scholarly journals Censored stable subordinators and fractional derivatives

2021 ◽  
Vol 24 (4) ◽  
pp. 1035-1068
Author(s):  
Qiang Du ◽  
Lorenzo Toniazzi ◽  
Zirui Xu
Keyword(s):  
Fractional Derivative ◽  
First Passage Time ◽  
Stable Process ◽  
Series Representation ◽  
Passage Time ◽  
Relaxation Equation ◽  
Algebraic Decay ◽  
Relaxation Solution ◽  
Completely Monotone ◽  
The Laplace Transform

Abstract Based on the popular Caputo fractional derivative of order β in (0, 1), we define the censored fractional derivative on the positive half-line ℝ+. This derivative proves to be the Feller generator of the censored (or resurrected) decreasing β-stable process in ℝ+. We provide a series representation for the inverse of this censored fractional derivative. We are then able to prove that this censored process hits the boundary in a finite time τ ∞, whose expectation is proportional to that of the first passage time of the β-stable subordinator. We also show that the censored relaxation equation is solved by the Laplace transform of τ ∞. This relaxation solution proves to be a completely monotone series, with algebraic decay one order faster than its Caputo counterpart, leading, surprisingly, to a new regime of fractional relaxation models. Lastly, we discuss how this work identifies a new sub-diffusion model.

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.

Transient behavior of the M/M/1 queue via Laplace transforms

1988 ◽  
Vol 20 (1) ◽  
pp. 145-178 ◽  
Author(s):  
Joseph Abate ◽  
Ward Whitt
Keyword(s):  
First Passage Time ◽  
Operational Calculus ◽  
Transient Behavior ◽  
Building Blocks ◽  
Passage Time ◽  
Reflected Brownian Motion ◽  
Dependent Behavior ◽  
The Laplace Transform ◽  
First Moment ◽  
Special Case

This paper shows how the Laplace transform analysis of Bailey (1954), (1957) can be continued to yield additional insights about the time-dependent behavior of the queue-length process in the M/M/1 model. A transform factorization is established that leads to a decomposition of the first moment as a function of time into two monotone components. This factorization facilitates developing approximations for the moments and determining their asymptotic behavior as . All descriptions of the transient behavior are expressed in terms of basic building blocks such as the first-passage-time distributions. The analysis is facilitated by appropriate scaling of space and time so that regulated or reflected Brownian motion (RBM) appears as the special case in which the traffic intensity ρ equals the critical value 1. An operational calculus is developed for obtaining M/M/1 results directly from corresponding RBM results as well as vice versa. The analysis thus provides useful insight about RBM approximations for queues.

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.

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.

Spectral structure of the first-passage-time densities for classes of Markov chains

1987 ◽  
Vol 24 (03) ◽  
pp. 631-643 ◽  
Author(s):  
Masaaki Kijima
Keyword(s):  
Markov Chains ◽  
First Passage Time ◽  
Passage Time ◽  
Death Process ◽  
Spectral Structure ◽  
First Passage ◽  
Completely Monotone ◽  
Spectral Representations ◽  
Absorbing States ◽  
Birth Death

Keilson [7] showed that for a birth-death process defined on non-negative integers with reflecting barrier at 0 the first-passage-time density from 0 to N (N to N + 1) has Pólya frequency of order infinity (is completely monotone). Brown and Chaganty [3] and Assaf et al. [1] studied the first-passage-time distribution for classes of discrete-time Markov chains and then produced the essentially same results as these through a uniformization. This paper addresses itself to an extension of Keilson's results to classes of Markov chains such as time-reversible Markov chains, skip-free Markov chains and birth-death processes with absorbing states. The extensions are due to the spectral representations of the infinitesimal generators governing these Markov chains. Explicit densities for those first-passage times are also given.

Spectral structure of the first-passage-time densities for classes of Markov chains

1987 ◽  
Vol 24 (3) ◽  
pp. 631-643 ◽  
Author(s):  
Masaaki Kijima
Keyword(s):  
Markov Chains ◽  
First Passage Time ◽  
Passage Time ◽  
Death Process ◽  
Spectral Structure ◽  
First Passage ◽  
Completely Monotone ◽  
Spectral Representations ◽  
Absorbing States ◽  
Birth Death

Keilson [7] showed that for a birth-death process defined on non-negative integers with reflecting barrier at 0 the first-passage-time density from 0 to N (N to N + 1) has Pólya frequency of order infinity (is completely monotone). Brown and Chaganty [3] and Assaf et al. [1] studied the first-passage-time distribution for classes of discrete-time Markov chains and then produced the essentially same results as these through a uniformization. This paper addresses itself to an extension of Keilson's results to classes of Markov chains such as time-reversible Markov chains, skip-free Markov chains and birth-death processes with absorbing states. The extensions are due to the spectral representations of the infinitesimal generators governing these Markov chains. Explicit densities for those first-passage times are also given.

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.

Sign in / Sign up

Export Citation Format

Share Document