In this paper we present the distribution of the maximum of the telegraph process in the cases where the initial velocity is positive or negative with an even and an odd number of velocity reversals. For the telegraph process with positive initial velocity a reflection principle is proved to be valid while in the case of an initial leftward displacement the conditional distributions are perturbed by a positive probability of never visiting the half positive axis.
Various relationships are established among the mentioned four classes of conditional distributions of the maximum.
The unconditional distributions of the maximum of the telegraph process are obtained for positive and negative initial steps as well as their limiting behaviour. Furthermore the cumulative distributions and the general moments of the conditional maximum are presented.
In this paper we study the joint distributions of the telegraph process and its maximum conditioned on the number of changes of direction and the initial velocity. We prove that in the case of positive starting velocity, a form of the reflection principle holds. We call it the negative reflection principle and we generalize it to the paths of the random motions moving with constant finite velocity. In the last section we obtain the conditional distribution of the first passage time of the telegraph process by using the known results on the maximum of the motion. Finally, we derive the distribu…
In this paper we present the distribution of the maximum of the asymmetric telegraph process in an arbitrary time interval $[0,t]$ under the conditions that the initial velocity $V(0)$ is either $c_1$ or $-c_2$ and the number of changes of direction is odd or even. For the case $V(0) = -c_2$ the singular component of the distribution of the maximum displays an unexpected cyclic behavior and depends only on $c_1$ and $c_2$, but not on the current time $t$. We obtain also the unconditional distribution of the maximum for either $V(0) = c_1$ or $V(0) = -c_2$ and its expression has the form of ser…
In this note, we present some ideas for describing the distributions of the running maximum/minimum, first passage times and telegraphic meanders. Explicit formulae for joint distribution of the extrema, the number of velocity switches and the terminal position are derived using coupled integral equations technique.
In this paper we consider a slight generalization of the damped telegraph process in Di Crescenzo and Martinucci (2010). We prove a large deviation principle for this process and an asymptotic result for its level crossing probabilities (as the level goes to infinity). Finally we compare our results with the analogous well-known results for the standard telegraph process.
We study the asymmetric one-dimensional telegraph process in the bounded domain. Lower boundary is absorbing and upper boundary is reflecting with delay. Point stays in the upper boundary until switch of regime occurs. We obtain the distribution of this process in terms of Laplace trasforms.
We study the long-time behavior of variants of the telegraph process with position-dependent jump-rates, which result in a monotone gradient-like drift toward the origin. We compute their invariant laws and obtain, via probabilistic couplings arguments, some quantitative estimates of the total variation distance to equilibrium. Our techniques extend ideas previously developed for a simplified piecewise deterministic Markov model of bacterial chemotaxis.
For the one-dimensional telegraph process, we obtain explicit distribution of the occupation time of the positive half-line. The long-term limiting distribution is then derived when the initial location of the process is in the range of sub-normal or normal deviations from the origin; in the former case, the limit is given by the arcsine law. These limit theorems are also extended to the case of more general occupation-type functionals.
A telegraph process with an elastic barrier at the origin was studied in [5]; in particular the number of visits of the origin before the absorption is a geometric distributed random variable M. Some asymptotic results (large and moderate deviations) for that model were obtained in [17]. In this paper we study large and moderate deviations for a generalized model where M is a light-tailed distributed random variable.
We consider the telegraph process with two velocities, $a_1>a_2\in\mathbb{R}$, and two rates of reversal, $λ_1,λ_2>0$. We study some of its features with respect to the conditional probability measure where both the initial speed and the number of changes of direction are known. We exhibit a new proof by induction of the (conditional) probability law and a detailed study of the distribution of the motion at time $t>0$ conditioned on its position at a previous time $0<s<t$. In the case of a symmetric process, we present some results on the joint distribution of the position of the motion at tim…
We study a family of memory-based persistent random walks and we prove weak convergences after space-time rescaling. The limit processes are not only Brownian motions with drift. We have obtained a continuous but non-Markov process $(Z_t)$ which can be easely expressed in terms of a counting process $(N_t)$. In a particular case the counting process is a Poisson process, and $(Z_t)$ permits to represent the solution of the telegraph equation. We study in detail the Markov process $((Z_t,N_t); t\ge 0)$.
