Stationary Independent Increments Research Articles

Semiparametric predictive inference for failure data using first-hitting-time threshold regression.

The progression of disease for an individual can be described mathematically as a stochastic process. The individual experiences a failure event when the disease path first reaches or crosses a critical disease level. This happening defines a failure event and a first hitting time or time-to-event, both of which are important in medical contexts. When the context involves explanatory variables then there is usually an interest in incorporating regression structures into the analysis and the methodology known as threshold regression comes into play. To date, most applications of threshold regression have been based on parametric families of stochastic processes. This paper presents a semiparametric form of threshold regression that requires the stochastic process to have only one key property, namely, stationary independent increments. As this property is frequently encountered in real applications, this model has potential for use in many fields. The mathematical underpinnings of this semiparametric approach for estimation and prediction are described. The basic data element required by the model is a pair of readings representing the observed change in time and the observed change in disease level, arising from either a failure event or survival of the individual to the end of the data record. An extension is presented for applications where the underlying disease process is unobservable but component covariate processes are available to construct a surrogate disease process. Threshold regression, used in combination with a data technique called Markov decomposition, allows the methods to handle longitudinal time-to-event data by uncoupling a longitudinal record into a sequence of single records. Computational aspects of the methods are straightforward. An array of simulation experiments that verify computational feasibility and statistical inference are reported in an online supplement. Case applications based on longitudinal observational data from The Osteoarthritis Initiative (OAI) study are presented to demonstrate the methodology and its practical use.

The Application of Stochastic Processes in Options Pricing Model

Since English botanist Brown first observed the inconsistent mobility of pollen floating in liquid in 1827, stochastic motion has become one of the primary research topics today, and the use of stochastic processes in financial markets has never stopped [1]. The change in the process in any time interval do not affect the change in any other time intervals that do not overlap with of Brownian Motion, which possesses stationary independent increments. Therefore, this paper obtains the Black-Scholes Merton formula from the stochastic process through layers of derivation and analyses its limitations in terms of its existence. Therefore, this paper will derive the BSM formula and the logical relationships between them, step by step, starting from the stochastic process and discuss the problems and limitations of the BSM formula. The experiential results demonstrate that the proposed method can achieve better performance.

A New Dynamic Fault Tree Analysis Method of Electromagnetic Brakes Based on Bayesian Network Accompanying Wiener Process

Product fault diagnosis has always been the focus of quality and reliability research. However, a failure–rate curve of some products is a symmetrical function, the fault analysis result is not true because the failure period of the products cannot be judged accurately. In order to solve the problem of fault diagnosis, this paper proposes a new Takagi-Sugeno (T-S) dynamic fault tree analysis method based on a Bayesian network accompanying the Wiener process. Firstly, the top event, middle event, and bottom event of the product failure mode are determined, and the T-S dynamic fault tree is constructed. Secondly, in order to form the Bayesian network diagram of the T-S dynamic fault tree, the events in the fault tree are transformed into nodes, and the T-S dynamic gate is also transformed into directed edges. Then, the Wiener process is used to model the performance degradation process of the stationary independent increment of the symmetric function distribution, and the maximum likelihood estimation method is applied to estimate the unknown parameters of the degradation model. Next, the product residual life prediction model is established based on the concept of first arrival time, and a symmetric function of failure–rate curve is obtained by using the product failure probability density function. According to the fault density function derived from the Wiener process, the reverse reasoning algorithm of the Bayesian network is established. Combined with the prior probability of the bottom event, the posterior probability of the root node is calculated and sorted as well. Finally, taking the insufficient braking force of electromagnetic brakes as an example, the practicability and objectivity of the new method are proved.

