A random Fredholm integral equation

Padgett, W. J.; Tsokos, Chris P.

doi:10.1090/S0002-9939-1972-0292197-4

A random Fredholm integral equation
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by W. J. Padgett and Chris P. Tsokos PDF

Proc. Amer. Math. Soc. 33 (1972), 534-542 Request permission

Abstract:

The aim of this paper is the study of a random or stochastic integral equation of the Fredholm type given by $x(t;\omega ) = h(t;\omega ) + \smallint _0^\infty {{k_0}(t,} \tau ;\omega )e(\tau ,x(\tau ;\omega ))\;d\tau , t \geqq 0$, where $\omega \in \Omega$, the supporting set of the probability measure space $(\Omega ,A,P)$. The existence and uniqueness of a random solution to the above stochastic integral equation is considered. A random solution, $x(t;\omega )$, of such a random equation is defined to be a random function which satisfies the equation almost surely. Several theorems and useful special cases are presented which give conditions such that a random solution exists.

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