Generalized telegraph process with random jumps
TLDR
In this article, the authors considered a generalized telegraph process which follows an alternating renewal process and is subject to random jumps, and developed 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.Abstract:
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.read more
Citations
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Telegraph Processes with Random Jumps and Complete Market Models
TL;DR: In this article, a generalisation of the jump-telegraph process with variable velocities and jumps is proposed, where the amplitude of 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.
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A New Model of Campi Flegrei Inflation and Deflation Episodes Based on Brownian Motion Driven by the Telegraph Process
TL;DR: In this article, a stochastic model to describe the vertical motions in the Campi Flegrei volcanic region is proposed, consisting of a Brownian motion process driven by a generalized telegraph process.
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On a jump-telegraph process driven by an alternating fractional Poisson process
TL;DR: This work analyses such a stochastic process when the interarrival times separating consecutive velocity changes (and jumps) have generalized Mittag-Leffler distributions, and constitute the random times of a fractional alternating Poisson process.
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First Crossing Times of Telegraph Processes with Jumps
TL;DR: In this article, exact formulae related to the distribution of the first passage time τx of the jump-telegraph process were presented, when a jump component is in the opposite direction to the crossing level x > 0.
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Piecewise linear process with renewal starting points
TL;DR: In this article, a Markovian piecewise linear process based on a continuous-time Markov chain with a finite state space is considered, which describes the movement of a particle that takes a new linear trend starting from a new random point (with statedependent distribution) after each trend switch.
References
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Tables of Integrals, Series, and Products
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Probability law, flow function, maximum distribution of wave-governed random motions and their connections with Kirchoff's laws
TL;DR: In this paper, the explicit form of the probability law and the associated flow function of a random motion governed by the telegraph equation are derived and connections of this law with the transition function of Brownian motion are explored.
Book
Lectures on Random Evolution
TL;DR: Random evolution as discussed by the authors is a class of stochastic processes which evolve according to a rule which varies in time according to jumps, in contrast to diffusion processes, which assume that the rule changes continuously with time.