A Damped Telegraph Random Process with Logistic Stationary Distribution

Crescenzo, Antonio Di; Martinucci, Barbara

doi:10.1017/s0021900200006410

Cited by 30 publications

(28 citation statements)

References 24 publications

(19 reference statements)

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“…We note that the right-hand side of (31) is a logistic density with mean m and variance π 2 s 2 /3. In addition, if p = 1 2 then the mean m vanishes, and the density identifies with the stationary probability density function of a damped telegraph process, as obtained in Corollary 3.3 of [17]. We note that if λv = μc then lim t→+∞ p(x, t) = 0,…”

Section: Corollary 2 Under the Assumptions Of Proposition 3 If λ V mentioning

confidence: 55%

“…With the aim of discussing a nontrivial case, and motivated by previous studies (see [17] and [18]) involving finite-velocity random motions with stochastically decreasing random intertimes, in the following we assume that the random variables U k and D k have exponential distributions with linear rates λk and μk. Hence, the tail distribution functions arē…”

Section: Particular Case For the Bernoulli Scheme (A = 0)mentioning

confidence: 99%

“…distribution with linearly increasing rate (see [17]). Moreover, a generalized telegraph process governed by an alternating renewal process was studied by Zacks [33], whereas Iacus [22] gave a rare example where an explicit probability law is obtained for an inhomogeneous telegraph process.…”

Section: Introductionmentioning

confidence: 99%

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A Generalized Telegraph Process with Velocity Driven by Random Trials

Crimaldi¹,

Crescenzo

Iuliano

et al. 2013

Advances in Applied Probability

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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.

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Section: Corollary 2 Under the Assumptions Of Proposition 3 If λ V mentioning

confidence: 55%

Section: Particular Case For the Bernoulli Scheme (A = 0)mentioning

confidence: 99%

See 1 more Smart Citation

A Generalized Telegraph Process with Velocity Driven by Random Trials

Crimaldi¹,

Crescenzo

Iuliano

et al. 2013

Advances in Applied Probability

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“…Probabilistic methods of solving Cauchy problems for the telegraph equation were developed in [11], [12], [14], and [25]. A generalization of the Goldstein-Kac model for the case of a damped telegraph process with logistic stationary distributions was given in [7]. A random motion with velocities alternating at Erlang-distributed random times was studied in [6].…”

Section: Introductionmentioning

confidence: 99%

Probability Distribution Function for the Euclidean Distance Between Two Telegraph Processes

Kolesnik

2014

Adv. Appl. Probab.

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Consider two independent Goldstein-Kac telegraph processes X 1 (t) and X 2 (t) on the real line R. The processes X k (t), k = 1, 2, describe stochastic motions at finite constant velocities c 1 > 0 and c 2 > 0 that start at the initial time instant t = 0 from the origin of R and are controlled by two independent homogeneous Poisson processes of rates λ 1 > 0 and λ 2 > 0, respectively. We obtain a closed-form expression for the probability distribution function of the Euclidean distance ρ(t) = |X 1 (t) − X 2 (t)|, t > 0, between these processes at an arbitrary time instant t > 0. Some numerical results are also presented.

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“…The motions with the velocities alternated in gamma-or Erlang-distributed random times have been considered in [6][7][8]. Telegraph processes with random velocities [9,10] and with random jumps [11] have been also studied including the applications in queuing theory.…”

Section: Introductionmentioning

confidence: 99%

Double Telegraph Processes and Complete Market Models

Ratanov¹

2014

Stochastic Analysis and Applications

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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.

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A Damped Telegraph Random Process with Logistic Stationary Distribution

Cited by 30 publications

References 24 publications

A Generalized Telegraph Process with Velocity Driven by Random Trials

A Generalized Telegraph Process with Velocity Driven by Random Trials

Probability Distribution Function for the Euclidean Distance Between Two Telegraph Processes

Double Telegraph Processes and Complete Market Models

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