Among all two-dimensional convex domains the disk is not optimal for the lifetime of a conditioned Brownian motion

2002 ◽  
Vol 86 (1) ◽  
pp. 335-357 ◽  
Author(s):  
Bernd Kawohl ◽  
Guido Sweers
Keyword(s):  
Brownian Motion ◽  
Two Dimensional ◽  
Convex Domains ◽  
Conditioned Brownian Motion

scholarly journals Two-dimensional stochastic modeling for predicting bankruptcy for manufacturing companies

2021 ◽  
Vol 54 (2) ◽  
pp. 123-129
Author(s):  
James C. Fu ◽  
Winnie H. W. Fu
Keyword(s):  
Brownian Motion ◽  
Real Data ◽  
Bankruptcy Prediction ◽  
Boundary Crossing ◽  
Time Interval ◽  
Two Dimensional ◽  
Manufacturing Companies ◽  
Boundary Crossing Probability ◽  
Data Set ◽  
Crossing Probability

Increasing accuracy of the model prediction on business bankruptcy helps reduce substantial losses for owners, creditors, investors and workers, and, further, minimize an economic and social problem frequently. In this study, we propose a stochastic model of financial working capital and cashflow as a two-dimensional Brownian motion X(t) = (X1(t),X2(t)) on the business bankruptcy prediction. The probability of bankruptcy occurring in a time interval [0,T] is defined by the boundary crossing probability of the two-dimensional Brownian motion entering a predetermined threshold domain. Mathematically, we extend the result in Fu and Wu (2016) on the boundary crossing probability of a high dimensional Brownian motion to an unbounded convex hull. The proposed model is applied to a real data set of companies in US and the numerical results show the proposed method performs well.

scholarly journals Serotonergic Axons as Fractional Brownian Motion Paths: Insights into the Self-organization of Regional Densities

2019 ◽  
Author(s):  
Skirmantas Janušonis ◽  
Nils Detering ◽  
Ralf Metzler ◽  
Thomas Vojta
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Adult Brain ◽  
The Self ◽  
Self Organization ◽  
Dense Matrix ◽  
Two Dimensional ◽  
Wide Range ◽  
Motion Paths ◽  
Key Features

ABSTRACTAll vertebrate brains contain a dense matrix of thin fibers that release serotonin (5-hydroxytryptamine), a neurotransmitter that modulates a wide range of neural, glial, and vascular processes. Perturbations in the density of this matrix have been associated with a number of mental disorders, including autism and depression, but its self-organization and plasticity remain poorly understood. We introduce a model based on reflected Fractional Brownian Motion (FBM), a rigorously defined stochastic process, and show that it recapitulates some key features of regional serotonergic fiber densities. Specifically, we use supercomputing simulations to model fibers as FBM-paths in two-dimensional brain-like domains and demonstrate that the resultant steady state distributions approximate the fiber distributions in physical brain sections immunostained for the serotonin transporter (a marker for serotonergic axons in the adult brain). We suggest that this framework can support predictive descriptions and manipulations of the serotonergic matrix and that it can be further extended to incorporate the detailed physical properties of the fibers and their environment.

4. Gelification and soapiness

2020 ◽  
pp. 63-90
Author(s):  
Tom McLeish
Keyword(s):  
Brownian Motion ◽  
Self Assembly ◽  
Soft Matter ◽  
Intermolecular Forces ◽  
Two Dimensional ◽  
Important Distinction ◽  
One Dimensional ◽  
The Third ◽  
Self Assembled ◽  
Weak Intermolecular Forces

‘Gelification and soapiness’ looks at the third class of soft matter: ‘self-assembly’. Like the colloids of inks and clays, and the polymers of plastics and rubbers, ‘self-assembled’ soft matter also emerges as a surprising consequence of Brownian motion combined with weak intermolecular forces. Like them, it also leads to explanations of a very rich world of materials and phenomena, such as gels, foams, soaps, and ultimately to many of the structures of biological life. There is an important distinction that needs to be made between one-dimensional and two-dimensional self-assembly.

First-passage times of two-dimensional Brownian motion

2016 ◽  
Vol 48 (4) ◽  
pp. 1045-1060 ◽  
Author(s):  
Steven Kou ◽  
Haowen Zhong
Keyword(s):  
Brownian Motion ◽  
Analytical Solutions ◽  
Laplace Transforms ◽  
First Passage Times ◽  
Absolute Difference ◽  
Two Dimensional ◽  
First Passage ◽  
Quantitative Finance ◽  
The 1960S ◽  
Passage Times

AbstractFirst-passage times (FPTs) of two-dimensional Brownian motion have many applications in quantitative finance. However, despite various attempts since the 1960s, there are few analytical solutions available. By solving a nonhomogeneous modified Helmholtz equation in an infinite wedge, we find analytical solutions for the Laplace transforms of FPTs; these Laplace transforms can be inverted numerically. The FPT problems lead to a class of bivariate exponential distributions which are absolute continuous but do not have the memoryless property. We also prove that the density of the absolute difference of FPTs tends to ∞ if and only if the correlation between the two Brownian motions is positive.

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