Telegraph Process Research Articles

Certain functionals of squared telegraph processes

We investigate the stochastic process defined as the square of the (integrated) symmetric telegraph process. Specifically, we obtain its probability law and a closed form expression of the moment generating function. Some results on the first-passage time through a fixed positive level are also provided. Moreover, we analyze some functionals [Formula: see text] of two independent squared telegraph processes, both in the case [Formula: see text] and [Formula: see text]. Starting from this study, we provide some results on the probability density functions of the two-dimensional radial telegraph process and of the product of two independent symmetric telegraph processes. Some of the expressions obtained are given in terms of new results about derivatives of hypergeometric functions with respect to parameters.

Piecewise linear processes with Poisson‐modulated exponential switching times

We consider the jump telegraph process when switching intensities depend on external shocks also accompanying with jumps. The incomplete financial market model based on this process is studied. The Esscher transform, which changes only unobservable parameters, is considered in detail. The financial market model based on this transform can price switching risks as well as jump risks of the model.

Stochastic Duffing equation in modelling of financial time series

In this study, we aim to introduce a new stochastic modelling and forecasting method for the chaotic noisy Duffing map series and then for the financial time series with similar properties. We examine the fractal properties of the noisy Duffing map by correlation dimension, wavelet analysis and MFDFA analysis. Then, it is shown that the non-linear deterministic multifractal Duffing map with additive white noise process exhibits the telegraph process modulated with the fractional Gaussian noise. The combination of these stochastic processes and the fractional Gaussian noise fit the Duffing map series with white noise which has regime switching and long-range dependent properties and then the real-world financial time series with similar properties. Numerical experiments show that the proposed model generates remarkable forecast result in different horizons when it is compared with the actual outcomes. The results put forward that the chaotic Duffing equation can be a fitting model to certain financial time series or the time series with the same characteristics.

Effect of noise on Bloch oscillations and Wannier-Stark localization

We calculate an exact expression for the probability propagator for a noisy electric field driven tight-binding lattice. The noise considered is a two-level jump process or a telegraph process (TP) which jumps randomly between two values $\pm\mu$. In the absence of a static field and in the limit of zero jump rate of the noisy field we find that the dynamics yield Bloch oscillations with frequency $\mu$, while with an additional static field $\epsilon$ we find oscillatory motion with a superposition of frequencies $(\epsilon \pm \mu)$. On the other hand, when the jump rate is `rapid', and in the absence of a static field, the stochastic field averages to zero if the two states of the TP are equally probable `a-priori'. In that case, we see a delocalization effect. The intimate relationship between the rapid relaxation case and the zero field case is a manifestation of what we call the `Muhammad Ali effect'. It is interesting to note that even for zero static field and rapid relaxation, Bloch oscillations ensue if there is a bias $\delta p$ in the probabilities of the two levels. Remarkably, the Wannier-Stark localization caused by an additional static field is destroyed if the latter is tuned to be exactly equal and opposite to the average stochastic field $\mu\delta p$. This is an example of \emph{incoherent} destruction of Wannier-Stark localization.

First Crossing Times of Telegraph Processes with Jumps

The paper presents exact formulae related to the distribution of the first passage time τx of the jump-telegraph process. In particular, the Laplace transform of τx is analysed, when a jump component is in the opposite direction to the crossing level x > 0. The case of double exponential jumps is also studied in detail.

The contribution of astrocytes to the neuroinflammatory response in multiple sclerosis and experimental autoimmune encephalomyelitis.

