Sticky Brownian motion as the limit of storage processes

1981 ◽  
Vol 18 (1) ◽  
pp. 216-226 ◽  
Author(s):  
J. Michael Harrison ◽  
Austin J. Lemoine
Keyword(s):  
Brownian Motion ◽  
Infinitesimal Generator ◽  
Boundary Behavior ◽  
Compound Poisson Process ◽  
Storage Process ◽  
Reflected Brownian Motion ◽  
Jump Intensity ◽  
Random Change Of Time ◽  
Random Change ◽  
Storage Processes

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.

Sticky Brownian motion as the limit of storage processes

1981 ◽  
Vol 18 (01) ◽  
pp. 216-226 ◽  
Author(s):  
J. Michael Harrison ◽  
Austin J. Lemoine
Keyword(s):  
Brownian Motion ◽  
Infinitesimal Generator ◽  
Boundary Behavior ◽  
Compound Poisson Process ◽  
Storage Process ◽  
Reflected Brownian Motion ◽  
Jump Intensity ◽  
Random Change Of Time ◽  
Random Change ◽  
Storage Processes

The paper considers a modified storage processwith state space [0,∞). Away from the origin,Wbehaves like an ordinary storage process with constant release rate and finite jump intensity A. In state 0, however, the jump intensity falls toIt is shown thatWcan 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 motionW∗on [0,∞). ThusW∗is the natural diffusion approximation forW, and it is shown thatWconverges in distribution toW∗under appropriate conditions. The boundary behavior ofW∗is discussed, its infinitesimal generator is calculated and its stationary distribution (which has an atom at the origin) is computed.

The first rendezvous time of Brownian motion and compound Poisson-type processes

2004 ◽  
Vol 41 (04) ◽  
pp. 1059-1070 ◽  
Author(s):  
D. Perry ◽  
W. Stadje ◽  
S. Zacks
Keyword(s):  
Brownian Motion ◽  
Stochastic Processes ◽  
Poisson Process ◽  
Compound Poisson Process ◽  
Reflected Brownian Motion ◽  
Compound Poisson ◽  
Brownian Motion With Drift ◽  
First Time ◽  
Random Jump ◽  
Poisson Type

The ‘rendezvous time’ of two stochastic processes is the first time at which they cross or hit each other. We consider such times for a Brownian motion with drift, starting at some positive level, and a compound Poisson process or a process with one random jump at some random time. We also ask whether a rendezvous takes place before the Brownian motion hits zero and, if so, at what time. These questions are answered in terms of Laplace transforms for the underlying distributions. The analogous problem for reflected Brownian motion is also studied.

The first rendezvous time of Brownian motion and compound Poisson-type processes

2004 ◽  
Vol 41 (4) ◽  
pp. 1059-1070 ◽  
Author(s):  
D. Perry ◽  
W. Stadje ◽  
S. Zacks
Keyword(s):  
Brownian Motion ◽  
Stochastic Processes ◽  
Poisson Process ◽  
Compound Poisson Process ◽  
Reflected Brownian Motion ◽  
Compound Poisson ◽  
Brownian Motion With Drift ◽  
First Time ◽  
Random Jump ◽  
Poisson Type

The ‘rendezvous time’ of two stochastic processes is the first time at which they cross or hit each other. We consider such times for a Brownian motion with drift, starting at some positive level, and a compound Poisson process or a process with one random jump at some random time. We also ask whether a rendezvous takes place before the Brownian motion hits zero and, if so, at what time. These questions are answered in terms of Laplace transforms for the underlying distributions. The analogous problem for reflected Brownian motion is also studied.

Moderate deviation principle for multivalued stochastic differential equations

2019 ◽  
Vol 20 (03) ◽  
pp. 2050015 ◽  
Author(s):  
Hua Zhang
Keyword(s):  
Brownian Motion ◽  
Weak Convergence ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Moderate Deviation ◽  
Reflected Brownian Motion ◽  
Moderate Deviation Principle ◽  
Deviation Principle ◽  
Weak Convergence Approach

In this paper, we prove a moderate deviation principle for the multivalued stochastic differential equations whose proof are based on recently well-developed weak convergence approach. As an application, we obtain the moderate deviation principle for reflected Brownian motion.

Development of algorithms for controlling and analyzing data of sugar beet storage processes based on the internet of things and artificial intelligence

2021 ◽  
Author(s):  
A.S. Travov ◽  
Keyword(s):  
Artificial Intelligence ◽  
Internet Of Things ◽  
Sugar Beet ◽  
The Internet ◽  
Storage Process ◽  
The Internet Of Things ◽  
Storage Processes

This article provides an overview of the decision to improve the field storage of sugar beet. The purpose of development is to preserve the crop. Methods of monitoring volumes of piles and microclimate inside them are considered. The method for obtaining data on volumes of piles and the further use thereof for optimizing the storage process is described.

On the Distribution of Multidimensional Reflected Brownian Motion

1981 ◽  
Vol 41 (2) ◽  
pp. 345-361 ◽  
Author(s):  
J. Michael Harrison ◽  
Martin I. Reiman
Keyword(s):  
Brownian Motion ◽  
Reflected Brownian Motion

scholarly journals The Randomized American Option as a Classical Solution to the Penalized Problem

2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
Guillaume Leduc
Keyword(s):  
Cauchy Problem ◽  
Brownian Motion ◽  
Classical Solution ◽  
Penalty Method ◽  
Infinitesimal Generator ◽  
American Option ◽  
Geometric Brownian Motion ◽  
Uniform Bound ◽  
Option Value ◽  
The Cauchy Problem

We connect the exercisability randomized American option to the penalty method by showing that the randomized American option valueuis the uniqueclassicalsolution to the Cauchy problem corresponding to thecanonicalpenalty problem for American options. We also establish a uniform bound forAu, whereAis the infinitesimal generator of a geometric Brownian motion.

Asymptotic variance parameters for the boundary local times of reflected Brownian motion on a compact interval

1992 ◽  
Vol 29 (04) ◽  
pp. 996-1002 ◽  
Author(s):  
R. J. Williams
Keyword(s):  
Brownian Motion ◽  
Asymptotic Variance ◽  
Hitting Time ◽  
Local Times ◽  
Compact Interval ◽  
Reflected Brownian Motion ◽  
Direct Derivation ◽  
One Dimensional ◽  
Brownian Motion With Drift ◽  
Variance Parameters

A direct derivation is given of a formula for the normalized asymptotic variance parameters of the boundary local times of reflected Brownian motion (with drift) on a compact interval. This formula was previously obtained by Berger and Whitt using an M/M/1/C queue approximation to the reflected Brownian motion. The bivariate Laplace transform of the hitting time of a level and the boundary local time up to that hitting time, for a one-dimensional reflected Brownian motion with drift, is obtained as part of the derivation.

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