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Sticky Brownian motion as the limit of storage processes

Published online by Cambridge University Press:  14 July 2016

J. Michael Harrison*
Affiliation:
Stanford University
Austin J. Lemoine*
Affiliation:
Systems Control, Inc.
*
Postal address: Graduate School of Business, Stanford University, Stanford, CA 94305, U.S.A.
∗∗Postal address: Systems Control Inc., 1801 Page Mill Rd., Palo Alto, CA 94304, U.S.A.

Abstract

The paper considers a modified storage process with state space [0,∞). Away from the origin, W behaves like an ordinary storage process with constant release rate and finite jump intensity A. In state 0, however, the jump intensity falls to It is shown that W can be obtained by applying first a reflection mapping and then a random change of time scale to a compound Poisson process with drift. When these same two transformations are applied to Brownian motion, one obtains sticky (or slowly reflected) Brownian motion W∗ on [0,∞). Thus W∗ is the natural diffusion approximation for W, and it is shown that W converges in distribution to W∗ under appropriate conditions. The boundary behavior of W∗ is discussed, its infinitesimal generator is calculated and its stationary distribution (which has an atom at the origin) is computed.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 

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