Stochastic Volterra equations with singular kernels
type: Item
instance of: scholarly article
title: Stochastic Volterra equations with singular kernels
zbMATH Open document ID: 0836.60049
DOI: 10.1016/0304-4149(94)00072-2
author: W. George Cochran
author: Jung-Soon Kim Lee
author: Jürgen Potthoff
published in: Stochastic Processes and their Applications
publication date: 1995-05-23T00:00:00Z
review text: Let us consider the following Volterra type equation \[ X(t) = Y(t) + \int_0^t\;K(t,s) X(s) dB(s),\quad t \geq 0, \] where \(Y\) is some possibly anticipating process, \(K\) is a deterministic kernel which can be singular at \(s = t\) and \(B(t)\) is the standard Brownian motion. Assume that \(|K(s,t) |\leq A (t - s)^{- 1/2}\) with a constant \(A\) and \(Y\) is a weakly measurable generalized process valued in Hida distribution \(({\mathcal S})^*\) and satisfies a boundedness condition. Then the above equation has a unique solution of generalized process in \(({\mathcal S})^*\). Assume that \(\int_0^{t \wedge t_n} |K(t,s) - K(t_n,s) |ds \to 0\), \(t_n \to t\), and the continuity of \(Y\). Then the solution \(X(t)\) is continuous in the strong topology of \(({\mathcal S})^*\).
Mathematics Subject Classification ID: 60G20
Mathematics Subject Classification ID: 60H20
zbMATH DE Number: 757535
zbMATH Keywords: white noise analysis
zbMATH Keywords: stochastic analysis
zbMATH Keywords: Volterra equations
zbMATH Keywords: unique solution
zbMATH Keywords: strong topology
MaRDI profile type: Publication
full work available at URL: 0304-4149(94)00072-2
OpenAlex ID: W2048662867
cites work: Q3899268
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cites work: A characterization of Hida distributions
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description: scientific article; zbMATH DE number 757535