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Levy Processes with Two-Sided Reflection

Andersen, Lars Norvang; Asmussen, Soren; Glynn, Peter W. and Pihlsgård, Mats LU (2015) In Lecture Notes in Mathematics 2149. p.67-182
Abstract
Let X be a Levy process and V the reflection at boundaries 0 and b > 0. A number of properties of V are studied, with particular emphasis on the behaviour at the upper boundary b. The process V can be represented as solution of a Skorokhod problem V(t) = V(0)+X(t)+L(t)-U(t) where L, U are the local times (regulators) at the lower and upper barrier. Explicit forms of V in terms of X are surveyed as well as more pragmatic approaches to the construction of V, and the stationary distribution pi is characterised in terms of a two-barrier first passage problem. A key quantity in applications is the loss rate l(b) at b, defined as E-pi U(1). Various forms of l(b) and various derivations are presented, and the asymptotics as b -> infinity is... (More)
Let X be a Levy process and V the reflection at boundaries 0 and b > 0. A number of properties of V are studied, with particular emphasis on the behaviour at the upper boundary b. The process V can be represented as solution of a Skorokhod problem V(t) = V(0)+X(t)+L(t)-U(t) where L, U are the local times (regulators) at the lower and upper barrier. Explicit forms of V in terms of X are surveyed as well as more pragmatic approaches to the construction of V, and the stationary distribution pi is characterised in terms of a two-barrier first passage problem. A key quantity in applications is the loss rate l(b) at b, defined as E-pi U(1). Various forms of l(b) and various derivations are presented, and the asymptotics as b -> infinity is exhibited in both the light-tailed and the heavy-tailed regime. The drift zero case EX(1) = 0 plays a particular role, with Brownian or stable functional limits being a key tool. Further topics include studies of the first hitting time of b, central limit theorems and large deviations results for U, and a number of explicit calculations for Levy processes where the jump part is compound Poisson with phase-type jumps. (Less)
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Contribution to journal
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published
subject
keywords
Applied probability, Central limit theorem, Finite buffer problem, First, passage problem, Functional limit theorem, Heavy tails, Integro-differential equation, Ito's formula, Linear equations, Local, time, Loss rate, Martingale, Overflow, Phase-type distribution, Poisson's equation, Queueing theory, Siegmund duality, Skorokhod, problem, Storage process
in
Lecture Notes in Mathematics
volume
2149
pages
67 - 182
publisher
Springer
external identifiers
  • wos:000368687700005
  • scopus:84946101024
ISSN
0075-8434
DOI
10.1007/978-3-319-23138-9_2
language
English
LU publication?
yes
id
455309c6-b259-4408-9c72-dc277599a6d2 (old id 8739094)
date added to LUP
2016-03-01 07:16:54
date last changed
2017-03-19 03:57:35
@article{455309c6-b259-4408-9c72-dc277599a6d2,
  abstract     = {Let X be a Levy process and V the reflection at boundaries 0 and b > 0. A number of properties of V are studied, with particular emphasis on the behaviour at the upper boundary b. The process V can be represented as solution of a Skorokhod problem V(t) = V(0)+X(t)+L(t)-U(t) where L, U are the local times (regulators) at the lower and upper barrier. Explicit forms of V in terms of X are surveyed as well as more pragmatic approaches to the construction of V, and the stationary distribution pi is characterised in terms of a two-barrier first passage problem. A key quantity in applications is the loss rate l(b) at b, defined as E-pi U(1). Various forms of l(b) and various derivations are presented, and the asymptotics as b -> infinity is exhibited in both the light-tailed and the heavy-tailed regime. The drift zero case EX(1) = 0 plays a particular role, with Brownian or stable functional limits being a key tool. Further topics include studies of the first hitting time of b, central limit theorems and large deviations results for U, and a number of explicit calculations for Levy processes where the jump part is compound Poisson with phase-type jumps.},
  author       = {Andersen, Lars Norvang and Asmussen, Soren and Glynn, Peter W. and Pihlsgård, Mats},
  issn         = {0075-8434},
  keyword      = {Applied probability,Central limit theorem,Finite buffer problem,First,passage problem,Functional limit theorem,Heavy tails,Integro-differential equation,Ito's formula,Linear equations,Local,time,Loss rate,Martingale,Overflow,Phase-type distribution,Poisson's equation,Queueing theory,Siegmund duality,Skorokhod,problem,Storage process},
  language     = {eng},
  pages        = {67--182},
  publisher    = {Springer},
  series       = {Lecture Notes in Mathematics},
  title        = {Levy Processes with Two-Sided Reflection},
  url          = {http://dx.doi.org/10.1007/978-3-319-23138-9_2},
  volume       = {2149},
  year         = {2015},
}