Multivariate Threshold Regression Models with Cure Rates: Identification and Estimation in the Presence of the Esscher Property

The first hitting time of a boundary or threshold by the sample path of a stochastic process is the central concept of threshold regression models for survival data analysis. Regression functions for the process and threshold parameters in these models are multivariate combinations of explanatory variates. The stochastic process under investigation may be a univariate stochastic process or a multivariate stochastic process. The stochastic processes of interest to us in this report are those that possess stationary independent increments (i.e., Lévy processes) as well as the Esscher property. The Esscher transform is a transformation of probability density functions that has applications in actuarial science, financial engineering, and other fields. Lévy processes with this property are often encountered in practical applications. Frequently, these applications also involve a ‘cure rate’ fraction because some individuals are susceptible to failure and others not. Cure rates may arise endogenously from the model alone or exogenously from mixing of distinct statistical populations in the data set. We show, using both theoretical analysis and case demonstrations, that model estimates derived from typical survival data may not be able to distinguish between individuals in the cure rate fraction who are not susceptible to failure and those who may be susceptible to failure but escape the fate by chance. The ambiguity is aggravated by right censoring of survival times and by minor misspecifications of the model. Slightly incorrect specifications for regression functions or for the stochastic process can lead to problems with model identification and estimation. In this situation, additional guidance for estimating the fraction of non-susceptibles must come from subject matter expertise or from data types other than survival times, censored or otherwise. The identifiability issue is confronted directly in threshold regression but is also present when applying other kinds of models commonly used for survival data analysis. Other methods, however, usually do not provide a framework for recognizing or dealing with the issue and so the issue is often unintentionally ignored. The theoretical foundations of this work are set out, which presents new and somewhat surprising results for the first hitting time distributions of Lévy processes that have the Esscher property.

Numerical solutions for asymmetric Lévy flights

Levy flights are generalised random walk processes where the independent stationary increments are drawn from a long-tailed α-stable jump length distribution. We consider the formulation of Levy flights, for 0 < α < 1, in terms of a space-fractional diffusion equation which fundamental solutions are the probability density functions. First, we present how to obtain the governing equation of Levy motion from the Fourier transform of the jump distribution. Then, we derive a family of implicit numerical methods to determine the numerical solutions and we study their consistency and stability. Although numerical algorithms for the case 1 < α < 2 have been widely discussed, very few works paid attention to the case we discuss here. We present numerical experiments to show the performance of the numerical methods and to highlight the advantages and disadvantages of the different approaches. In the end we determine the numerical solutions of an initial value problem, that considers an approximation of the Dirac delta function as the initial condition, in order to obtain approximations of the probability density functions.

First-passage properties of asymmetric Lévy flights

Lévy flights are paradigmatic generalised random walk processes, in which the independent stationary increments—the ‘jump lengths’—are drawn from an -stable jump length distribution with long-tailed, power-law asymptote. As a result, the variance of Lévy flights diverges and the trajectory is characterised by occasional extremely long jumps. Such long jumps significantly decrease the probability to revisit previous points of visitation, rendering Lévy flights efficient search processes in one and two dimensions. To further quantify their precise property as random search strategies we here study the first-passage time properties of Lévy flights in one-dimensional semi-infinite and bounded domains for symmetric and asymmetric jump length distributions. To obtain the full probability density function of first-passage times for these cases we employ two complementary methods. One approach is based on the space-fractional diffusion equation for the probability density function, from which the survival probability is obtained for different values of the stable index and the skewness (asymmetry) parameter . The other approach is based on the stochastic Langevin equation with -stable driving noise. Both methods have their advantages and disadvantages for explicit calculations and numerical evaluation, and the complementary approach involving both methods will be profitable for concrete applications. We also make use of the Skorokhod theorem for processes with independent increments and demonstrate that the numerical results are in good agreement with the analytical expressions for the probability density function of the first-passage times.

