scholarly journals The First Passage Time Problem for Mixed-Exponential Jump Processes with Applications in Insurance and Finance

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Chuancun Yin ◽  
Yuzhen Wen ◽  
Zhaojun Zong ◽  
Ying Shen
Keyword(s):  
Risk Model ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Laplace Transforms ◽  
First Passage Times ◽  
Jump Processes ◽  
Closed Form Expression ◽  
First Passage ◽  
Time Problem ◽  
Passage Times

This paper studies the first passage times to constant boundaries for mixed-exponential jump diffusion processes. Explicit solutions of the Laplace transforms of the distribution of the first passage times, the joint distribution of the first passage times and undershoot (overshoot) are obtained. As applications, we present explicit expression of the Gerber-Shiu functions for surplus processes with two-sided jumps, present the analytical solutions for popular path-dependent options such as lookback and barrier options in terms of Laplace transforms, and give a closed-form expression on the price of the zero-coupon bond under a structural credit risk model with jumps.

scholarly journals Explicit asymptotics on first passage times of diffusion processes

2020 ◽  
Vol 52 (2) ◽  
pp. 681-704
Author(s):  
Angelos Dassios ◽  
Luting Li
Keyword(s):  
Diffusion Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Perturbation Series ◽  
First Passage Times ◽  
Unified Framework ◽  
First Passage ◽  
Single Side ◽  
Probability Densities ◽  
Passage Times

AbstractWe introduce a unified framework for solving first passage times of time-homogeneous diffusion processes. Using potential theory and perturbation theory, we are able to deduce closed-form truncated probability densities, as asymptotics or approximations to the original first passage time densities, for single-side level crossing problems. The framework is applicable to diffusion processes with continuous drift functions; in particular, for bounded drift functions, we show that the perturbation series converges. In the present paper, we demonstrate examples of applying our framework to the Ornstein–Uhlenbeck, Bessel, exponential-Shiryaev, and hypergeometric diffusion processes (the latter two being previously studied by Dassios and Li (2018) and Borodin (2009), respectively). The purpose of this paper is to provide a fast and accurate approach to estimating first passage time densities of various diffusion 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.

On first passage times of sticky reflecting diffusion processes with double exponential jumps

2020 ◽  
Vol 57 (1) ◽  
pp. 221-236 ◽  
Author(s):  
Shiyu Song ◽  
Yongjin Wang
Keyword(s):  
Diffusion Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Explicit Solutions ◽  
First Passage ◽  
Uhlenbeck Process ◽  
Double Exponential ◽  
Upper Level ◽  
First Passage Problem ◽  
Passage Times

AbstractWe explore the first passage problem for sticky reflecting diffusion processes with double exponential jumps. The joint Laplace transform of the first passage time to an upper level and the corresponding overshoot is studied. In particular, explicit solutions are presented when the diffusion part is driven by a drifted Brownian motion and by an Ornstein–Uhlenbeck process.

First-passage times of two-dimensional Brownian motion

2016 ◽  
Vol 48 (4) ◽  
pp. 1045-1060 ◽  
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.

First-passage times with PFr densities

1985 ◽  
Vol 22 (1) ◽  
pp. 185-196 ◽  
Author(s):  
David Assaf ◽  
Moshe Shared ◽  
J. George shanthikumar
Keyword(s):  
First Passage Time ◽  
Compound Poisson Process ◽  
Passage Time ◽  
First Passage Times ◽  
Embedded Markov Chain ◽  
First Passage ◽  
Completely Monotone ◽  
Totally Positive ◽  
Finite State ◽  
Passage Times

It is shown that if a finite-state continuous-time Markov process can be uniformized such that the embedded Markov chain has a TPr (totally positive of order r) transition matrix, then the first-passage time from state 0 to any other state has a PFr (Polya frequency of order r) density. As a consequence, results of Keilson (1971), Esary, Marshall and Proschan (1973), Ghosh and Ebrahimi (1982) and Derman, Ross and Schechner (1983) are strengthened. It is also shown that some cumulative damage shock models, with an underlying compound Poisson process and ‘damages' which are not necessarily non-negative, are associated with wear processes having PFr first-passage times to any threshold. First-passage times with completely monotone densities are also discussed.

A note on first passage times in birth and death and nonnegative diffusion processes

1983 ◽  
Vol 30 (2) ◽  
pp. 283-285 ◽  
Author(s):  
Cyrus Derman ◽  
Sheldon M. Ross ◽  
Zvi Schechner
Keyword(s):  
Diffusion Processes ◽  
First Passage Times ◽  
First Passage ◽  
Passage Times

scholarly journals The Protein Hourglass: Exact First Passage Time Distributions for Protein Thresholds

2020 ◽  
Author(s):  
Krishna Rijal ◽  
Ashok Prasad ◽  
Dibyendu Das
Keyword(s):  
First Passage Time ◽  
Self Regulation ◽  
Threshold Value ◽  
Passage Time ◽  
First Passage Times ◽  
Lambda Phage ◽  
First Passage ◽  
Analytical Expression ◽  
Protein Levels ◽  
Passage Times

Protein thresholds have been shown to act as an ancient timekeeping device, such as in the time to lysis of E. coli infected with bacteriophage lambda. The time taken for protein levels to reach a particular threshold for the first time is defined as the first passage time of the protein synthesis system, which is a stochastic quantity. The first few moments of the distribution of first passage times were known earlier, but an analytical expression for the full distribution was not available. In this work, we derive an analytical expression for the first passage times for a long-lived protein. This expression allows us to calculate the full distribution not only for cases of no self-regulation, but also for both positive and negative self-regulation of the threshold protein. We show that the shape of the distribution matches previous experimental data on lambda-phage lysis time distributions. We also provide analytical expressions for the FPT distribution with non-zero degradation in Laplace space. Furthermore, we study the noise in the precision of the first passage times described by coefficient of variation (CV) of the distribution as a function of the protein threshold value. We show that under conditions of positive self-regulation, the CV declines monotonically with increasing protein threshold, while under conditions of linear negative self-regulation, there is an optimal protein threshold that minimizes the noise in the first passage times.

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