# Stochastic Analysis with Financial ApplicationsStability of a Nonlinear Equation Related to a Spatially-inhomogeneous Branching Process

Stochastic Analysis with Financial Applications: Stability of a Nonlinear Equation Related to a... [Consider the nonlinear equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\frac{\partial}{\partial t}u(x,t)=\Delta_\alpha u (x, t) + a(x) \sum\limits_{k=2}^{\infty} pk^{{u^k}} (x, t)+(p0 + p1 u(x, t))\phi(x), x\in \mathbb{R}^d,$$ \end{document} where α ∈ (0, 2], u (x, 0) is nonnegative, {pk, k = 0, 1,...} is a probability distribution on ℤ+, and a and φ are positive functions satisfying certain growth conditions. We prove existence of non-trivial positive global solutions when p0, p1 and u(x, 0) are small.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# Stochastic Analysis with Financial ApplicationsStability of a Nonlinear Equation Related to a Spatially-inhomogeneous Branching Process

Part of the Progress in Probability Book Series (volume 65)
Editors: Kohatsu-Higa, Arturo; Privault, Nicolas; Sheu, Shuenn-Jyi
12 pages

/lp/springer-journals/stochastic-analysis-with-financial-applications-stability-of-a-i93M1w9J5M

# References (11)

Publisher
Springer Basel
ISBN
978-3-0348-0096-9
Pages
21 –32
DOI
10.1007/978-3-0348-0097-6_2
Publisher site
See Chapter on Publisher Site

### Abstract

[Consider the nonlinear equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\frac{\partial}{\partial t}u(x,t)=\Delta_\alpha u (x, t) + a(x) \sum\limits_{k=2}^{\infty} pk^{{u^k}} (x, t)+(p0 + p1 u(x, t))\phi(x), x\in \mathbb{R}^d,$$ \end{document} where α ∈ (0, 2], u (x, 0) is nonnegative, {pk, k = 0, 1,...} is a probability distribution on ℤ+, and a and φ are positive functions satisfying certain growth conditions. We prove existence of non-trivial positive global solutions when p0, p1 and u(x, 0) are small.]

Published: Jul 12, 2011

Keywords: Semi-linear partial differential equation; positive solutions; global solutions