Brownian Motion and Stochastic Flow Systems (J. Michael Harrison)

SIAM Review ◽  
1986 ◽  
Vol 28 (3) ◽  
pp. 424-426
Author(s):  
Frank B. Knight
Keyword(s):  
Brownian Motion ◽  
Stochastic Flow ◽  
Flow Systems

Transient behavior of regulated Brownian motion, I: Starting at the origin

1987 ◽  
Vol 19 (03) ◽  
pp. 560-598 ◽  
Author(s):  
Joseph Abate ◽  
Ward Whitt
Keyword(s):  
Brownian Motion ◽  
Steady State ◽  
First Passage Time ◽  
Cumulative Distribution ◽  
Stochastic Flow ◽  
Completely Monotone ◽  
Constant Diffusion ◽  
Flow Systems ◽  
The Moment ◽  
First Moment

A natural model for stochastic flow systems is regulated or reflecting Brownian motion (RBM), which is Brownian motion on the positive real line with constant negative drift and constant diffusion coefficient, modified by an impenetrable reflecting barrier at the origin. As a basis for understanding how stochastic flow systems approach steady state, this paper provides relatively simple descriptions of the moments of RBM as functions of time. In Part I attention is restricted to the case in which RBM starts at the origin; then the moment functions are increasing. After normalization by the steady-state limits, these moment c.d.f.&s (cumulative distribution functions) coincide with gamma mixtures of inverse Gaussian c.d.f.&s. The first moment c.d.f. thus coincides with the first-passage time to the origin starting in steady state with the exponential stationary distribution. From this probabilistic characterization, it follows that thekth-moment c.d.f is thek-fold convolution of the first-moment c.d.f. As a consequence, it is easy to see that the (k +1)th moment approaches its steady-state limit more slowly than thekthmoment. It is also easy to derive the asymptotic behavior ast→∞. The first two moment c.d.f.&s have completely monotone densities, supporting approximation by hyperexponential (H2)c.d.f.&s (mixtures of two exponentials). TheH2approximations provide easily comprehensible descriptions of the first two moment c.d.f.&s suitable for practical purposes. The two exponential components of theH2approximation yield simple exponential approximations in different regimes. On the other hand, numerical comparisons show that the limit related to the relaxation time does not predict the approach to steady state especially well in regions of primary interest. In Part II (Abate and Whitt (1987a)), moments of RBM with non-zero initial conditions are treated by representing them as the difference of two increasing functions, one of which is the moment function starting at the origin studied here.

Brownian Motion and Stochastic Flow Systems.

1987 ◽  
Vol 36 (1) ◽  
pp. 66
Author(s):  
J. F. C. Kingman ◽  
J. Michael Harrison
Keyword(s):  
Brownian Motion ◽  
Stochastic Flow ◽  
Flow Systems

Transient behavior of regulated Brownian motion, I: Starting at the origin

1987 ◽  
Vol 19 (3) ◽  
pp. 560-598 ◽  
Author(s):  
Joseph Abate ◽  
Ward Whitt
Keyword(s):  
Brownian Motion ◽  
Steady State ◽  
First Passage Time ◽  
Cumulative Distribution ◽  
Stochastic Flow ◽  
Completely Monotone ◽  
Constant Diffusion ◽  
Flow Systems ◽  
The Moment ◽  
First Moment

