Dependence on the boundary condition for linear stochastic differential equations in the plane
type: Item
instance of: scholarly article
title: Dependence on the boundary condition for linear stochastic differential equations in the plane
zbMATH Open document ID: 0693.60051
DOI: 10.1016/0304-4149(89)90065-3
author: N. E. Zubov
published in: Stochastic Processes and their Applications
publication date: 1989-01-01T00:00:00Z
review text: The following linear stochastic differential equation is considered: \[ X(z,f)=f(s,0)+f(0,t)-f(0,0)+a\int_{(0,z]}X(z_ 1,f) W(dz_ 1), \] where \(z=(s,t)\in R^ 2_+\), \(a\in R\), W is a two-parameter Wiener process, and f is a real-valued continuous function on \(\partial R^ 2_+\). The strong solution of this equation is represented as a sum of series of multiple Ito-Wiener integrals of the function f. The main result of the paper is the following: If f(0,\(\cdot)\) and f(\(\cdot,0)\) are functions of bounded variation on every finite interval then there exists a version of the strong solution X(z,f) such that for any \(z_ 0\in R^ 2_+\) the mapping \(f(\cdot)\to X(z_ 0+\cdot,f)\) is linear and continuous with respect to the family of total variation seminorms as well as to families of \(\alpha\)-Hölder seminorms for \(\alpha\in (0,1/2).As\) a corollary, the existence of such a version of X(z,f) that the mapping (z,f)\(\to X(z,f)\) is jointly continuous is proven.
Mathematics Subject Classification ID: 60H20
Mathematics Subject Classification ID: 60H10
zbMATH DE Number: 4135146
zbMATH Keywords: boundary condition
zbMATH Keywords: stochastic differential equation
zbMATH Keywords: strong solution
zbMATH Keywords: multiple Ito-Wiener integrals
zbMATH Keywords: total variation seminorms
MaRDI profile type: Publication
full work available at URL: 0304-4149(89)90065-3
OpenAlex ID: W2009210621
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description: scientific article; zbMATH DE number 4135146