scholarly journals A Note on First Passage Functionals for Lévy Processes with Jumps of Rational Laplace Transforms

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Djilali Ait-Aoudia
Keyword(s):  
Boundary Value Problem ◽  
Laplace Transform ◽  
Explicit Formula ◽  
First Passage Time ◽  
Boundary Value ◽  
Jump Process ◽  
Passage Time ◽  
Laplace Transforms ◽  
First Passage ◽  
Exit Problem

This paper investigates the two-sided first exit problem for a jump process having jumps with rational Laplace transform. The corresponding boundary value problem is solved to obtain an explicit formula for the first passage functional. Also, we derive the distribution of the first passage time to two-sided barriers and the value at the first passage time.

scholarly journals Exit Problems for Jump Processes Having Double-Sided Jumps with Rational Laplace Transforms

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Yuzhen Wen ◽  
Chuancun Yin
Keyword(s):  
Laplace Transform ◽  
Joint Distribution ◽  
First Passage Time ◽  
Jump Process ◽  
Passage Time ◽  
Jump Processes ◽  
Threshold Strategy ◽  
First Passage ◽  
Exit Problem ◽  
Exit Problems

We consider the two-sided first-exit problem for a jump process having jumps with rational Laplace transform. We derive the joint distribution of the first passage time to two-sided barriers and the value of process at the first passage time. As applications, we present explicit expressions of the dividend formulae for barrier strategy and threshold strategy.

scholarly journals Occupation Times for Markov-Modulated Brownian Motion

2012 ◽  
Vol 49 (02) ◽  
pp. 549-565 ◽  
Author(s):  
Lothar Breuer
Keyword(s):  
Brownian Motion ◽  
First Passage Time ◽  
Passage Time ◽  
Laplace Transforms ◽  
Occupation Times ◽  
First Passage ◽  
Scale Functions ◽  
Positive Variation ◽  
Markov Modulated

In this paper we determine the distributions of occupation times of a Markov-modulated Brownian motion (MMBM) in separate intervals before a first passage time or an exit from an interval. We derive the distributions in terms of their Laplace transforms, and we also distinguish between occupation times in different phases. For MMBMs with strictly positive variation parameters, we further propose scale functions.

scholarly journals Occupation Times for Markov-Modulated Brownian Motion

2012 ◽  
Vol 49 (2) ◽  
pp. 549-565 ◽  
Author(s):  
Lothar Breuer
Keyword(s):  
Brownian Motion ◽  
First Passage Time ◽  
Passage Time ◽  
Laplace Transforms ◽  
Occupation Times ◽  
First Passage ◽  
Scale Functions ◽  
Positive Variation ◽  
Markov Modulated

In this paper we determine the distributions of occupation times of a Markov-modulated Brownian motion (MMBM) in separate intervals before a first passage time or an exit from an interval. We derive the distributions in terms of their Laplace transforms, and we also distinguish between occupation times in different phases. For MMBMs with strictly positive variation parameters, we further propose scale functions.

Evaluating first-passage probabilities for spectrally one-sided Lévy processes

2000 ◽  
Vol 37 (04) ◽  
pp. 1173-1180 ◽  
Author(s):  
L. C. G. Rogers
Keyword(s):  
Time Distribution ◽  
First Passage Time ◽  
Passage Time ◽  
Levy Processes ◽  
Laplace Transforms ◽  
Levy Process ◽  
Basic Algorithm ◽  
First Passage ◽  
Original Algorithm ◽  
Numerical Performance

Fast stable methods for inverting multidimensional Laplace transforms have been developed in recent years by Abate, Whitt and others. We use these methods here to compute numerically the first-passage-time distribution for a spectrally one-sided Lévy process; the basic algorithm is not easy to apply, and we have to develop a variant of it. The numerical performance is as good as the original algorithm.

