stopping problem when the dynamics of the system are a general adapted stochastic process. We use backward stochastic differential equations (BSDEs) and Snell envelopes. Finally, we give some numerical results. [ABSTRACT FROM AUTHOR]

Many researchers have investigated first hitting times as models for survival data. First hitting times arise naturally in many types of stochastic processes, ranging from Wiener processes to Markov chains. In a survival context, the state of the underlying process represents the strength of an item or the health of an individual. The item fails or the individual experiences a clinical endpoint when the process reaches an adverse threshold state for the first time. The time scale can be calendar time or some other operational measure of degradation or disease progression. In many applications, the process is latent (i.e., unobservable). Threshold regression refers to first-hitting-time models with regression structures that accommodate covariate data. The parameters of the process, threshold state and time scale may depend on the covariates. This paper reviews aspects of this topic and discusses fruitful avenues for future research. [ABSTRACT FROM AUTHOR]

Methods using gambling teams and martingales are developed and applied to find formulae for the expected value and the generating function of the waiting time to observation of an element of a finite collection of patterns in a sequence generated by a two-state Markov chain of first, or higher, order. [ABSTRACT FROM AUTHOR]

In this paper, we use filtering techniques to estimate the occurrence time of an event in a financial market. The occurrence time is being viewed as a Markov stopping time with respect to the σ-field generated by a hidden Markov process. We also generalize our result to the N th occurrence time of that event. [ABSTRACT FROM AUTHOR]

The first passage time of a random walk to a barrier (constant or concave) is of great importance in many areas, such as insurance, finance, and sequential analysis. Here, we consider a sum of independent, identically distributed random variables and the convex barrier cb(n/c), where c is a scale parameter and n is time. It is shown, using large-deviation techniques, that the limit distribution of the first passage time decays exponentially in c. Under a tilt of measure, which changes the drift, four properties are proved: the limit distribution of the overshoot is distributed as an overshoot over a linear barrier; the stopping time is asymptotically normally distributed when it is properly normalized; the overshoot and the asymptotic normal part are asymptotically independent; and the overshoot over a linear bather is bounded by an exponentially distributed random variable. The determination of the function that multiplies the exponential part is guided by consideration of these properties. [ABSTRACT FROM AUTHOR]

We deal with reflected backward stochastic differential equations with right continuous and left limited barrier. We show the existence and uniqueness of the solution and we give a comparison theorem. As an application, we study the link between such an equations with stochastic mixed control problems. [ABSTRACT FROM AUTHOR]

We establish, for various scenarios, whether or not interruptible exact stationary sampling is possible when a finite-state Markov chain can only be viewed passively. In particular, we prove that such sampling is not possible using a single copy of the chain. Such sampling is possible when enough copies of the chain are available, and we provide an algorithm that terminates with probability one. [ABSTRACT FROM AUTHOR]

Glazebrook [1] has given an account of improved procedures for strategy evaluation for resource allocation in a stochastic environment. These methods are extended in the paper in such a way that they can be applied to problems which, for example, have precedence constraints and/or an arrivals process of new jobs. Theoretical results, backed up by numerical studies, show that quasi-myopic heuristics often perform well. [ABSTRACT FROM AUTHOR]

Robbins' problem of optimal stopping is that of minimising the expected rank of an observation chosen by some nonanticipating stopping rule. We settle a conjecture regarding the value of the stopped variable under the rule that yields the minimal expected rank, by embedding the problem in a much more general context of selection problems with the nonanticipation constraint lifted, and with the payoff growing like a power function of the rank. [ABSTRACT FROM AUTHOR]