Moments and Correlation Functions of Solutions of a Stochastic Differential Equation

Abstract

This paper shows how to obtain exact, closed‐form expressions for various moments and correlation functions of the solutions of the stochastic, ordinary differential equation d 2 u dz 2 +β 0 2 [1+ηT(z)]u=0 ,where T(z) is the so‐called ``random telegraph'' wave and β0 2 and η are positive real constants. These moments and correlation functions are calculated by two different methods, one a phase space method and the other a matrix method familiar from optics. It is found that the moments are sums of exponentials. The first‐order moments decay exponentially but the second‐order moments grow exponentially. The correlation functions are also sums of exponentials and show that the solutions do not form a stationary process. An important application of these results is obtained in the problem of a plane electromagnetic wave normally incident on a randomly stratified dielectric plate. It is shown that, if S is the amplitude transmission coefficient of the plate, then 〈1/ SS *〉 can be expressed in terms of the second‐order moments of the solutions and derivatives of solutions of the stochastic differential equation.