A general model system related to affine stochastic differential equations

We link a general method for modeling random phenomena using systems of stochastic differential equations (SDEs) to the class of affine SDEs. This general construction emphasizes the central role of the Duffie-Kan system [Duffie and Kan, A yield-factor model of interest rates, Math. Finance 6 (1996) 379-406] as a model for first-order approximations of a wide class of nonlinear systems perturbed by noise. We also specialize to a two-dimensional framework and propose a direct proof of the Duffie-Kan theorem which does not passes through the comparison with an auxiliary process. Our proof produces a scheme to obtain an explicit representation of the solution once the one-dimensional square root process is assigned.