A Damped Telegraph Random Process with Logistic Stationary Distribution

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.

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Fractional compound Poisson processes with multiple internal states

Mathematical Modelling of Natural Phenomena ◽

10.1051/mmnp/2018001 ◽

2018 ◽

Vol 13 (1) ◽

pp. 10 ◽

Cited By ~ 8

Author(s):

Pengbo Xu ◽

Weihua Deng

Keyword(s):

Stochastic Process ◽

Random Walk ◽

Waiting Time ◽

Anomalous Diffusion ◽

First Passage Time ◽

Passage Time ◽

Poisson Processes ◽

First Passage ◽

Internal States ◽

Functional Distribution

For the particles undergoing the anomalous diffusion with different waiting time distributions for different internal states, we derive the Fokker-Planck and Feymann-Kac equations, respectively, describing positions of the particles and functional distributions of the trajectories of particles; in particular, the equations governing the functional distribution of internal states are also obtained. The dynamics of the stochastic processes are analyzed and the applications, calculating the distribution of the first passage time and the distribution of the fraction of the occupation time, of the equations are given. For the further application of the newly built models, we make very detailed discussions on the none-immediately-repeated stochastic process, e.g., the random walk of smart animals.

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On the First-Passage Distribution for the Envelope of a Nonstationary Narrow-Band Stochastic Process

Journal of Applied Mechanics ◽

10.1115/1.3423390 ◽

1974 ◽

Vol 41 (3) ◽

pp. 793-797 ◽

Cited By ~ 31

Author(s):

W. C. Lennox ◽

D. A. Fraser

Keyword(s):

Stochastic Process ◽

Hypergeometric Functions ◽

Narrow Band ◽

First Passage Time ◽

Wide Band ◽

Passage Time ◽

First Passage ◽

Eigenvalues And Eigenfunctions ◽

One Dimensional ◽

Backward Equation

A narrow-band stochastic process is obtained by exciting a lightly damped linear oscillator by wide-band stationary noise. The equation describing the envelope of the process is replaced, in an asymptotic sense, by a one-dimensional Markov process and the modified Kolmogorov (backward) equation describing the first-passage distribution function is solved exactly using classical methods by reducing the problem to that of finding the related eigenvalues and eigenfunctions; in this case degenerate hypergeometric functions. If the exciting process is white noise, the analysis is exact. The method also yields reasonable approximations for the first-passage time of the actual narrow-band process for either a one-sided or a symmetric two-sided barrier.

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First Passage Time Analysis of Stochastic Process Algebra Using Partial Orders

Tools and Algorithms for the Construction and Analysis of Systems - Lecture Notes in Computer Science ◽

10.1007/3-540-45319-9_16 ◽

2001 ◽

pp. 220-235 ◽

Cited By ~ 5

Author(s):

Theo C. Ruys ◽

Rom Langerak ◽

Joost-Pieter Katoen ◽

Diego Latella ◽

Mieke Massink

Keyword(s):

Stochastic Process ◽

Process Algebra ◽

First Passage Time ◽

Passage Time ◽

Partial Orders ◽

Time Analysis ◽

First Passage ◽

Stochastic Process Algebra

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Weak convergence of first passage time processes

Journal of Applied Probability ◽

10.2307/3211913 ◽

1971 ◽

Vol 8 (2) ◽

pp. 417-422 ◽

Cited By ~ 18

Author(s):

Ward Whitt

Keyword(s):

Stochastic Process ◽

Weak Convergence ◽

First Passage Time ◽

Continuous Mapping ◽

Passage Time ◽

Continuous Functions ◽

First Passage ◽

Image Position ◽

Supremum Process

Let D = D[0, ∞) be the space of all real-valued right-continuous functions on [0, ∞) with limits from the left. For any stochastic process X in D, let the associated supremum process be S(X), wherefor any x ∊ D. It is easy to verify that S: D → D is continuous in any of Skorohod's (1956) topologies extended from D[0,1] to D[0, ∞) (cf. Stone (1963) and Whitt (1970a, c)). Hence, weak convergence Xn ⇒ X in D implies weak convergence S(Xn) ⇒ S(X) in D by virtue of the continuous mapping theorem (cf. Theorem 5.1 of Billingsley (1968)).

