Approximate Solution Methods for Linear Stochastic Difference Equations. II. Inhomogeneous Equations, Multi-Time Averages, Biological Application

The cumulant expansion for linear stochastic difference equations introduced in part I is applied to the general case, where the equation contains multiplicative, additive and initial value terms which are all random and statistically interdependent. Also the two-time correlation functions of the solution are discussed. Finally the expansion for the probability density functions is studied. In the case that the coefficient matrix constitutes a Markov process, an exact equation for the joint probability density function of the solution of the difference equation and the random coefficient matrix is derived. From this equation the moments of the solution can be obtained in a simple way. As an application we consider the growth of a biological population with two age classes in a random environment, which itself is modelled by a two-state Markov chain. The exact results are compared with those of the cumulant expansion and with previous findings.