Recurrence-transience criteria for storage processes
type: Item
instance of: scholarly article
title: Recurrence-transience criteria for storage processes
zbMATH Open document ID: 0970.60088
DOI: 10.1007/BF02674096
published in: Journal of Mathematical Sciences (New York)
publication date: 2001-10-16T00:00:00Z
review text: The paper considers storage processes governed by stochastic integral equations of the form \(X(t)=x- \int^t_0 r(X(s))ds+A(t)\), where \(r(x)\) is a function defined on \([0,\infty)\) with values in \([0,\infty)\), \(r(0)=0\), \(r(x)>0\) for \(x>0\) and left continuous with positive right limits on \((0,\infty)\), and \(A(t)\) is a subordinator with \[ E(e^{-\theta A(t)})= \exp \left\{\int^\infty_0 t(e^{-\theta y}-1) \nu(dy) \right\}. \] Storage processes are classified into three types: positive recurrent, null recurrent, and transient processes. The main results of the paper are: Theorem 3.2. If for some \(c>0\) \[ \int^\infty_c {\nu[x,\infty) \over r(x)}dx=\infty \quad\text{and} \quad\int^\infty_c {1\over r(x)}\exp \left\{-\int^x_c {\nu[y,\infty)\over r(y)} dy\right\} dx<\infty, \] then \(X(t)\) is transient. Theorem 3.3. Assume that \(\inf_{x\geq 1}r(x)>0\). If there is \(c>0\) such that \(\int^x_c{\nu(x-y,\infty) \over r(y)} dy\leq 1\) for all \(x\geq c\), then \(X(t)\) is recurrent.
Mathematics Subject Classification ID: 60J25
Mathematics Subject Classification ID: 60K30
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 60H20
zbMATH DE Number: 1529334
zbMATH Keywords: recurrence-transience criteria
zbMATH Keywords: storage processes
zbMATH Keywords: stochastic integral equations
author: Makoto Yamazato
reviewed by: Oleg K. Zakusilo
MaRDI profile type: Publication
cites work: Storage processes with general release rule and additive inputs
cites work: A local time for a storage process
cites work: The Stationary Distribution and First Exit Probabilities of a Storage Process with General Release Rule
cites work: The Recurrence Classification of Risk and Storage Processes
cites work: Q4698320
cites work: Operator-selfdecomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type
cites work: A recurrence criterion for Markov processes of Ornstein-Uhlenbeck type
cites work: Sato's conjecture on recurrence conditions for multidimensional processes of Ornstein-Uhlenbeck type
cites work: General Storage Processes with Additive Inputs
cites work: Q4350399
Recommended article: Recurrence-transience criteria for storage processes and their applications
Recommended article: Q4578301
Recommended article: Storage impulsive processes on increasing time intervals
Recommended article: Criteria for recurrence and transience of semistable processes
Recommended article: Q4667292
Recommended article: On moments and tail behaviors of storage processes
Recommended article: Q3077827
Recommended article: A discrete-time storage process with a general release rule
Recommended article: Q5488442
description: scientific article; zbMATH DE number 1529334