We study a one-dimensional Markov modulated random walk with jumps. It is assumed that amplitudes of jumps as well as a chosen velocity regime are random and depend on a time spent by the process at a previous state of the underlying Markov process. Equations for the distribution and equations for its moments are derived. We characterise the martingale distributions in terms of observable proportions between jump and velocity regimes.
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.…
We investigate the one-dimensional telegraph random process in the presence of an elastic boundary at the origin. This process describes a finite-velocity random motion that alternates between two possible directions of motion (positive or negative). When the particle hits the origin, it is either absorbed, with probability $α$, or reflected upwards, with probability $1-α$. In the case of exponentially distributed random times between consecutive changes of direction, we obtain the distribution of the renewal cycles and of the absorption time at the origin. This investigation is performed both…
We present the stochastic solution to a generalized fractional partial differential equation involving a regularized operator related to the so-called Prabhakar operator and admitting, amongst others, as specific cases the fractional diffusion equation and the fractional telegraph equation. The stochastic solution is expressed as a Lévy process time-changed with the inverse process to a linear combination of (possibly subordinated) independent stable subordinators of different indices. Furthermore a related SDE is derived and discussed.
Consider two independent Goldstein-Kac telegraph processes $X_1(t)$ and $X_2(t)$ on the real line $\Bbb R$. The processes $X_k(t), \; k=1,2,$ are performed by stochastic motions at finite constant velocities $c_1>0, \; c_2>0,$ that start at the initial time instant $t=0$ from the origin of the real line $\Bbb R$ and are controlled by two independent homogeneous Poisson processes of rates $λ_1>0, \; λ_2>0$, respectively. Closed-form expression for the probability distribution function of the Euclidean distance $$ρ(t) = |X_1(t) - X_2(t) |, \qquad t>0,$$ between these processes at arbitrary time …
We consider the random evolution described by the motion of a particle moving on a circle alternating the angular velocities $ \pm c $ and changing rotation at Poisson random times, resulting in a telegraph process over the circle. We study the analytic properties of the semigroup it generates as well as its probability distribution. The asymptotic behavior of the wrapped process is also studied in terms of circular Brownian motion. Besides, it is possible to derive a stochastic model for harmonic oscillators with random changes in direction and we give a diffusive approximation of this proces…
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 whi…
We consider two independent Goldstein-Kac telegraph processes $X_1(t)$ and $X_2(t)$ on the real line $\Bbb R$, both developing with finite constant speed $c>0$, that, at the initial time instant $t=0$, simultaneously start from the origin $0\in\Bbb R$ and whose evolutions are controlled by two independent homogeneous Poisson processes of the same rate $λ>0$. Closed-form expressions for the transition density $p(x,t)$ and the probability distribution function $Φ(x,t)=\text{Pr} \{ S(t)<x \}, \; x\in\Bbb R, \; t>0,$ of the sum $S(t)=X_1(t)+X_2(t)$ of these processes at arbitrary time instant $t>0…
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 non-constant 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 velocity …
Consider $n$ independent Goldstein-Kac telegraph processes $X_1(t), \dots ,X_n(t), \; n\ge 2, \; t\ge 0,$ on the real line $\Bbb R$. Each the process $X_k(t), \; k=1,\dots,n,$ describes a stochastic motion at constant finite speed $c_k>0$ of a particle that, at the initial time instant $t=0$, starts from some initial point $x_k^0=X_k(0)\in\Bbb R$ and whose evolution is controlled by a homogeneous Poisson process $N_k(t)$ of rate $λ_k>0$. The governing Poisson processes $N_k(t), \; k=1,\dots,n,$ are supposed to be independent as well. Consider the linear form of the processes $X_1(t), \dots ,X_…