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The First Passage Time and Estimation of the Number of Level-Crossings for a Telegraph Process

Ukrainian Mathematical Journal ◽

10.1007/s11253-015-1132-y ◽

2015 ◽

Vol 67 (7) ◽

pp. 998-1007 ◽

Cited By ~ 3

Author(s):

A. A. Pogorui ◽

R. M. Rodríguez-Dagnino ◽

T. Kolomiets

Keyword(s):

First Passage Time ◽

Passage Time ◽

First Passage ◽

Level Crossings ◽

Telegraph Process

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On a jump-telegraph process driven by an alternating fractional Poisson process

Journal of Applied Probability ◽

10.1017/jpr.2018.8 ◽

2018 ◽

Vol 55 (1) ◽

pp. 94-111 ◽

Cited By ~ 1

Author(s):

Antonio Di Crescenzo ◽

Alessandra Meoli

Keyword(s):

Poisson Process ◽

First Passage Time ◽

Renewal Theory ◽

General Setting ◽

Passage Time ◽

Telegraph Process ◽

Transition Densities ◽

In Series ◽

Interarrival Times ◽

Velocity Changes

AbstractThe basic jump-telegraph process with exponentially distributed interarrival times deserves interest in various applied fields such as financial modelling and queueing theory. Aiming to propose a more general setting, we analyse 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. By means of renewal theory-based issues we obtain the forward and backward transition densities of the motion in series form, and prove their uniform convergence. Specific attention is then given to the case of jumps with constant size, for which we also obtain the mean of the process. Finally, we investigate the first-passage time of the process through a constant positive boundary, providing its formal distribution and suitable lower bounds.

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Weak convergence of first passage time processes

Journal of Applied Probability ◽

10.1017/s0021900200035440 ◽

1971 ◽

Vol 8 (02) ◽

pp. 417-422 ◽

Cited By ~ 4

Author(s):

Ward Whitt

Keyword(s):

Stochastic Process ◽

Weak Convergence ◽

First Passage Time ◽

Continuous Mapping ◽

Passage Time ◽

Continuous Functions ◽

First Passage ◽

Image Position ◽

Supremum Process

LetD = D[0, ∞) be the space of all real-valued right-continuous functions on [0, ∞) with limits from the left. For any stochastic processXinD,let the associatedsupremum processbeS(X), wherefor anyx ∊ D. It is easy to verify thatS:D→Dis continuous in any of Skorohod's (1956) topologies extended fromD[0,1] toD[0, ∞) (cf. Stone (1963) and Whitt (1970a, c)). Hence, weak convergenceXn⇒XinDimplies weak convergenceS(Xn) ⇒S(X) inDby virtue of the continuous mapping theorem (cf. Theorem 5.1 of Billingsley (1968)).

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COMPUTATIONAL ASPECTS OF MONTE-CARLO SIMULATIONS OF THE FIRST PASSAGE TIME FOR MULTIVARIATE TRANSFORMED BROWNIAN MOTIONS WITH JUMPS

International Journal of Computational Methods ◽

10.1142/s0219876213500266 ◽

2013 ◽

Vol 10 (05) ◽

pp. 1350026 ◽

Cited By ~ 1

Author(s):

DI ZHANG ◽

RODERICK V. N. MELNIK

Keyword(s):

Stochastic Process ◽

Monte Carlo ◽

Option Pricing ◽

First Passage Time ◽

Jump Diffusion ◽

Passage Time ◽

First Passage ◽

Efficient Tool ◽

Default Rates ◽

Computational Aspects

Many problems in science, engineering, and finance require the information on the first passage time (FPT) of a stochastic process. Mathematically, such problems are often reduced to the evaluation of the probability density of the time for such a process to cross a certain level, a boundary, or to enter a certain region. While in other areas of applications the FPT problem can often be solved analytically, in finance we usually have to resort to the application of numerical procedures, in particular when we deal with jump-diffusion stochastic processes (JDP). In this paper, we propose a Monte-Carlo-based methodology for the solution of the FPT problem in the context of multivariate (and correlated) JDPs. The developed technique provides an efficient tool for a number of applications, including credit risk and option pricing. We demonstrate its applicability to the analysis of default rates and default correlations of several different, but correlated firms via a set of empirical data.

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First passage time for the state of exact chemical equilibrium

Collection of Czechoslovak Chemical Communications ◽

10.1135/cccc19800777 ◽

1980 ◽

Vol 45 (3) ◽

pp. 777-782 ◽

Cited By ~ 1

Author(s):

Milan Šolc

Keyword(s):

Chemical Equilibrium ◽

First Passage Time ◽

Passage Time ◽

The State ◽

Recurrence Time ◽

First Passage ◽

First Order ◽

The Mean ◽

Passage Times ◽

Number Of Particles

The establishment of chemical equilibrium in a system with a reversible first order reaction is characterized in terms of the distribution of first passage times for the state of exact chemical equilibrium. The mean first passage time of this state is a linear function of the logarithm of the total number of particles in the system. The equilibrium fluctuations of composition in the system are characterized by the distribution of the recurrence times for the state of exact chemical equilibrium. The mean recurrence time is inversely proportional to the square root of the total number of particles in the system.

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