Neuroinflammation is the coordinated response of the central nervous system (CNS) to threats to its integrity posed by a variety of conditions, including autoimmunity, pathogens and trauma. Activated astrocytes, in concert with other cellular elements of the CNS and immune system, are important players in the modulation of the neuroinflammatory response. During neurological disease, they produce and respond to cellular signals that often lead to dichotomous processes, which can promote further damage or contribute to repair. This occurs also in multiple sclerosis (MS), where astrocytes are now recognized as key components of its immunopathology. Evidence supporting this role has emerged not only from studies in MS patients, but also from animal models, among which the experimental autoimmune encephalomyelitis (EAE) model has proved especially instrumental. Based on this premise, the purpose of the present review is to summarize the current knowledge of astrocyte behavior in MS and EAE. Following a brief description of the pathological characteristics of the two diseases and the main functional roles of astrocytes in CNS physiology, we will delve into the specific responses of this cell population, analyzing MS and EAE in parallel. We will define the temporal and anatomical profile of astroglial activation, then focus on key processes they participate in. These include: (1) production and response to soluble mediators (e.g., cytokines and chemokines), (2) regulation of oxidative stress, and (3) maintenance of BBB integrity and function. Finally, we will review the state of the art on the available methods to measure astroglial activation in vivo in MS patients, and how this could be exploited to optimize diagnosis, prognosis and treatment decisions. Ultimately, we believe that integrating the knowledge obtained from studies in MS and EAE may help not only better understand the pathophysiology of MS, but also uncover new signals to be targeted for therapeutic intervention.

High-temperature ratchets driven by deterministic and stochastic fluctuations.

We consider the overdamped dynamics of a Brownian particle in an arbitrary spatial periodic and time-dependent potential on the basis of an exact solution for the probability density in the form of a power series in the inverse friction coefficient. The expression for the average velocity of a Brownian ratchet is simplified in the high-temperature consideration when only the first terms of the series can be used. For the potential of an additive-multiplicative form (a sum of a time-independent contribution and a time-dependent multiplicative perturbation), general explicit expressions are obtained which allow comparative analysis of frequency dependencies of the average velocity, implying deterministic and stochastic potential energy fluctuations. For qualitative and quantitative analysis of these dependences, we choose illustrative examples for spatial harmonic fluctuations: with deterministic time dependences of a relaxation type and stochastic time dependences describing Markovian dichotomous and harmonic noise processes. We explore the influence of fluctuation types on the ratchet effect and demonstrate its enhancement in the case of harmonic noise.

Probabilistic analysis of systems alternating for state-dependent dichotomous noise.

Aiming to construct a simple stochastic model able to describe systems alternating due to state-dependent dichotomous noise, we consider a generalized telegraph process whose sample-paths fluctuates around the zero state. Indeed, the latter process describes the motion of a particle on the real line, which is characterized by constant velocities and state-dependent intensities that vanish when the motion is toward the origin. This assumption allows to adopt an approach based on renewal theory to obtain formal expressions of the forward and backward transition densities of the process. The special case when certain random times of the motion possess gamma distribution leads to closed-form expressions of the transition densities, given in terms of the generalized Mittag-Leffler function. We also analyze a first-passage-time problem for the considered process in the presence of two constant boundaries.

In Russian

Stochastic non-equilibrium processes can serve as a driving force for a Brownian motor. In this case, the simplest description of an induced directional motion is carried out under the approximation of small changes in potential energy of a particle compared to the thermal energy (the high-temperature or low-energy approximation). Within this approximation, characteristics of the stochastic process are included into the final analytical expressions for the average velocity of a Brownian motor only through lower correlation functions. The purpose of this article is to obtain these correlation functions for Markov stochastic processes of a general form. The Pauli master equation has been considered, which describes the kinetics of an N states system in terms of given transition rate constants. The general solution of this equation, expressed in terms of the eigenvalues and eigenfunctions of the transition rate matrix, has been specified for the cases of dichotomous fluctuations (N = 2) and fluctuations of transitions from the ground state into two symmetric excited ones (N = 3). To study the influence of a type of a correlation function on the average velocity of a Brownian motor, a spatially harmonic controlled signal is considered, with which the average motor velocity is proportional to the second-order correlation function. The introduction of the additional state allowed us to obtain a bell-shaped frequency dependence of the average motor velocity, with the maximum and the width of the bell easily controllable by the transition rate constants and with the diffusion coefficient which depends on temperature, a particle size, and medium viscosity. Dichotomous process, which is a particular case of the model considered, is characterized by the highest values of the average velocity. A three-level model is preferable in case of a narrower width of the bell-shaped frequency dependence is required.

