Brownian Motion on a Symmetric Space of Non-Compact Type: Asymptotic Behaviour in Polar Coordinates

1991 ◽  
Vol 43 (5) ◽  
pp. 1065-1085 ◽  
Author(s):  
J. C. Taylor
Keyword(s):  
Brownian Motion ◽  
Symmetric Space ◽  
Asymptotic Behaviour ◽  
Polar Coordinates ◽  
Skew Product ◽  
Compact Type ◽  
Product Representation

AbstractThe results of Orihara [10] and Malliavin2 [7] on the asymptotic behaviour in polar coordinates of Brownian motion on a symmetric space of non-compact type are obtained by means of a skew product representation on K/M x A+of the Brownian motion on the set of regular points of X. Results of Norris, Rogers, and Williams [9] are interpreted in this context.

scholarly journals The fixed point set of a holomorphic isometry, the intersection of two real forms in a Hermitian symmetric space of compact type and symmetric triads

2015 ◽  
Vol 26 (06) ◽  
pp. 1541005 ◽  
Author(s):  
Osamu Ikawa ◽  
Makiko Sumi Tanaka ◽  
Hiroyuki Tasaki
Keyword(s):  
Fixed Point ◽  
Symmetric Space ◽  
Hermitian Symmetric Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Type ◽  
Fixed Point Set ◽  
Point Set ◽  
Necessary And Sufficient ◽  
Real Forms

We show a necessary and sufficient condition that the fixed point set of a holomorphic isometry and the intersection of two real forms of a Hermitian symmetric space of compact type are discrete and prove that they are antipodal sets in the cases. We also consider some relations between the intersection of two real forms and the fixed point set of a certain holomorphic isometry.

scholarly journals Windings of planar processes, exponential functionals and Asian options

2018 ◽  
Vol 50 (3) ◽  
pp. 726-742 ◽  
Author(s):  
Wissem Jedidi ◽  
Stavros Vakeroudis
Keyword(s):  
Brownian Motion ◽  
Exit Time ◽  
Random Variable ◽  
Infinite Divisibility ◽  
Mathematical Finance ◽  
Asian Options ◽  
Product Representation ◽  
Exponential Functional ◽  
Planar Brownian Motion ◽  
Divisibility Properties

Abstract Motivated by a common mathematical finance topic, we discuss the reciprocal of the exit time from a cone of planar Brownian motion which also corresponds to the exponential functional of Brownian motion in the framework of planar Brownian motion. We prove a conjecture of Vakeroudis and Yor (2012) concerning infinite divisibility properties of this random variable and present a novel simple proof of the result of DeBlassie (1987), (1988) concerning the asymptotic behavior of the distribution of the Bessel clock appearing in the skew-product representation of planar Brownian motion, as t→∞. We use the results of the windings approach in order to obtain results for quantities associated to the pricing of Asian options.

ASYMPTOTIC BEHAVIOUR OF A CLASS OF RESOURCE COMPETITION BIOLOGY SPECIES SYSTEM BY THE FRACTIONAL BROWNIAN MOTION

2017 ◽  
Vol 58 (3-4) ◽  
pp. 491-499
Author(s):  
Q. ZHANG ◽  
M. YE ◽  
H. LEI ◽  
Q. JIN
Keyword(s):  
Brownian Motion ◽  
Asymptotic Behaviour ◽  
Fractional Brownian Motion ◽  
Lyapunov Functional ◽  
Resource Competition ◽  
Mortality Rates ◽  
Mean Square ◽  
Chemostat Model ◽  
Extrinsic Noise ◽  
The Mean

We analyse the asymptotic behaviour of a biological system described by a stochastic competition model with $n$ species and $k$ resources (chemostat model), in which the species mortality rates are influenced by the fractional Brownian motion of the extrinsic noise environment. By constructing a Lyapunov functional, the persistence and extinction criteria are derived in the mean square sense. Some examples are given to illustrate the effectiveness of the theoretical result.

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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