A sample treatment of Langevin-type stochastic differential equations

A sample treatment of stochastic differential equations of the Langevin type is presented. It is shown that the formal mean square solutions of these equations are, in a certain sense, consistent with the solutions obtained based upon a sample theoretic approach. The procedure adopted in this development is to first construct an explicit sequence of Gaussian processes which approach in the limit the Brownian motion process. With derivatives of these processes as inputs to linear differential equations, we show that these equations possess solutions in the sample sense. These solutions are then shown to converge to the formal mean square solution of a differential equation having as input the white noise or the formal derivative of the Brownian motion process.