Evaluating first-passage probabilities for spectrally one-sided Lévy processes

2000 ◽  
Vol 37 (4) ◽  
pp. 1173-1180 ◽  
Author(s):  
L. C. G. Rogers
Keyword(s):  
Time Distribution ◽  
First Passage Time ◽  
Passage Time ◽  
Levy Processes ◽  
Laplace Transforms ◽  
Levy Process ◽  
Basic Algorithm ◽  
First Passage ◽  
Original Algorithm ◽  
Numerical Performance

Fast stable methods for inverting multidimensional Laplace transforms have been developed in recent years by Abate, Whitt and others. We use these methods here to compute numerically the first-passage-time distribution for a spectrally one-sided Lévy process; the basic algorithm is not easy to apply, and we have to develop a variant of it. The numerical performance is as good as the original algorithm.

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.

A note on first-passage time and some related problems

1985 ◽  
Vol 22 (2) ◽  
pp. 346-359 ◽  
Author(s):  
A. G. Nobile ◽  
L. M. Ricciardi ◽  
L. Sacerdote
Keyword(s):  
Laplace Transform ◽  
Diffusion Process ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage

Expansions for the first-passage-time p.d.f. through a constant boundary and for its Laplace transform are derived in terms of probability currents for a temporally homogeneous diffusion process. Ultimate absorption and recurrence problems are also considered. The moments of the first-passage time are finally explicitly obtained.

scholarly journals Statistical Characteristics of the First Passage Time Analysis for the Gene Regulatory Circuit in Bacillus subtilis by Cell Mapping Method

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-10 ◽  
Author(s):  
Liang Wang ◽  
Minjuan Yuan ◽  
Shichao Ma ◽  
Xiaole Yue ◽  
Ying Zhang
Keyword(s):  
First Passage Time ◽  
Vegetative State ◽  
Passage Time ◽  
Mapping Method ◽  
Cell Mapping ◽  
First Passage ◽  
Generalized Cell Mapping ◽  
Exit Problem ◽  
Stochastic Exit Problem ◽  
Stochastic Exit

In this paper, we will explore the stochastic exit problem for the gene regulatory circuit in B. subtilis affected by colored noise. The stochastic exit problem studies the state transition in B. subtilis (from competent state to vegetative state in this case) through three different quantities: the probability density function of the first passage time, the mean of first passage time, and the reliability function. To satisfy the Markov nature, we convert the colored noise system into the equivalent white noise system. Then, the stochastic generalized cell mapping method can be used to explore the stochastic exit problem. The results indicate that the intensity of noise and system parameters have the effect on the transition from competent to vegetative state in B. subtilis. In addition, the effectiveness of the stochastic generalized cell mapping method is verified by Monte Carlo simulation.

A note on first-passage time and some related problems

1985 ◽  
Vol 22 (02) ◽  
pp. 346-359 ◽  
Author(s):  
A. G. Nobile ◽  
L. M. Ricciardi ◽  
L. Sacerdote
Keyword(s):  
Laplace Transform ◽  
Diffusion Process ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage

Expansions for the first-passage-time p.d.f. through a constant boundary and for its Laplace transform are derived in terms of probability currents for a temporally homogeneous diffusion process. Ultimate absorption and recurrence problems are also considered. The moments of the first-passage time are finally explicitly obtained.

scholarly journals On functionals of a marked Poisson process observed by a renewal process

2001 ◽  
Vol 26 (7) ◽  
pp. 427-436 ◽  
Author(s):  
Jewgeni H. Dshalalow ◽  
Jean-Baptiste Bacot
Keyword(s):  
Laplace Transform ◽  
Poisson Process ◽  
Renewal Process ◽  
Stochastic Models ◽  
First Passage Time ◽  
Passage Time ◽  
Time Dependent ◽  
First Passage ◽  
Time Dependent Solution ◽  
Time Dependent Analysis

We study the functionals of a Poisson marked processΠobserved by a renewal process. A sequence of observations continues untilΠcrosses some fixed level at one of the observation epochs (the first passage time). In various stochastic models applications (such as queueing withN-policy combined with multiple vacations), it is necessary to operate with the value ofΠprior to the first passage time, or prior to the first passage time plus some random time. We obtain a time-dependent solution to this problem in a closed form, in terms of its Laplace transform. Many results are directly applicable to the time-dependent analysis of queues and other stochastic models via semi-regenerative techniques.

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