Levy Processes with TwoSided Reflection
(2015) In Lecture Notes in Mathematics 2149. p.67182 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 twobarrier first passage problem. A key quantity in applications is the loss rate l(b) at b, defined as Epi 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 twobarrier first passage problem. A key quantity in applications is the loss rate l(b) at b, defined as Epi U(1). Various forms of l(b) and various derivations are presented, and the asymptotics as b > infinity is exhibited in both the lighttailed and the heavytailed 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 phasetype jumps. (Less)
Please use this url to cite or link to this publication:
http://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, Integrodifferential equation, Ito's formula, Linear equations, Local, time, Loss rate, Martingale, Overflow, Phasetype 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
 00758434
 DOI
 10.1007/9783319231389_2
 language
 English
 LU publication?
 yes
 id
 455309c6b25944089c72dc277599a6d2 (old id 8739094)
 date added to LUP
 20160301 07:16:54
 date last changed
 20180318 04:22:17
@article{455309c6b25944089c72dc277599a6d2, 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 twobarrier first passage problem. A key quantity in applications is the loss rate l(b) at b, defined as Epi U(1). Various forms of l(b) and various derivations are presented, and the asymptotics as b > infinity is exhibited in both the lighttailed and the heavytailed 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 phasetype jumps.}, author = {Andersen, Lars Norvang and Asmussen, Soren and Glynn, Peter W. and Pihlsgård, Mats}, issn = {00758434}, keyword = {Applied probability,Central limit theorem,Finite buffer problem,First,passage problem,Functional limit theorem,Heavy tails,Integrodifferential equation,Ito's formula,Linear equations,Local,time,Loss rate,Martingale,Overflow,Phasetype distribution,Poisson's equation,Queueing theory,Siegmund duality,Skorokhod,problem,Storage process}, language = {eng}, pages = {67182}, publisher = {Springer}, series = {Lecture Notes in Mathematics}, title = {Levy Processes with TwoSided Reflection}, url = {http://dx.doi.org/10.1007/9783319231389_2}, volume = {2149}, year = {2015}, }