In a recent work we introduced a semi-Markovian discrete-time generalization of the telegraph process. We referred this random walk to as squirrel random walk (SRW). The SRW is a discrete-time random walk on the one-dimensional infinite lattice where the step direction is reversed at arrival times of a discrete-time renewal process and remains unchanged at uneventful time instants. We first recall general notions of the SRW. The main subject of the paper is the study of the SRW where the step direction switches at the arrival times of a generalization of the Sibuya discrete-time renewal proces…
In this paper we consider a class of non-local in time telegraph equations. Recently, it has been proved that the fundamental solutions of such equations can be interpreted as the probability density function of a stochastic process. We study the asymptotic behavior of the variance of this process at large and short times. In this context, we develop a method to construct new examples such the variance has a slowly growth behavior, extending the earlier results. Finally, we show that our approach can be adapted to define new integro-differential operators which are interesting in sub-diffusion…
Brownian motion whose infinitesimal variance changes according to a three-state continuous time Markov Chain is studied. This Markov Chain can be viewed as a telegraph process with one on state and two off states. We first derive the distribution of occupation time of the on state. Then the result is used to develop a likelihood estimation procedure when the stochastic process at hand is observed at discrete, possibly irregularly spaced time points. The likelihood function is evaluated with the forward algorithm in the general framework of hidden Markov models. The analytic results are confirm…
We analyze the one-dimensional telegraph random process confined by two boundaries, 0 and $H>0$. The process experiences hard reflection at the boundaries (with random switching to full absorption). Namely, when the process hits the origin (the threshold $H$) it is either absorbed, with probability $α$, or reflected upwards (downwards), with probability $1-α$, for $0<α<1$. We provide various results on the expected values of the renewal cycles and of the absorption time. The adopted approach is based on the analysis of the first-crossing times of a suitable compound Poisson process through lin…
The telegraph process $\{X(t), t>0\}$, is supposed to be observed at $n+1$ equidistant time points $t_i=iΔ_n,i=0,1,..., n$. The unknown value of $λ$, the underlying rate of the Poisson process, is a parameter to be estimated. The asymptotic framework considered is the following: $Δ_n \to 0$, $nΔ_n = T \to \infty$ as $n \to \infty$. We show that previously proposed moment type estimators are consistent and asymptotically normal but not efficient. We study further an approximated moment type estimator which is still not efficient but comes in explicit form. For this estimator the additional assu…
The telegraph process models a random motion with finite velocity and it is usually proposed as an alternative to diffusion models. The process describes the position of a particle moving on the real line, alternatively with constant velocity $+ v$ or $-v$. The changes of direction are governed by an homogeneous Poisson process with rate $λ>0.$ In this paper, we consider a change point estimation problem for the rate of the underlying Poisson process by means of least squares method. The consistency and the rate of convergence for the change point estimator are obtained and its asymptotic dist…
The telegraph process $X(t)$, $t>0$, (Goldstein, 1951) and the geometric telegraph process $S(t) = s_0 \exp\{(μ-\frac12σ^2)t + σX(t)\}$ with $μ$ a known constant and $σ>0$ a parameter are supposed to be observed at $n+1$ equidistant time points $t_i=iΔ_n,i=0,1,..., n$. For both models $λ$, the underlying rate of the Poisson process, is a parameter to be estimated. In the geometric case, also $σ>0$ has to be estimated. We propose different estimators of the parameters and we investigate their performance under the high frequency asymptotics, i.e. $Δ_n \to 0$, $nΔ= T<\infty$ as $n \to \infty$, w…
