The expected first-passage time for a sufficient statistic arising in the Poisson disorder problem

1979 ◽  
Vol 16 (2) ◽  
pp. 274-286 ◽  
Author(s):  
K. Wickwire
Keyword(s):  
Differential Equation ◽  
Mixed Type ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage ◽  
Constant Amount ◽  
Sufficient Statistic ◽  
State Dependent ◽  
Backward Equation ◽  
Differential Difference Equation

In the Poisson disorder problem the probability that the parameter of a Poisson process y, has increased by a constant amount, given observations of ys, s ≦ t, is a Markov process of mixed type with jumps of variable (state-dependent) magnitude superimposed upon a drift which satisfies an ordinary differential equation. Using a likelihood-ratio transformation, one can reduce the backward equation satisfied by the expected first-passage time to a constant level for the mixed process to a differential-difference equation with a constant retardation. We discuss a method for solving this equation and present some numerical results on its solution. The accuracy of some approximations which are easier to calculate is investigated.

The expected first-passage time for a sufficient statistic arising in the Poisson disorder problem

1979 ◽  
Vol 16 (02) ◽  
pp. 274-286
Author(s):  
K. Wickwire
Keyword(s):  
Differential Equation ◽  
Mixed Type ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage ◽  
Constant Amount ◽  
Sufficient Statistic ◽  
State Dependent ◽  
Backward Equation ◽  
Differential Difference Equation

In the Poisson disorder problem the probability that the parameter of a Poisson process y, has increased by a constant amount, given observations of ys, s ≦ t, is a Markov process of mixed type with jumps of variable (state-dependent) magnitude superimposed upon a drift which satisfies an ordinary differential equation. Using a likelihood-ratio transformation, one can reduce the backward equation satisfied by the expected first-passage time to a constant level for the mixed process to a differential-difference equation with a constant retardation. We discuss a method for solving this equation and present some numerical results on its solution. The accuracy of some approximations which are easier to calculate is investigated.

Moments of the first-passage time of a Wiener process with drift between two elastic barriers

1995 ◽  
Vol 32 (4) ◽  
pp. 1007-1013 ◽  
Author(s):  
Marco Dominé
Keyword(s):  
Wiener Process ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage ◽  
One Dimensional ◽  
Special Cases ◽  
Backward Equation ◽  
The One ◽  
First Passage Problem ◽  
Reflecting Barriers

The first-passage problem for the one-dimensional Wiener process with drift in the presence of elastic boundaries is considered. We use the Kolmogorov backward equation with corresponding boundary conditions to derive explicit closed-form expressions for the expected value and the variance of the first-passage time. Special cases with pure absorbing and/or reflecting barriers arise for a certain choice of a parameter constellation.

The final size of a nearly critical epidemic, and the first passage time of a Wiener process to a parabolic barrier

1998 ◽  
Vol 35 (3) ◽  
pp. 671-682 ◽  
Author(s):  
Anders Martin-Löf
Keyword(s):  
Differential Equation ◽  
Wiener Process ◽  
Critical Parameter ◽  
Spectral Representation ◽  
First Passage Time ◽  
Passage Time ◽  
Initial Number ◽  
Airy Functions ◽  
Final Size ◽  
First Passage

The distribution of the final size, K, in a general SIR epidemic model is considered in a situation when the critical parameter λ is close to 1. It is shown that with a ‘critical scaling’ λ ≈ 1 + a / n1/3, m ≈ bn1/3, where n is the initial number of susceptibles and m is the initial number of infected, then K / n2/3 has a limit distribution when n → ∞. It can be described as that of T, the first passage time of a Wiener process to a parabolic barrier b + at − t2/2. The proof is based on a diffusion approximation. Moreover, it is shown that the distribution of T can be expressed analytically in terms of Airy functions using the spectral representation connected with Airy's differential equation.

Sufficient Conditions for a First Passage Time Process to be that of Brownian Motion

1969 ◽  
Vol 6 (01) ◽  
pp. 218-223
Author(s):  
M.T. Wasan
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
First Passage Time ◽  
Sufficient Conditions ◽  
Passage Time ◽  
Kolmogorov Equation ◽  
First Passage ◽  
State Variable ◽  
Strong Markov ◽  
Passage Times

In this paper we assign a set of conditions to a strong Markov process and arrive at a differential equation analogous to the Kolmogorov equation. However, in this case the duration variable is the net distance travelled and the state variable is a time, a situation precisely opposite to that of Brownian motion. Solving this differential equation under certain boundary conditions produces the density function of the first passage times of Brownian motion with positive drift (see [1]), with the aid of which we define a new stochastic process.

