Generalized Telegraph Process with Random Jumps

2013 ◽  
Vol 50 (02) ◽  
pp. 450-463 ◽  
Author(s):  
Antonio Di Crescenzo ◽  
Antonella Iuliano ◽  
Barbara Martinucci ◽  
Shelemyahu Zacks
Keyword(s):  
Real Line ◽  
Fixed Time ◽  
Exponential Distributions ◽  
Telegraph Process ◽  
Velocity Reversal ◽  
Alternating Renewal Process ◽  
Alternating Direction ◽  
Random Jumps ◽  
Random Times ◽  
Random Jump

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.

scholarly journals Generalized Telegraph Process with Random Jumps

2013 ◽  
Vol 50 (2) ◽  
pp. 450-463 ◽  
Author(s):  
Antonio Di Crescenzo ◽  
Antonella Iuliano ◽  
Barbara Martinucci ◽  
Shelemyahu Zacks
Keyword(s):  
Real Line ◽  
Fixed Time ◽  
Exponential Distributions ◽  
Telegraph Process ◽  
Velocity Reversal ◽  
Alternating Renewal Process ◽  
Alternating Direction ◽  
Random Jumps ◽  
Random Times ◽  
Random Jump

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.

On random motions with velocities alternating at Erlang-distributed random times

2001 ◽  
Vol 33 (03) ◽  
pp. 690-701 ◽  
Author(s):  
Antonio Di Crescenzo
Keyword(s):  
Stochastic Process ◽  
Random Process ◽  
Bessel Function ◽  
Real Line ◽  
Renewal Process ◽  
The Real ◽  
Alternating Renewal Process ◽  
Transition Densities ◽  
Random Motions ◽  
Random Times

We analyse a non-Markovian generalization of the telegrapher's random process. It consists of a stochastic process describing a motion on the real line characterized by two alternating velocities with opposite directions, where the random times separating consecutive reversals of direction perform an alternating renewal process. In the case of Erlang-distributed interrenewal times, explicit expressions of the transition densities are obtained in terms of a suitable two-index pseudo-Bessel function. Some results on the distribution of the maximum of the process are also disclosed.

scholarly journals A Generalized Telegraph Process with Velocity Driven by Random Trials

2013 ◽  
Vol 45 (04) ◽  
pp. 1111-1136 ◽  
Author(s):  
Irene Crimaldi ◽  
Antonio Di Crescenzo ◽  
Antonella Iuliano ◽  
Barbara Martinucci
Keyword(s):  
Real Line ◽  
Counting Process ◽  
Bernoulli Trials ◽  
Exponential Distributions ◽  
The Real ◽  
Telegraph Process ◽  
Nonzero Mean ◽  
The Mean ◽  
Moving Particle ◽  
Random Trial

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.

scholarly journals A Generalized Telegraph Process with Velocity Driven by Random Trials

2013 ◽  
Vol 45 (4) ◽  
pp. 1111-1136 ◽  
Author(s):  
Irene Crimaldi ◽  
Antonio Di Crescenzo ◽  
Antonella Iuliano ◽  
Barbara Martinucci
Keyword(s):  
Real Line ◽  
Counting Process ◽  
Bernoulli Trials ◽  
Exponential Distributions ◽  
The Real ◽  
Telegraph Process ◽  
Nonzero Mean ◽  
The Mean ◽  
Moving Particle ◽  
Random Trial

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.

scholarly journals A Damped Telegraph Random Process with Logistic Stationary Distribution

2010 ◽  
Vol 47 (01) ◽  
pp. 84-96 ◽  
Author(s):  
Antonio Di Crescenzo ◽  
Barbara Martinucci
Keyword(s):  
Stochastic Process ◽  
Real Line ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage ◽  
Finite Velocity ◽  
Telegraph Process ◽  
Velocity Changes ◽  
Random Times ◽  
Special Case

We introduce a stochastic process that describes a finite-velocity damped motion on the real line. Differently from the telegraph process, the random times between consecutive velocity changes have exponential distribution with linearly increasing parameters. We obtain the probability law of the motion, which admits a logistic stationary limit in a special case. Various results on the distributions of the maximum of the process and of the first passage time through a constant boundary are also given.