On double-boundary non-crossing probability for a class of compound processes with applications

We develop an efficient method for computing the probability that a non-decreasing, pure jump (compound) stochastic process stays between arbitrary upper and lower boundaries (i.e., deterministic functions, possibly discontinuous) within a finite time period. The compound process is composed of a process modelling the arrivals of certain events (e.g., demands for a product in inventory systems, customers in queuing, or claims/capital gains in insurance/dual risk models), and a sequence of independent and identically distributed random variables modelling the sizes of the events. The events arrival process is assumed to belong to the wide class of point processes with conditional stationary independent increments which includes (non-)homogeneous Poisson, binomial, negative binomial, mixed Poisson and doubly stochastic Poisson (i.e., Cox) processes as special cases. The proposed method is based on expressing the non-exit probability through Chapman–Kolmogorov equations, re-expressing them in terms of a circular convolution of two vectors which is then computed applying fast Fourier transform (FFT). We further demonstrate that our FFT-based method is computationally efficient and can be successfully applied in the context of inventory management (to determine an optimal replenishment policy), ruin theory (to evaluate ruin probabilities and related quantities) and double-barrier option pricing or simply computing non-exit probabilities for Brownian motion with general boundaries.

A new class of survival distribution for degradation processes subject to shocks

Many systems experience gradual degradation while simultaneously being exposed to a stream of random shocks of varying magnitudes that eventually cause failure when a shock exceeds the residual strength of the system. In this paper, we present a family of stochastic processes, called shock-degradation processes, that describe this failure mechanism. In our failure model, system strength follows a geometric degradation process. The degradation process itself is any Lévy process, that is, any stochastic process with stationary independent increments. The shock stream is a Fréchet stochastic process, a process derived from the Fréchet extreme-value distribution. Finally, the shock-degradation process is a convolution of the Fréchet shock process and any one of the candidate degradation processes. The system fails at the first occasion when a shock takes system strength across a threshold at zero. The paper presents results for Wiener diffusion processes and gamma processes as examples of Lévy degradation processes. The paper develops key statistical properties of the process model and its survival distribution, including several that are important for its practical application. As the failure mechanism is a first hitting time event, applications that require regression structures fall within the domain of threshold regression methodology.

The fractional and mixed-fractional CEV model

The continuous observation of the financial markets has identified some ‘stylized facts’ which challenge the conventional assumptions, promoting the born of new approaches. On the one hand, the long-range dependence has been faced replacing the traditional Gauss–Wiener process (Brownian motion), characterized by stationary independent increments, by a fractional version. On the other hand, the CEV model addresses the Leverage effect and smile-skew phenomena, efficiently. In this paper, these two insights are merging and both the fractional and mixed-fractional extensions for the CEV model, are developed. Using the fractional versions of both the Itô’s calculus and the Fokker–Planck equation, the transition probability density function of the asset price is obtained as the solution of a non-stationary Feller process with time-varying coefficients, getting an analytical valuation formula for a European Call option. Besides, the Greeks are computed and compared with the standard case.

Classical robots perturbed by L\xe9vy processes: analysis and L\xe9vy disturbance rejection methods

The stability and convergence of state, disturbance and parametric estimates of a robot have been analyzed using the Lyapunov method in the existing literature. In this paper, we analyze the problem of stochastic stability and also prove some results regarding behavior of statistically averaged Lyapunov energy function in the presence of jerk noise modeled as the sum of independent random variables hitting the robot at Poisson times. This type of noise is also called jerk noise in contrast to white Gaussian noise. Jerk noise is a Lévy process, i.e., a process with stationary independent increments, and is the natural non-Gaussian generalization of white Gaussian noise. Jerk noise can successfully be used to model hand tremor occurring at sporadic time intervals. We also study here the problem of time delay estimation in Lévy noise.

