Existence and unique of the mild solution of stochastic integro differential equation

Abstract

This paper is devoted to show the existence of the mild solution of the stochastic integro-differential equation by employing the successive approximation with standard Brownian motion  W(t).

Using theorem (2.1), to find the mild solution

For

This is done by applying the stochastic differential equation (2.1) and that provides the solution (2.2) ² and by compensation (2.2) for (2.1) we get:

It proves its existence by the successive approximation and its equivalent integral stochastic in equation (1.1), then we can say:

For

To find the mild solution of the stochastic integro-differential equation, we used the descriptive and experimental approach. Also, we used Gronwall inequality to ensure the uniqueness of the solution.