# Some stochastic integral and discrete equations of the volterra and fredholm types with applications

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## Abstract

Random or stochastic integral equations occur frequently in the mathematical description of random phenomena in engineering, physics, biology, and oceanography. The present study is concerned with random or stochastic integral equations of the Volterra type in the form

x(t;w) = h(tiW) + fa k(t,T~w)f(T,x(TjwÂ»dT, t > 0,

and of the Fredholm type in the form

00 x(tjw) = h(t:w) + fa ko(t,T;w)e(T,x(T;wÂ»dT, t ~ 0,

where w Â£ Q, the supporting set of a complete probability measure space (n,A,p). A random function x(t:w} is said to be a random solution of an equation such as those above if it satisfies the equation with probability one. It is also required that X(tiW) be a second order stochastic process.

The purpose of this dissertation is to investigate the existence, uniqueness, and stochastic stability properties of a random solution of these Volterra and Fredholm stochastic integral equations using the "theory of admissibility" and probabilistic functional analysis. The techniques of successive approximations and stochastic approximation are employed to approximate the random solution of the stochastic Volterra integral equation, and the convergence of the approximations to the unique random solution in mean square and with probability one is proven.

Problems in telephone traffic theory, hereditary mechanics, population growth, and stochastic control theory are formulated, and some of the results of the investigation are applied.

Finally, a discrete version of the above random integral equations is given, and several theorems concerning the existence, uniqueness, and stochastic stability of a random solution of the discrete equation are proven. Approximation of the random solution of the discrete version is obtained, and its convergence to the random solution is studied.

This work extends and generalizes the work done by C. P. Tsokos in Mathematical Systems Theory 3 {1969}, pages 222-231, and M. W. Anderson in his Ph.D. dissertation at the University of Tennessee, 1966, among others. Extensions:of this research to several areas of application are proposed.