Existence of solutions of a stochastic integral equation with an application from the theory of epidemics. (English) Zbl 0967.60065

This paper proves an existence theorem for the stochastic integral equation \[ x(t;w)= \int^t_{t-u} f(s,x(s;w); w)ds, \] arising in epidemic theory. Here \(x(t;w)\) is a random variable, and \(f\) a real-valued function from \(R^+\times R\times \Omega\), with \(\Omega\) the supporting set of probability measure space \((\Omega,A,P)\). The authors use the contractor concept of A. C. H. Lee and W. J. Padgett [Nonlinear Anal., Theory Methods Appl. 3, 707-715 (1979; Zbl 0417.60071)] to attack the problem, and prove that under certain conditions, there exists a unique random solution for the above random integral equation.


60H20 Stochastic integral equations


Zbl 0417.60071