On random motions with velocities alternating at Erlang-distributed random times

2001 ◽  
Vol 33 (3) ◽  
pp. 690-701 ◽  
Author(s):  
Antonio Di Crescenzo
Keyword(s):  
Stochastic Process ◽  
Random Process ◽  
Bessel Function ◽  
Real Line ◽  
Renewal Process ◽  
The Real ◽  
Alternating Renewal Process ◽  
Transition Densities ◽  
Random Motions ◽  
Random Times

We analyse a non-Markovian generalization of the telegrapher's random process. It consists of a stochastic process describing a motion on the real line characterized by two alternating velocities with opposite directions, where the random times separating consecutive reversals of direction perform an alternating renewal process. In the case of Erlang-distributed interrenewal times, explicit expressions of the transition densities are obtained in terms of a suitable two-index pseudo-Bessel function. Some results on the distribution of the maximum of the process are also disclosed.

scholarly journals A Damped Telegraph Random Process with Logistic Stationary Distribution

2010 ◽  
Vol 47 (1) ◽  
pp. 84-96 ◽  
Author(s):  
Antonio Di Crescenzo ◽  
Barbara Martinucci
Keyword(s):  
Stochastic Process ◽  
Real Line ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage ◽  
Finite Velocity ◽  
Telegraph Process ◽  
Velocity Changes ◽  
Random Times ◽  
Special Case

We introduce a stochastic process that describes a finite-velocity damped motion on the real line. Differently from the telegraph process, the random times between consecutive velocity changes have exponential distribution with linearly increasing parameters. We obtain the probability law of the motion, which admits a logistic stationary limit in a special case. Various results on the distributions of the maximum of the process and of the first passage time through a constant boundary are also given.

scholarly journals Option Pricing Driven by a Telegraph Process with Random Jumps

2012 ◽  
Vol 49 (3) ◽  
pp. 838-849 ◽  
Author(s):  
Oscar López ◽  
Nikita Ratanov
Keyword(s):  
Option Pricing ◽  
Financial Market ◽  
Option Prices ◽  
Martingale Measures ◽  
Market Models ◽  
Telegraph Process ◽  
Equivalent Martingale Measures ◽  
Hedging Strategies ◽  
Random Jumps ◽  
Equivalent Martingale

In this paper we propose a class of financial market models which are based on telegraph processes with alternating tendencies and jumps. It is assumed that the jumps have random sizes and that they occur when the tendencies are switching. These models are typically incomplete, but the set of equivalent martingale measures can be described in detail. We provide additional suggestions which permit arbitrage-free option prices as well as hedging strategies to be obtained.

scholarly journals On a Model for the Storage of Files on a Hardware: Statistics at a Fixed Time and Asymptotic Regimes

2009 ◽  
Vol 2009 ◽  
pp. 1-24
Author(s):  
Vincent Bansaye
Keyword(s):  
Point Process ◽  
Real Line ◽  
Continuous Time ◽  
Poisson Point Process ◽  
Fixed Time ◽  
The Real ◽  
Parking Problem ◽  
The Right

We consider a version in continuous time of the parking problem of Knuth. Files arrive following a Poisson point process and are stored on a hardware identified with the real line, in the closest free portions at the right of the arrival location. We specify the distribution of the space of unoccupied locations at a fixed time and give asymptotic regimes when the hardware is becoming full.

scholarly journals Generalized Telegraph Process with Random Delays

2012 ◽  
Vol 49 (03) ◽  
pp. 850-865 ◽  
Author(s):  
Daoud Bshouty ◽  
Antonio Di Crescenzo ◽  
Barbara Martinucci ◽  
Shelemyahu Zacks
Keyword(s):  
Exact Distribution ◽  
Exponential Distributions ◽  
Absolutely Continuous ◽  
Telegraph Process ◽  
Random Delays ◽  
Explicit Formulae

In this paper we study the distribution of the location, at time t, of a particle moving U time units upwards, V time units downwards, and W time units of no movement (idle). These are repeated cyclically, according to independent alternating renewals. The distributions of U, V, and W are absolutely continuous. The velocities are v = +1 upwards, v = -1 downwards, and v = 0 during idle periods. Let Y +(t), Y −(t), and Y 0(t) denote the total time in (0, t) of movements upwards, downwards, and no movements, respectively. The exact distribution of Y +(t) is derived. We also obtain the probability law of X(t) = Y +(t) - Y −(t), which describes the particle's location at time t. Explicit formulae are derived for the cases of exponential distributions with equal rates, with different rates, and with linear rates (leading to damped processes).

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