A natural model for stochastic flow systems is regulated or reflecting Brownian motion (RBM), which is Brownian motion on the positive real line with constant negative drift and constant diffusion coefficient, modified by an impenetrable reflecting barrier at the origin. As a basis for understanding how stochastic flow systems approach steady state, this paper provides relatively simple descriptions of the moments of RBM as functions of time. In Part I attention is restricted to the case in which RBM starts at the origin; then the moment functions are increasing. After normalization by the steady-state limits, these moment c.d.f.&s (cumulative distribution functions) coincide with gamma mixtures of inverse Gaussian c.d.f.&s. The first moment c.d.f. thus coincides with the first-passage time to the origin starting in steady state with the exponential stationary distribution. From this probabilistic characterization, it follows that the kth-moment c.d.f is the k-fold convolution of the first-moment c.d.f. As a consequence, it is easy to see that the (k + 1)th moment approaches its steady-state limit more slowly than the kth moment. It is also easy to derive the asymptotic behavior as t →∞. The first two moment c.d.f.&s have completely monotone densities, supporting approximation by hyperexponential (H2) c.d.f.&s (mixtures of two exponentials). The H2 approximations provide easily comprehensible descriptions of the first two moment c.d.f.&s suitable for practical purposes. The two exponential components of the H2 approximation yield simple exponential approximations in different regimes. On the other hand, numerical comparisons show that the limit related to the relaxation time does not predict the approach to steady state especially well in regions of primary interest. In Part II (Abate and Whitt (1987a)), moments of RBM with non-zero initial conditions are treated by representing them as the difference of two increasing functions, one of which is the moment function starting at the origin studied here.

scholarly journals Regularity properties of the stochastic flow of a skew fractional Brownian motion

2020 ◽  
Vol 23 (01) ◽  
pp. 2050005
Author(s):  
Oussama Amine ◽  
David R. Baños ◽  
Frank Proske
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Solution Process ◽  
Variational Calculus ◽  
Stochastic Flow ◽  
Intrinsic Properties ◽  
Regularity Properties ◽  
Fractional Brownian Noise ◽  
Unknown Solution ◽  
Variation Part

In this paper we prove, for small Hurst parameters, the higher-order differentiability of a stochastic flow associated with a stochastic differential equation driven by an additive multi-dimensional fractional Brownian noise, where the bounded variation part is given by the local time of the unknown solution process. The proof of this result relies on Fourier analysis-based variational calculus techniques and on intrinsic properties of the fractional Brownian motion.

An isometric embedding of the g(t)-Brownian motion with application in stability and homotopy group

2019 ◽  
Vol 19 (06) ◽  
pp. 1950045
Author(s):  
Claudia Luque Justo ◽  
Diego Sebastian Ledesma ◽  
Fabiano Borges da Silva
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
Laplace Operator ◽  
Homotopy Group ◽  
Isometric Embedding ◽  
Time Dependent ◽  
Homotopy Groups ◽  
Stochastic Flow ◽  
Tensor Fields ◽  
The Laplace Operator

In this work, we construct a [Formula: see text]-Brownian motion via an isometric embedding, whose approach permit to define the Laplace operator associated with parametrized metric [Formula: see text], for every [Formula: see text]. We present an Itô formula for stochastic flow acting on time-dependent tensor fields, in particular to the metric [Formula: see text] and consequently to the norm of a stochastic process in [Formula: see text]. We use this approach to study stability by [Formula: see text]th moment exponent of [Formula: see text]-Brownian motion and its applications on homotopy groups.

An asymptotically exact decomposition of coupled Brownian systems

1993 ◽  
Vol 30 (4) ◽  
pp. 819-834
Author(s):  
Kerry W. Fendick
Keyword(s):  
Brownian Motion ◽  
Asymptotic Behaviour ◽  
Queue Length ◽  
Inventory Systems ◽  
Input Process ◽  
Simple Description ◽  
Flow Systems ◽  
Component Processes ◽  
Brownian Models ◽  
Brownian Flow Systems

Brownian flow systems, i.e. multidimensional Brownian motion with regulating barriers, can model queueing and inventory systems in which the behavior of different queues is correlated because of shared input processes. The behavior of such systems is typically difficult to describe exactly. We show how Brownian models of such systems, conditioned on one queue length exceeding a large value, decompose asymptotically into smaller subsystems. This conditioning induces a change in drift of the system's net input process and its components. The results here are analogous to results for jump-Markov queues recently obtained by Shwartz and Weiss. The Brownian setting leads to a simple description of the component processes' asymptotic behaviour, as well as to explicit distributional results.

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