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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On a nonlinear stochastic integral equation of the Hammerstein type


Author: W. J. Padgett
Journal: Proc. Amer. Math. Soc. 38 (1973), 625-631
MSC: Primary 45G99
DOI: https://doi.org/10.1090/S0002-9939-1973-0320663-2
MathSciNet review: 0320663
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Abstract: A nonlinear stochastic integral equation of the Hammerstein type in the form

$\displaystyle x(t;\omega ) = h(t;\omega ) + \int_s {k(t,s;\omega )f(s,x(s;\omega )} )d\mu (s)$

is studied where $ t \in S,a$, a $ \sigma $-finite measure space with certain properties, $ \omega \in \Omega $, the supporting set of a probability measure space $ (\Omega ,A,P)$, and the integral is a Bochner integral. A random solution of the equation is defined to be a second order vector-valued stochastic process $ x(t;\omega )$ on $ S$ which satisfies the equation almost certainly. Using certain spaces of functions, which are spaces of second order vector-valued stochastic processes on $ S$, and fixed point theory, several theorems are proved which give conditions such that a unique random solution exists.

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DOI: https://doi.org/10.1090/S0002-9939-1973-0320663-2
Keywords: Stochastic Hammerstein integral equation, nonlinear stochastic integral equation, random solutions, second order processes
Article copyright: © Copyright 1973 American Mathematical Society