On a nonlinear stochastic integral equation of the Hammerstein type

Source abstract

A nonlinear stochastic integral equation of the Hammerstein type in the form $$x ( t ; ω ) = h ( t ; ω ) + ∫ s k ( t , s ; ω ) f ( s , x ( s ; ω ) ) d μ ( s ) x(t;\omega ) = h(t;\omega ) + \int _s {k(t,s;\omega )f(s,x(s;\omega )} )d\mu (s)$$ is studied where t ∈ S , a t \in S,a , a σ \sigma -finite measure space with certain properties, ω ∈ Ω \omega \in \Omega , the supporting set of a probability measure space ( Ω , A , P ) (\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 ; ω ) x(t;\omega ) on S S which satisfies the equation almost certainly. Using certain spaces of functions, which are spaces of second order vector-valued stochastic processes on S S , and fixed point theory, several theorems are proved which give conditions such that a unique random solution exists.