Fourth-Order Differential Equation for a Two-Stage Absorbing Markov Chain with a Stochastic Forward Transition Probability

The problem of averaging the kinetics of a two-stage absorbing Markov chain over random fluctuations in its forward transition probability approximated by the symmetric dichotomous stochastic process is solved exactly. It is shown that the temporal behavior of the population of chain’s transient state obeys a fourth-order differential equation with the tetra-exponential form of a solution given the finite frequency and mean amplitude of fluctuations. In the limit of frequent fluctuations, this tetra-exponential solution reduces to a simple bi-exponential form typical of the deterministic two-stage decay process lacking fluctuations in its transition probability. Rather, in the limit of rare fluctuations, the tetra-exponential solution, while simplifying to the tri- and bi-exponential solutions, becomes specific both for the low amplitude and the resonance amplitude fluctuations, respectively. Furthermore, there is a stochastic resonance point, where the forward transition probability is in resonance with the mean fluctuation amplitude, whereas the backward transition probability, decay transition probability, and fluctuation frequency are negligibly small. In result, the stochastic immobilization of the two-stage absorbing Markov chain in its initial state occurs at this point.

Interaction of particles governed by generalized integrated telegraph processes

Abstract In this paper we introduce the analysis of the interaction of two or more particles, where each of them is governed by a generalized integrated telegraph process. We present systematically the distribution of the meeting instant of two telegraph particles on the line (or collision phenomenon) that started their motion at the same time but from different locations. We also investigate the distribution of a switching time such that there exists the first moment of the first instant of two particles collision. Asymptotic results of particle collisions for a system of interacting particles with Markov switching, as time goes to infinity, are considered.

Linear combinations of the telegraph random processes driven by partial differential equations

Consider [Formula: see text] independent Goldstein–Kac telegraph processes [Formula: see text] on the real line [Formula: see text]. Each process [Formula: see text] describes a stochastic motion at constant finite speed [Formula: see text] of a particle that, at the initial time instant [Formula: see text], starts from some initial point [Formula: see text] and whose evolution is controlled by a homogeneous Poisson process [Formula: see text] of rate [Formula: see text]. The governing Poisson processes [Formula: see text] are supposed to be independent as well. Consider the linear combination of the processes [Formula: see text] defined by [Formula: see text] where [Formula: see text] are arbitrary real nonzero constant coefficients. We obtain a hyperbolic system of [Formula: see text] first-order partial differential equations for the joint probability densities of the process [Formula: see text] and of the directions of motions at arbitrary time [Formula: see text]. From this system we derive a partial differential equation of order [Formula: see text] for the transition density of [Formula: see text] in the form of a determinant of a block matrix whose elements are the differential operators with constant coefficients. Initial-value problems for the transition densities of the sum and difference [Formula: see text] of two independent telegraph processes with arbitrary parameters, are also posed.

On parameter estimation of hidden telegraph process

The problem of parameter estimation is considered for the two-state telegraph process, observed in the white Gaussian observation noise. An online one-step Maximum Likelihood Estimator process is constructed, using a preliminary Method of Moments Estimator. The obtained estimation procedure is shown to be asymptotically normal and asymptotically efficient in the large sample regime.

A New Model of Campi Flegrei Inflation and Deflation Episodes Based on Brownian Motion Driven by the Telegraph Process

A stochastic model to describe the vertical motions in the Campi Flegrei volcanic region is proposed herein, consisting of a Brownian motion process driven by a generalized telegraph process. Knowledge on the probability law of this process enables quantitative investigation of some basic parameters regulating the inflation/deflation processes, such as velocities and time constants. Statistical analysis was carried out based on linear regression with constraints. Predictions of ground displacements and their changing tendency at future time instants were also made. Finally, a statistical test on the Brownian component of the process confirmed the goodness of the model.

