Levy Processes with Two-Sided Reflection

• 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.

Ämnesord

NATURVETENSKAP  -- Matematik -- Sannolikhetsteori och statistik (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Probability Theory and Statistics (hsv//eng)

Nyckelord

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

Publikations- och innehållstyp

art (ämneskategori)

ref (ämneskategori)