Levy Processes with Two-Sided Reflection
(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)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/8739094
- author
- Andersen, Lars Norvang ; Asmussen, Soren ; Glynn, Peter W. and Pihlsgård, Mats LU
- organization
- publishing date
- 2015
- type
- Contribution to journal
- publication status
- 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
- 2016-04-01 14:46:05
- date last changed
- 2022-04-14 19:40:08
@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}},
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}},
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}},
doi = {{10.1007/978-3-319-23138-9_2}},
volume = {{2149}},
year = {{2015}},
}