Telegraph noise in Markovian master equation for electron transport through molecular junctions.

We present a theoretical approach to solve the Markovian master equation for quantum transport with stochastic telegraph noise. Considering probabilities as functionals of a random telegraph process, we use Novikov's functional method to convert the stochastic master equation to a set of deterministic differential equations. The equations are then solved in the Laplace space, and the expression for the probability vector averaged over the ensemble of realisations of the stochastic process is obtained. We apply the theory to study the manifestations of telegraph noise in the transport properties of molecular junctions. We consider the quantum electron transport in a resonant-level molecule as well as polaronic regime transport in a molecular junction with electron-vibration interaction.

Stochastic differential equation in a random environment

Solutions of the Ito stochastic differential equation in a random environment are considered. The random environment is formed by the generalized telegraph process. It is proved that the initial problem is equivalent to a system of two stochastic differential equations with nonrandom coefficients. The first equation is the Ito equation, and the initial process is its solution. The second equation is an equation with Poisson process, and its solution is a generalized telegraph process. The theorems of existence and uniqueness of strong and weak solutions are proved.

Automatic Anatomy Recognition using Neural Network Learning of Object Relationships via Virtual Landmarks.

The recently developed body-wide Automatic Anatomy Recognition (AAR) methodology depends on fuzzy modeling of individual objects, hierarchically arranging objects, constructing an anatomy ensemble of these models, and a dichotomous object recognition-delineation process. The parent-to-offspring spatial relationship in the object hierarchy is crucial in the AAR method. We have found this relationship to be quite complex, and as such any improvement in capturing this relationship information in the anatomy model will improve the process of recognition itself. Currently, the method encodes this relationship based on the layout of the geometric centers of the objects. Motivated by the concept of virtual landmarks (VLs), this paper presents a new one-shot AAR recognition method that utilizes the VLs to learn object relationships by training a neural network to predict the pose and the VLs of an offspring object given the VLs of the parent object in the hierarchy. We set up two neural networks for each parent-offspring object pair in a body region, one for predicting the VLs and another for predicting the pose parameters. The VL-based learning/prediction method is evaluated on two object hierarchies involving 14 objects. We utilize 54 computed tomography (CT) image data sets of head and neck cancer patients and the associated object contours drawn by dosimetrists for routine radiation therapy treatment planning. The VL neural network method is found to yield more accurate object localization than the currently used simple AAR method.

Generalized Poisson--Kac Processes and the Regularity of Laws of Nature

Generalized Poisson--Kac Processes and the Regularity of Laws of Nature

Mean first passage time of active Brownian particle in one dimension

ABSTRACTWe investigate the mean first passage time of an active Brownian particle in one dimension using numerical simulations. The activity in one dimension is modelled as a two state model; the particle moves with a constant propulsion strength but its orientation switches from one state to other as in a random telegraphic process. We study the influence of a finite resetting rate r on the mean first passage time to a fixed target of a single free active Brownian particle and map this result using an effective diffusion process. As in the case of a passive Brownian particle, we can find an optimal resetting rate r* for an active Brownian particle for which the target is found with the minimum average time. In the case of the presence of an external potential, we find good agreement between the theory and numerical simulations using an effective potential approach.

Heat currents in electronic junctions driven by telegraph noise

The energy and charge fluxes carried by electrons in a two-terminal junction subjected to a random telegraph noise, produced by a single electronic defect, are analyzed. The telegraph processes are imitated by the action of a stochastic electric field that acts on the electrons in the junction. Upon averaging over all random events of the telegraph process, it is found that this electric field supplies, on the average, energy to the electronic reservoirs, which is distributed unequally between them: the stronger is the coupling of the reservoir with the junction, the more energy it gains. Thus the noisy environment can lead to a temperature gradient across an un-biased junction.