Telegraph Process Research Articles

Double Telegraph Processes and Complete Market Models

The traditional jump-telegraph processes are based on a Poisson process with alternating intensities. We develop a new model based on an alternating doubly stochastic Poisson process with random intensities of jumps. Moreover, in this model the jump-telegraph process performs additional jumps each time when the intensity changes at random. Martingale measures for this type of processes are described by using Girsanov’s transformation. The market model based on the doubly stochastic jump-telegraph process is studied. A class of complete models is also considered.

On the Asymmetric Telegraph Processes

We study the one-dimensional random motionX=X(t),t≥ 0, which takes two different velocities with two different alternating intensities. The closed-form formulae for the density functions ofXand for the moments of any order, as well as the distributions of the first passage times, are obtained. The limit behaviour of the moments is analysed under nonstandard Kac's scaling.

On the Asymmetric Telegraph Processes

We study the one-dimensional random motion X = X(t), t ≥ 0, which takes two different velocities with two different alternating intensities. The closed-form formulae for the density functions of X and for the moments of any order, as well as the distributions of the first passage times, are obtained. The limit behaviour of the moments is analysed under nonstandard Kac's scaling.

Fractional Klein–Gordon Equations and Related Stochastic Processes

This paper presents finite-velocity random motions driven by fractional Klein-Gordon equations of order $\alpha \in (0,1]$. A key tool in the analysis is played by the McBride's theory which converts fractional hyper-Bessel operators into Erdelyi-Kober integral operators. Special attention is payed to the fractional telegraph process whose space-dependent distribution solves a non-homogeneous fractional Klein-Gordon equation. The distribution of the fractional telegraph process for $\alpha = 1$ coincides with that of the classical telegraph process and its driving equation converts into the homogeneous Klein-Gordon equation. Fractional planar random motions at finite velocity are also investigated, the corresponding distributions obtained as well as the explicit form of the governing equations. Fractionality is reflected into the underlying random motion because in each time interval a binomial number of deviations $B(n,\alpha)$ (with uniformly-distributed orientation) are considered. The parameter $n$ of $B(n,\alpha)$ is itself a random variable with fractional Poisson distribution, so that fractionality acts as a subsampling of the changes of directions. Finally the behaviour of each coordinate of the planar motion is examined and the corresponding densities obtained. Extensions to $N$-dimensional fractional random flights are envisaged as well as the fractional counterpart of the Euler-Poisson-Darboux equation to which our theory applies.

Dynamical decoupling of random telegraph noise in a two-qubit gate

Controlling the dynamics of entanglement and preventing its disappearance are central requisites for any implementation of quantum information processing. Solid state qubits are frequently affected by random telegraph noise due to bistable impurities of different nature coupled to the device. In this paper, we investigate the possibility to achieve an efficient universal two-qubit gate in the presence of random telegraph noise by periodic dynamical decoupling. We find an analytic form of the gate error as a function of the number of applied pulses valid when the gate time is much shorter then the telegraphic process correlation time. The analysis is further supplemented by exact numerical results demonstrating the feasibility of a highly-efficient universal two-qubit gate.

The dichotomous Markov process with nonparametric test application; a decision support method in long-term river behavioural analysis: the Zayandeh Rud River; a case study from central Iran

We use the Dichotomous Markov Noise model with constant transition rates to describe the dynamics of fluctuations in the water level as a stochastic process, which is imposed on river discharge changes. By applying this model, two different regimes are determined for the long-term behaviour of the river. Based on these regimes, we define two nonparametric classes of the overall increasing/decreasing nature of the water level in the long-term behaviour, which are separated by an exponential steady state regime. In this paper, we develop a nonparametric testing procedure to test exponentially (steady state regime) against an alternative overall decreasing level distribution. The proposed test predicts the long-term regime behaviour of the river. The mathematical tools introduced to handle the problem should be of general use and the testing procedure can be considered as a new mathematical tool in the study of water level dynamics. Under conditions of data austerity and as a case of study, we examine the stochastic characteristics of the Zayandeh Rud (Zāyande-Rūd or Zāyanderūd, also spelled as Zayandeh-Rood or Zayanderood) River (Isfahan, Iran) water level.

Large deviations for integrals of telegraph processes type

Abstract. We consider the sequence of processes η n ( t ) = λ n ϕ ( n ) ∫ 0 t cos ( α ν ( n s ) + β ) d s , $ \eta _n(t)= \frac{\sqrt{\lambda n}}{\varphi (n)}\int _0^t\cos (\alpha \nu (ns)+\beta )ds, $ where ν ( t ) ${\nu (t) }$ is a Poisson process with parameter λ t ${\lambda t}$ , ϕ ( n ) ${\varphi (n)}$ is a monotonically increasing function, n ∈ ℕ ${n\in \mathbb {N}}$ . The aim of this paper is to prove a large deviations principle for the processes η n ${\eta _n}$ .