To hit or to pass it over—remarkable transient behavior of first arrivals and passages for Lévy flights in finite domains

The term ‘Lévy flights’ was coined by Benoit Mandelbrot, who thus poeticized α-stable Lévy random motion, a Markovian process with stationary independent increments distributed according to the α-stable Lévy probability law. Contrary to the Brownian motion, the trajectories of the α-stable Lévy motion are discontinous, that is exhibit jumps. This feature implies that the process of first passage through the boundary of a given space domain, or the first escape, is different from the process of first arrival (hit) at the boundary. Here we investigate the properties of first escapes and first arrivals for Lévy flights and explore how the asymptotic behavior of the corresponding (passage and hit) probabilities is sensitive to the size of the domain. In particular, we find that the survival probability to stay in a large enough, finite domain has a universal Sparre Andersen temporal scaling , which is transient and changes to an exponential non-universal decay at longer times. Also, the probability to arrive at a finite domain possesses a similar transient Sparre Andersen universality that turns into a non-universal and slower power-law decay in course of time. Finally, we demonstrate that the probability density of the leapover length ℓ over the boundary, related to overshooting events, has an intermediate asymptotics () which is inherent for the escape from a semi-infinite domain. However, for larger leapovers the probability density decays faster according to the law. Thus, we find that the laws derived for the α-stable processes on the semi-infinite domain, manifest themselves as transients for Lévy flights on the finite domain.

Optimal periodic inspection policy for systems generating environmental damage over a finite time horizon

Purpose The authors consider systems that generate damage to environment as they get older and degrade. The purpose of this paper is to develop an optimal condition-based maintenance strategy for such systems in situations where they have a finite operational time requirement. The authors determine simultaneously the optimal number of inspections and the threshold level of environmental damage which minimize the total expected cost over the considered finite time horizon. Design/methodology/approach The environmental degradation level is monitored through periodic inspections. The authors model the environmental degradation process due to the equipment’s degradation by the Wiener process. A mathematical model and a numerical procedure are developed. Numerical calculations are performed and the influence of the variation of key parameters on the optimal solution is investigated. Findings Numerical tests indicate that as the cost of the penalty related to the generation of an excess damage to environment increases, inspections should become more frequent and the threshold level should be lowered in order to favor preventive actions reducing the probability to pay the penalty. Research limitations/implications Given the complexity of the cost function to be minimized, it is difficult to derive analytically the optimal solution. A numerical procedure is designed to obtain the optimal condition-based maintenance policy. Also, the developed model is based on the assumption that the degradation follows a process with stationary independent increments. This may not be appropriate for all types of degradation processes. Practical implications The proposed optimal maintenance policy may be relevant and very useful in the perspective of green operations. In fact, this paper offers to decision-makers a comprehensive approach to implement a green maintenance policy and to rapidly understand the net effect of the maintenance policy with respect to environmental regulation requirements. Originality/value The main contribution consists in the modeling and optimization of the condition-based maintenance policy over a finite time horizon. Indeed, existing condition-based maintenance models over an infinite time horizon are not applicable for systems with a finite operational time requirement.

Nonlinear Lévy processes and their characteristics

We develop a general construction for nonlinear Levy processes with given characteristics. More precisely, given a set of Levy triplets, we construct a sublinear expectation on Skorohod space under which the canonical process has stationary independent increments and a nonlinear generator corresponding to the supremum of all generators of classical Levy processes with triplets in . The nonlinear Levy process yields a tractable model for Knightian uncertainty about the distribution of jumps for which expectations of Markovian functionals can be calculated by means of a partial integro-dierenti al equation.

Classical definitions of the Poisson process do not coincide in the case of generalized convolutions

In the paper we consider a generalizations of the notion of Poisson process to the case when classical convolution is replaced by generalized convolution in the sense of K. Urbanik [16] following two classical definitions of Poisson process. First, for every generalized convolution $\diamond$ we define $\diamond$-generalized Poisson process type I as a Markov process with the $\diamond$-generalized Poisson distribution. Such processes have stationary independent increments in the sense of generalized convolution, but usually they do not live on $\mathbb{N}_0$. The $\diamond$-generalized Poisson process type II is defined as a renewal process based on the sequence $S_n$, which is a Markov process with the step with the lack of memory property. Such processes take values in $\mathbb{N}_0$, however they do not have to be Markov processes, do not have to have independent increments, even in generalized convolution sense. It turns out that the second construction is possible only for monotonic generalized convolutions which admit the existence of distributions with lack of memory, thus we also study these properties.

