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).

Spectral Theory for Skip-Free Markov Chains

1989 ◽  
Vol 3 (1) ◽  
pp. 77-88 ◽  
Author(s):  
Joseph Abate ◽  
Ward Whitt
Keyword(s):  
Markov Chains ◽  
Spectral Theory ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage Times ◽  
Simple Relationship ◽  
First Passage ◽  
Birth And Death Processes ◽  
The Laplace Transform ◽  
Passage Times

The distribution of upward first passage times in skip-free Markov chains can be expressed solely in terms of the eigenvalues in the spectral representation, without performing a separate calculation to determine the eigenvectors. We provide insight into this result and skip-free Markov chains more generally by showing that part of the spectral theory developed for birth-and-death processes extends to skip-free chains. We show that the eigenvalues and eigenvectors of skip-free chains can be characterized in terms of recursively defined polynomials. Moreover, the Laplace transform of the upward first passage time from 0 to n is the reciprocal of the nth polynomial. This simple relationship holds because the Laplace transforms of the first passage times satisfy the same recursion as the polynomials except for a normalization.

Hitting-Time Densities of a Two-Dimensional Markov Process

1992 ◽  
Vol 6 (4) ◽  
pp. 561-580
Author(s):  
C. H. Hesse
Keyword(s):  
Global Analysis ◽  
First Passage Time ◽  
Type Equation ◽  
Passage Time ◽  
Two Dimensional ◽  
First Passage ◽  
Wide Range ◽  
The Laplace Transform ◽  
Mean And Variance

This paper deals with the two-dimensional stochastic process (X(t), V(t)) where dX(t) = V(t)dt, V(t) = W(t) + ν for some constant ν and W(t) is a one-dimensional Wiener process with zero mean and variance parameter σ2= 1. We are interested in the first-passage time of (X(t), V(t)) to the plane X = 0 for a process starting from (X(0) = −x, V(0) = ν) with x > 0. The partial differential equation for the Laplace transform of the first-passage time density is transformed into a Schrödinger-type equation and, using methods of global analysis, such as the method of dominant balance, an approximation to the first-passage density is obtained. In a series of simulations, the quality of this approximation is checked. Over a wide range of x and ν it is found to perform well, globally in t. Some applications are mentioned.

scholarly journals The First Passage Time and the Dividend Value Function for One-Dimensional Diffusion Processes between Two Reflecting Barriers

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Chuancun Yin ◽  
Huiqing Wang
Keyword(s):  
Boundary Conditions ◽  
Value Function ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage ◽  
One Dimensional ◽  
The Laplace Transform ◽  
The Value Function ◽  
Reflecting Barriers

We consider the general one-dimensional time-homogeneous regular diffusion process between two reflecting barriers. An approach based on the Itô formula with corresponding boundary conditions allows us to derive the differential equations with boundary conditions for the Laplace transform of the first passage time and the value function. As examples, the explicit solutions of them for several popular diffusions are obtained. In addition, some applications to risk theory are considered.

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 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.

scholarly journals FIRST PASSAGE TIME PROBLEM FOR BIASED CONTINUOUS-TIME RANDOM WALKS

Fractals ◽  
2000 ◽  
Vol 08 (02) ◽  
pp. 139-145 ◽  
Author(s):  
GOVINDAN RANGARAJAN ◽  
MINGZHOU DING
Keyword(s):  
Random Walks ◽  
Density Function ◽  
Continuous Time ◽  
First Passage Time ◽  
Hermite Polynomials ◽  
Passage Time ◽  
First Passage ◽  
Time Problem ◽  
The Laplace Transform ◽  
Continuous Time Random Walks

We study the first passage time (FPT) problem for biased continuous time random walks. Using the recently formulated framework of fractional Fokker-Planck equations, we obtain the Laplace transform of the FPT density function when the bias is constant. When the bias depends linearly on the position, the full FPT density function is derived in terms of Hermite polynomials and generalized Mittag-Leffler functions.

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

1995 ◽  
Vol 32 (04) ◽  
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 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.

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