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

1974 ◽  
Vol 41 (3) ◽  
pp. 793-797 ◽  
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.

scholarly journals First-passage time of Markov processes to moving barriers

1984 ◽  
Vol 21 (4) ◽  
pp. 695-709 ◽  
Author(s):  
Henry C. Tuckwell ◽  
Frederic Y. M. Wan
Keyword(s):  
Differential Equation ◽  
Markov Processes ◽  
First Passage Time ◽  
Exit Time ◽  
Passage Time ◽  
First Exit Time ◽  
First Passage ◽  
Sample Paths ◽  
First Exit Times ◽  
First Passage Time Problems

The first-passage time of a Markov process to a moving barrier is considered as a first-exit time for a vector whose components include the process and the barrier. Thus when the barrier is itself a solution of a differential equation, the theory of first-exit times for multidimensional processes may be used to obtain differential equations for the moments and density of the first-passage time of the process to the barrier. The procedure is first illustrated for first-passage-time problems where the solutions are known. The mean first-passage time of an Ornstein–Uhlenbeck process to an exponentially decaying barrier is then found by numerical solution of a partial differential equation. Extensions of the method to problems involving Markov processes with discontinuous sample paths and to cases where the process is confined between two moving barriers are also discussed.

The final size of a nearly critical epidemic, and the first passage time of a Wiener process to a parabolic barrier

1998 ◽  
Vol 35 (03) ◽  
pp. 671-682 ◽  
Author(s):  
Anders Martin-Löf
Keyword(s):  
Differential Equation ◽  
Wiener Process ◽  
Critical Parameter ◽  
Spectral Representation ◽  
First Passage Time ◽  
Passage Time ◽  
Initial Number ◽  
Airy Functions ◽  
Final Size ◽  
First Passage

The distribution of the final size, K, in a general SIR epidemic model is considered in a situation when the critical parameter λ is close to 1. It is shown that with a ‘critical scaling’ λ ≈ 1 + a / n 1/3, m ≈ bn 1/3, where n is the initial number of susceptibles and m is the initial number of infected, then K / n 2/3 has a limit distribution when n → ∞. It can be described as that of T, the first passage time of a Wiener process to a parabolic barrier b + at − t 2/2. The proof is based on a diffusion approximation. Moreover, it is shown that the distribution of T can be expressed analytically in terms of Airy functions using the spectral representation connected with Airy's differential equation.

FIRST PASSAGE TIMES OF REFLECTED GENERALIZED ORNSTEIN–UHLENBECK PROCESSES

2012 ◽  
Vol 13 (01) ◽  
pp. 1250014 ◽  
Author(s):  
LIJUN BO ◽  
GUIJUN REN ◽  
YONGJIN WANG ◽  
XUEWEI YANG
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Laplace Transform ◽  
First Passage Time ◽  
Integral Functional ◽  
Stable Process ◽  
Passage Time ◽  
First Passage ◽  
Dynkin’S Formula ◽  
Passage Times

We study first passage problems of a class of reflected generalized Ornstein–Uhlenbeck processes without positive jumps. By establishing an extended Dynkin's formula associated with the process, we derive that the joint Laplace transform of the first passage time and an integral functional stopped at the time satisfies a truncated integro-differential equation. Two solvable examples are presented when the driven Lévy process is a drifted-Brownian motion and a spectrally negative stable process with index α ∈ (1, 2], respectively. Finally, we give two applications in finance.

scholarly journals First passage upwards for state-dependent-killed spectrally negative Lévy processes

2019 ◽  
Vol 56 (2) ◽  
pp. 472-495 ◽  
Author(s):  
Matija Vidmar
Keyword(s):  
Branching Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Additive Functional ◽  
First Passage ◽  
State Dependent ◽  
Continuous State ◽  
The Laplace Transform ◽  
Exit Probability ◽  
The One

AbstractFor a spectrally negative Lévy process X, killed according to a rate that is a function ω of its position, we complement the recent findings of [12] by analysing (in greater generality) the exit probability of the one-sided upwards passage problem. When ω is strictly positive, this problem is related to the determination of the Laplace transform of the first passage time upwards for X that has been time-changed by the inverse of the additive functional $$\int_0^ \cdot \omega ({X_u}){\kern 1pt} {\rm{d}}u$$. In particular, our findings thus shed extra light on related results concerning first passage times downwards (resp. upwards) of continuous-state branching processes (resp. spectrally negative positive self-similar Markov processes).

scholarly journals First-passage time of Markov processes to moving barriers

1984 ◽  
Vol 21 (04) ◽  
pp. 695-709 ◽  
Author(s):  
Henry C. Tuckwell ◽  
Frederic Y. M. Wan
Keyword(s):  
Differential Equation ◽  
Markov Processes ◽  
First Passage Time ◽  
Exit Time ◽  
Passage Time ◽  
First Exit Time ◽  
First Passage ◽  
Sample Paths ◽  
First Exit Times ◽  
First Passage Time Problems

The first-passage time of a Markov process to a moving barrier is considered as a first-exit time for a vector whose components include the process and the barrier. Thus when the barrier is itself a solution of a differential equation, the theory of first-exit times for multidimensional processes may be used to obtain differential equations for the moments and density of the first-passage time of the process to the barrier. The procedure is first illustrated for first-passage-time problems where the solutions are known. The mean first-passage time of an Ornstein–Uhlenbeck process to an exponentially decaying barrier is then found by numerical solution of a partial differential equation. Extensions of the method to problems involving Markov processes with discontinuous sample paths and to cases where the process is confined between two moving barriers are also discussed.

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