Half-region depth for stochastic processes

We study the concept of half-region depth, introduced in López-Pintado and Romo (2011). We show that for a wide variety of standard stochastic processes, such as Brownian motion and other symmetric stable processes with stationary independent increments tied down at 0, half-region depth assigns depth zero to all sample functions. To alleviate this difficulty we introduce a method of smoothing, which often not only eliminates the problem of zero depth, but allows us to extend the theoretical results on consistency in that paper up to the n level for many smoothed processes.

Markov chain approximations to scale functions of Lévy processes

We introduce a general algorithm for the computation of the scale functions of a spectrally negative Lévy process X, based on a natural weak approximation of X via upwards skip-free continuous-time Markov chains with stationary independent increments. The algorithm consists of evaluating a finite linear recursion with its (nonnegative) coefficients given explicitly in terms of the Lévy triplet of X. Thus it is easy to implement and numerically stable. Our main result establishes sharp rates of convergence of this algorithm providing an explicit link between the semimartingale characteristics of X and its scale functions, not unlike the one-dimensional Itô diffusion setting, where scale functions are expressed in terms of certain integrals of the coefficients of the governing SDE.

Renormalization Fixed Point of the KPZ Universality Class

The one dimensional Kardar-Parisi-Zhang universality class is believed to describe many types of evolving interfaces which have the same characteristic scaling exponents. These exponents lead to a natural renormalization/rescaling on the space of such evolving interfaces. We introduce and describe the renormalization fixed point of the Kardar-Parisi-Zhang universality class in terms of a random nonlinear semigroup with stationary independent increments, and via a variational formula. Furthermore, we compute a plausible formula the exact transition probabilities using replica Bethe ansatz. The semigroup is constructed from the Airy sheet, a four parameter space-time field which is the Airy2 process in each of its two spatial coordinates. Minimizing paths through this field describe the renormalization group fixed point of directed polymers in a random potential. At present, the results we provide do not have mathematically rigorous proofs, and they should at most be considered proposals.

An Approximation Model of the Collective Risk Model with INAR(1) Claim Process

Cossette et al. (2010, 2011) gave a novel collective risk model where the total numbers of claims satisfy the first-order integer-valued autoregressive process. For a risk model, it is interesting to investigate the upper bound of ruin probability. However, the loss increments of the above model are dependent; it is difficult to derive the upper bound of ruin probability. In this article, we propose an approximation model with stationary independent increments. The upper bound of ruin probability and the adjustment coefficient are derived. The approximation model is illustrated via four simulated examples. Results show that the gap of the approximation model and dependent model can be ignored by adjusting values of parameters.

Lévy Processes, Phase-Type Distributions, and Martingales

Lévy processes are defined as processes with stationary independent increments and have become increasingly popular as models in queueing, finance, etc.; apart from Brownian motion and compound Poisson processes, some popular examples are stable processes, variance gamma processes, CGMY Lévy processes (tempered stable processes), NIG (normal inverse Gaussian) Lévy processes, and hyperbolic Lévy processes. We consider here a dense class of Lévy processes, compound Poisson processes with phase-type jumps in both directions and an added Brownian component. Within this class, we survey how to explicitly compute a number of quantities that are traditionally studied in the area of Lévy processes, in particular two-sided exit probabilities and associated Laplace transforms, the closely related scale function, one-sided exit probabilities and associated Laplace transforms coming up in queueing problems, and similar quantities for a Lévy process with reflection in 0. The solutions are in terms of roots to polynomials, and the basic equations are derived by purely probabilistic arguments using martingale optional stopping; a particularly useful martingale is the so-called Kella-Whitt martingale. Also, the relation to fluid models with a Brownian component is discussed.

Stability in p-th moment for uncertain differential equation

An canonical process is stationary independent increment uncertain process whose increments are normal uncertain variables. Uncertain differential equation is a type of differential equation driven...