Occupation time distributions for the telegraph process (original) (raw)
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On the asymmetric telegraph processes
Journal of Applied Probability, 2014
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 non-standard Kac's scaling.
On the time a diffusion process spends along a line
Stochastic Processes and Their Applications, 1993
For an arbitrary diffusion process X with time-homogeneous drift and variance parameters μ(x) and σ2(x), let Vε be times the total time X(t) spends in the strip . The limit V as ε → 0 is the full halfline version of the local time of X(t) − a − bt at zero, and can be thought of as the time X spends along the straight line x = a + bt. We prove that V is either infinite with probability 1 or distributed as a mixture of an exponential and a unit point mass at zero, and we give formulae for the parameters of this distribution in terms of μ(·), σ(·), a, b, and the starting point X(0). The special case of a Brownian motion is studied in more detail, leading in particular to a full process V(b) with continuous sample paths and exponentially distributed marginals. This construction leads to new families of bivariate and multivariate exponential distributions. Truncated versions of such ‘total relative time’ variables are also studied. A relation is pointed out to a second order asymptotics problem in statistical estimation theory, recently investigated in Hjort and Fenstad (1992a, 1992b).
Stochastic Processes and their Applications, 2009
The long time asymptotics of the time spent on the positive side are discussed for one-dimensional diffusion processes in random environments. The limiting distributions under the log-log scale are obtained for the diffusion processes in the stable medium as well as for the Brox model. Similar problems are discussed for random walks in random environments and it is proved that the limiting laws are the same as in the case of diffusions.
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.
Some Results on the Telegraph Process Confined by Two Non-Standard Boundaries
Methodology and Computing in Applied Probability, 2020
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 linear boundaries. Our analysis includes also some comparisons between suitable stopping times of the considered telegraph process and of the corresponding diffusion process obtained under the classical Kac's scaling conditions.
Illinois Journal of Mathematics
It is well known that a class of subordinators can be represented using the local time of Brownian motions. An extension of such a representation is given for a class of Lévy processes which are not necessarily of bounded variation. This class can be characterized by the complete monotonicity of the Lévy measures. The asymptotic behavior of such processes is also discussed and the results are applied to the generalized arc-sine law, an occupation time problem on the positive side for one-dimensional diffusion processes.
Occupation Time of Exclusion Processes with Conductances
Journal of Statistical Physics, 2014
We obtain the fluctuations for the occupation time of one-dimensional symmetric exclusion processes with speed change, where the transition rates (conductances) are driven by a general function W. The approach does not require sharp bounds on the spectral gap of the system nor the jump rates to be bounded from above or below. We present some examples and for one of them, we observe that the fluctuations of the current are trivial, but the fluctuations of the occupation time are given by a fractional Brownian Motion. This shows that, in general, the fluctuations of the current and of the occupation time are not of same order.
Cox-Based and Elliptical Telegraph Processes and Their Applications
Risks
This paper studies two new models for a telegraph process: Cox-based and elliptical telegraph processes. The paper deals with the stochastic motion of a particle on a straight line and on an ellipse with random positive velocity and two opposite directions of motion, which is governed by a telegraph–Cox switching process. A relevant result of our analysis on the straight line is obtaining a linear Volterra integral equation of the first kind for the characteristic function of the probability density function (PDF) of the particle position at a given time. We also generalize Kac’s condition for the telegraph process to the case of a telegraph–Cox switching process. We show some examples of random velocity where the distribution of the coordinate of a particle is expressed explicitly. In addition, we present some novel results related to the switched movement evolution of a particle according to a telegraph–Cox process on an ellipse. Numerical examples and applications are presented f...
Occupation times of refracted L\'evy processes
2012
A refracted Lévy process is a Lévy process whose dynamics change by subtracting off a fixed linear drift (of suitable size) whenever the aggregate process is above a pre-specified level. More precisely, whenever it exists, a refracted Lévy process is described by the unique strong solution to the stochastic differential equation