Ornstein-Uhlenbeck process of bounded variation is introduced as a solution of an analogue of the Langevin equation with an integrated telegraph process replacing a Brownian motion. There is an interval $I$ such that the process starting from the internal point of $I$ always remains within $I$. Starting outside, this process a. s. reaches this interval in a finite time. The distribution of the time for which the process falls into this interval is obtained explicitly. The certain formulae for the mean and the variance of this process are obtained on the basis of the joint distribution of the t…
Stochastic processes are proposed whose master equations coincide with classical wave, telegraph, and Klein-Gordon equations. Similar to predecessors based on the Goldstein-Kac telegraph process, the model describes the motion of particles with constant speed and transitions between discreet allowed velocity directions. A new ingredient is that transitions into a given velocity state depend on spatial derivatives of other states populations, rather than on populations themselves. This feature requires the sacrifice of the single-particle character of the model, but allows to imitate the Huygen…
This paper presents finite-velocity random motions driven by fractional Klein-Gordon equations of order $α\in (0,1]$. 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 $α= 1$ coincides with that of the classical telegraph process and its driving equation converts into the homogeneous Kl…
In this paper we consider the general fractional equation \sum_{j=1}^m λ_j \frac{\partial^{ν_j}}{\partial t^{ν_j}} w(x_1,..., x_n ; t) = -c^2 (-Δ)^βw(x_1,..., x_n ; t), for ν_j \in (0,1], β\in (0,1] with initial condition w(x_1,..., x_n ; 0)= \prod_{j=1}^n δ(x_j). The solution of the Cauchy problem above coincides with the distribution of the n-dimensional process \bm{S}_n^{2β} \mathcal{L} c^2 {L}^{ν_1,..., ν_m} (t) \r, t>0, where \bm{S}_n^{2β} is an isotropic stable process independent from {L}^{ν_1,..., ν_m}(t) which is the inverse of {H}^{ν_1,..., ν_m} (t) = \sum_{j=1}^m λ_j^{1/ν_j} H^{ν_j}…
We study planar random motions with finite velocities, of norm $c>0$, along orthogonal directions and changing at the instants of occurrence of a non-homogeneous Poisson process with rate function $λ(t),\ t\ge0$. We focus on the distribution of the current position $\bigl(X(t), Y(t)\bigr),\ t\ge0$, in the case where the motion has orthogonal deviations and where also reflection is admitted. In all the cases the process is located within the closed square $S_{ct}=\{(x,y)\in \mathbb{R}^2\,:\,|x|+|y|\le ct\}$ and we obtain the probability law inside $S_{ct}$, on the edge $\partial S_{ct}$ and on …
In this paper we consider a telegraph equation with time-dependent coefficients, governing the persistent random walk of a particle moving on the line with a time-varying velocity $c(t)$ and changing direction at instants distributed according to a non-stationary Poisson distribution with rate $λ(t)$. We show that, under suitable assumptions, we are able to find the exact form of the probability distribution. We also consider the space-fractional counterpart of this model, finding the characteristic function of the related process. A conclusive discussion is devoted to the potential applicatio…
The zigzag process is a variant of the telegraph process with position dependent switching intensities. A characterization of the $L^2$-spectrum for the generator of the one-dimensional zigzag process is obtained in the case where the marginal stationary distribution on $\R$ is unimodal and the refreshment intensity is zero. Sufficient conditions are obtained for a spectral mapping theorem, mapping the spectrum of the generator to the spectrum of the corresponding Markov semigroup. Furthermore results are obtained for symmetric stationary distributions and for perturbations of the spectrum, in…
The joint distribution of the maximum loss and the maximum gain is obtained for a spectrally negative Levy process until the passage time of a given level. Their marginal distributions up to an independent exponential time are also provided. The existing formulas for Brownian motion with drift are recovered using the particular scale functions.Keywords Maximum drawdown Maximum drawup Spectrally negative Reflected process Fluctuation theory
Gaussian random processes which variances reach theirs maximum values at unique points are considered. Exact asymptotic behaviors of probabilities of large absolute maximums of theirs trajectories have been evaluated using Double Sum Method under the widest possible conditions.