Effort in acute traumatic brain injury: considering more than pass/fail.

Performance validity is often conceptualized as a dichotomous process. Effort likely lies on a continuum, however, and psychologists' tendency to rely on pass/fail descriptors of one's effort may not be the only approach. The current study aims to show that when performance validity is considered on a continuum, it may provide clinical information related to cognitive functioning. Forty-four patients with moderate or severe traumatic brain injury were evaluated with the Repeatable Battery for the Assessment of Neuropsychological Status upon their emergence from posttraumatic amnesia. From this data, previously developed effort index scores and "other cognitive functions" index scores were calculated. Performance on the effort index significantly accounted for the patients' performance on a cognitive composite score after considering education and severity of injury. Findings suggest that more in-depth analysis of validity test performance is beneficial to gauge a patient's level of effort and is important to consider when interpreting results and in treatment planning.

Probability Law and Flow Function of Brownian Motion Driven by a Generalized Telegraph Process

We consider a standard Brownian motion whose drift alternates randomly between a positive and a negative value, according to a generalized telegraph process. We first investigate the distribution of the occupation time, i.e. the fraction of time when the motion moves with positive drift. This allows to obtain explicitly the probability law and the flow function of the random motion. We discuss three special cases when the times separating consecutive drift changes have (i) exponential distribution with constant rates, (ii) Erlang distribution, and (iii) exponential distribution with linear rates. In conclusion, in view of an application in environmental sciences we evaluate the density of a Wiener process with infinitesimal moments alternating at inverse Gaussian distributed random times.

A Generalized Telegraph Process with Velocity Driven by Random Trials

We consider a random trial-based telegraph process, which describes a motion on the real line with two constant velocities along opposite directions. At each epoch of the underlying counting process the new velocity is determined by the outcome of a random trial. Two schemes are taken into account: Bernoulli trials and classical Pólya urn trials. We investigate the probability law of the process and the mean of the velocity of the moving particle. We finally discuss two cases of interest: (i) the case of Bernoulli trials and intertimes having exponential distributions with linear rates (in which, interestingly, the process exhibits a logistic stationary density with nonzero mean), and (ii) the case of Pólya trials and intertimes having first gamma and then exponential distributions with constant rates.

A Generalized Telegraph Process with Velocity Driven by Random Trials

We consider a random trial-based telegraph process, which describes a motion on the real line with two constant velocities along opposite directions. At each epoch of the underlying counting process the new velocity is determined by the outcome of a random trial. Two schemes are taken into account: Bernoulli trials and classical Pólya urn trials. We investigate the probability law of the process and the mean of the velocity of the moving particle. We finally discuss two cases of interest: (i) the case of Bernoulli trials and intertimes having exponential distributions with linear rates (in which, interestingly, the process exhibits a logistic stationary density with nonzero mean), and (ii) the case of Pólya trials and intertimes having first gamma and then exponential distributions with constant rates.

Telegraph Processes with Random Jumps and Complete Market Models

We propose a new generalisation of jump-telegraph process with variable velocities and jumps. Amplitude of the jumps and velocity values are random, and they depend on the time spent by the process in the previous state of the underlying Markov process. This construction is applied to markets modelling. The distribution densities and the moments satisfy some integral equations of the Volterra type. We use them for characterisation of the equivalent risk-neutral measure and for the expression of historical volatility in various settings. The fundamental equation is derived by similar arguments. Historical volatilities are computed numerically.

Quantifying the speed, growth modes, and landscape pattern changes of urbanization: a hierarchical patch dynamics approach

Urbanization transforms landscape structure and profoundly affects biodiversity and ecological processes. To understand and solve these ecological problems, at least three aspects of spatiotemporal patterns of urbanization need to be quantified: the speed, urban growth modes, and resultant changes in landscape pattern. In this study, we quantified these spatiotemporal patterns of urbanization in the central Yangtze River Delta region, China from 1979 to 2008, based on a hierarchical patch dynamics framework that guided the research design and the analysis with landscape metrics. Our results show that the urbanized area in the study region increased exponentially during the 30 years at the county, prefectural, and regional levels, with increasing speed down the urban hierarchy. Three growth modes—infilling, edge-expanding, and leapfrogging—operated concurrently and their relative dominance shifted over time. As urbanization progressed, patch density and edge density generally increased, and the connectivity of urban patches in terms of the average nearest neighbor distance also increased. While landscape-level structural complexity also tended to increase, the shape of individual patches became increasingly regular. Our results suggest that whether urban landscapes are becoming more homogenous or heterogeneous may be dependent on scale in time and space as well as landscape metrics used. The speed, growth modes, and landscape pattern are related to each other in complicated fashions. This complex relationship can be better understood by conceptualizing urbanization not simply as a dichotomous diffusion-coalescence switching process, but as a spiraling process of shifting dominance among multiple growth modes: the wax and wane of infilling, edge-expansion, and leapfrog across the landscape.

