Stochastic bifurcation models
type: Item
instance of: scholarly article
title: Stochastic bifurcation models
zbMATH Open document ID: 0943.60087
author: Richard F. Bass
author: K. Burdzy
published in: The Annals of Probability
publication date: 1999-11-09T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/9802045
review text: It is studied an ordinary differential equation controlled by a stochastic process. The results are encompassing the existence and uniqueness of solutions for the following ordinary differential equations \[ \frac{-dx_t}{dt}= \begin{cases} \beta_1 &\text{if }x_t< B_t,\\ \beta_2 &\text{if }x_t> B_t,\;t\in \mathbb{R},\;x(t_0)= x_0. \end{cases} \] There are proved: 1) If \(B_t\) is a typical Brownian motion path, then there will be a unique Lipschitz solution; 2) if \(\beta_1< 0\) and \(\beta_2> 0\), then a bifurcation situation appears provided \(B_t\) is a Brownian motion. There are given seven theorems including local time, time and direction of bifurcation. The results are illustrated by several examples.
Mathematics Subject Classification ID: 60J70
Mathematics Subject Classification ID: 60H10
zbMATH DE Number: 1359653
zbMATH Keywords: Brownian motion
zbMATH Keywords: differential equations
zbMATH Keywords: stochastic differential equations
reviewed by: Constantin Vârsan
MaRDI profile type: Publication
OpenAlex ID: W2115461361
arXiv ID: math/9802045
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DOI: 10.1214/AOP/1022677254
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description: scientific article