Projection operator method applied to stochastic linear differential equations

Abstract

Abstract Under appropriate conditions the expectation value of a stochastic process defined by a linear differential equation with stochastic coefficients and a fixed initial condition satisfies an approximate markovian equation, as was recently shown by Van Kampen. In this article the markovian equation is derived in a very direct and concise way with the help of a projection operator technique. A projection operator which reduces quantities to their expectation value is introduced and an exact integro-differential equation for the expectation value of the stochastic process is derived. The kernel which occurs in this equation is shown to be short-lived under appropriate conditions. Its series expansion, combined with iteration of the integro-differential equation then leads to the approximate markovian equation. The results are compared with those of Van Kampen and Bourret and the methods is applied to two examples.