Stochastic resonance in a non-Poissonian dichotomous process: a new analytical approach

In this paper we present a new approach to evaluate the average of a function of a stochastic variable in the case of a non-Poissonian dichotomous process. We show that using a two-point correlation function approximation we can explore the asymptotic regime with great precision. We apply our approach to study the phenomenon of stochastic resonance. As an example we consider a resistor–capacitor circuit with a stochastic capacitance C and driven by a periodic voltage. We provide an analytical expression for the average charge in the stationary regime and we show that the amplitude of the average charge, and consequently of the average current, displays the phenomenon of stochastic resonance.

Generalized Telegraph Process with Random Jumps

We consider a generalized telegraph process which follows an alternating renewal process and is subject to random jumps. More specifically, consider a particle at the origin of the real line at timet=0. Then it goes along two alternating velocities with opposite directions, and performs a random jump toward the alternating direction at each velocity reversal. We develop the distribution of the location of the particle at an arbitrary fixed timet, and study this distribution under the assumption of exponentially distributed alternating random times. The cases of jumps having exponential distributions with constant rates and with linearly increasing rates are treated in detail.

Generalized Telegraph Process with Random Jumps

We consider a generalized telegraph process which follows an alternating renewal process and is subject to random jumps. More specifically, consider a particle at the origin of the real line at time t=0. Then it goes along two alternating velocities with opposite directions, and performs a random jump toward the alternating direction at each velocity reversal. We develop the distribution of the location of the particle at an arbitrary fixed time t, and study this distribution under the assumption of exponentially distributed alternating random times. The cases of jumps having exponential distributions with constant rates and with linearly increasing rates are treated in detail.

Spin-echo entanglement protection from random telegraph noise

We analyze local spin-echo procedures for protecting entanglement between two non-interacting qubits, each subject to pure-dephasing random telegraph noise. For superconducting qubits, this simple model captures the characteristic features of the effect of bistable impurities coupled to the device. An analytic expression for the entanglement dynamics is reported. Peculiar features related to the non-Gaussian nature of the noise already observed in the single-qubit dynamics also occur in the entanglement dynamics for proper values of the ratio g = v/γ, between the qubit–impurity coupling strength and the switching rate of the random telegraph process, and of the separation between the pulses Δt. We found that the echo procedure may delay the disappearance of entanglement, cancel the dynamical structure of entanglement revivals and dark periods and induce peculiar plateau-like behaviors of the concurrence.

Coarse-grained kinetic equations for quantum systems

The nonequilibrium density matrix method is employed to derive a master equation for the averaged state populations of an open quantum system subjected to an external high frequency stochastic field. It is shown that if the characteristic time τstoch of the stochastic process is much lower than the characteristic time τsteady of the establishment of the system steady state populations, then on the time scale Δt ∼ τsteady, the evolution of the system populations can be described by the coarse-grained kinetic equations with the averaged transition rates. As an example, the exact averaging is carried out for the dichotomous Markov process of the kangaroo type.

Probabilistic estimation of irrigation requirement under climate uncertainty using dichotomous and marked renewal processes

This study addresses estimation of net irrigation requirement over a growing season under climate uncertainty. An ecohydrological model, building upon the stochastic differential equation of soil moisture dynamics, is employed as a basis to derive new analytical expressions for estimating seasonal net irrigation requirement probabilistically. Two distinct irrigation technologies are considered. For micro irrigation technology, probability density function of seasonal net irrigation depth (SNID) is derived assessing transient behavior of a stochastic process which is time integral of dichotomous Markov process. Probability mass function of SNID which is a discrete random variable for traditional irrigation technology is also presented using a marked renewal process with quasi-exponentially-distributed time intervals. Comparing the results obtained from the presented models with those resulted from a Monte Carlo approach verified the significance of the probabilistic expressions derived and assumptions made.

Quantitative Estimates for the Long-Time Behavior of an Ergodic Variant of the Telegraph Process

Motivated by stability questions on piecewise-deterministic Markov models of bacterial chemotaxis, we study the long-time behavior of a variant of the classic telegraph process having a nonconstant jump rate that induces a drift towards the origin. We compute its invariant law and show exponential ergodicity, obtaining a quantitative control of the total variation distance to equilibrium at each instant of time. These results rely on an exact description of the excursions of the process away from the origin and on the explicit construction of an original coalescent coupling for both the velocity and position. Sharpness of the obtained convergence rate is discussed.