Investigation of Laplace transforms for Erlangen distribution of the first passage of zero level of the semi-Markov random process with positive tendency and negative jump

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.

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First-passage times of two-dimensional Brownian motion

Advances in Applied Probability ◽

10.1017/apr.2016.64 ◽

2016 ◽

Vol 48 (4) ◽

pp. 1045-1060 ◽

Cited By ~ 7

Author(s):

Steven Kou ◽

Haowen Zhong

Keyword(s):

Brownian Motion ◽

Analytical Solutions ◽

Laplace Transforms ◽

First Passage Times ◽

Absolute Difference ◽

Two Dimensional ◽

First Passage ◽

Quantitative Finance ◽

The 1960S ◽

Passage Times

AbstractFirst-passage times (FPTs) of two-dimensional Brownian motion have many applications in quantitative finance. However, despite various attempts since the 1960s, there are few analytical solutions available. By solving a nonhomogeneous modified Helmholtz equation in an infinite wedge, we find analytical solutions for the Laplace transforms of FPTs; these Laplace transforms can be inverted numerically. The FPT problems lead to a class of bivariate exponential distributions which are absolute continuous but do not have the memoryless property. We also prove that the density of the absolute difference of FPTs tends to ∞ if and only if the correlation between the two Brownian motions is positive.

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

Abstract and Applied Analysis ◽

10.1155/2016/5914657 ◽

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.

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A Simulation-Based Random Process Method for Time-Dependent Reliability of Dynamic Systems

Volume 1: 36th Design Automation Conference, Parts A and B ◽

10.1115/detc2010-28131 ◽

2010 ◽

Cited By ~ 3

Author(s):

Amandeep Singh ◽

Zissimos P. Mourelatos ◽

Efstratios Nikolaidis

Keyword(s):

Random Process ◽

Operating Conditions ◽

Time Dependent ◽

Process Time ◽

Cumulative Probability ◽

Repairable Systems ◽

First Passage ◽

True Value ◽

Warranty Costs ◽

Short Time

Reliability is an important engineering requirement for consistently delivering acceptable product performance through time. As time progresses, a product may fail due to time-dependent operating conditions and material properties, and component degradation. The reliability degradation with time may significantly increase the lifecycle cost due to potential warranty costs, repairs and loss of market share. In this work, we consider the first-passage reliability, which accounts for the first time failure of non-repairable systems. Methods are available that provide an upper bound to the true reliability, but they may overestimate the true value considerably. This paper proposes a methodology to calculate the cumulative probability of failure (probability of first passage or upcrossing) of a dynamic system with random properties, driven by an ergodic input random process. Time series modeling is used to characterize the input random process based on data from a “short” time period (e.g. seconds) from only one sample function of the random process. Sample functions of the output random process are calculated for the same “short” time because it is usually impractical to perform the calculation for a “long” duration (e.g. hours). The proposed methodology calculates the time-dependent reliability, at a “long” time using an accurate “extrapolation” procedure of the failure rate. A representative example of a quarter car model subjected to a stochastic road excitation demonstrates the improved accuracy of the proposed method compared with available methods.

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