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.

中文翻译：

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与仿射随机微分方程相关的一般模型系统

我们将使用随机微分方程 (SDE) 系统建模随机现象的一般方法与仿射 SDE 类联系起来。这种一般结构强调了 Duffie-Kan 系统的核心作用 [Duffie and Kan, A yield-factor model of interest rates, Math. Finance 6 (1996) 379–406] 作为一类受噪声干扰的非线性系统的一阶近似模型。我们还专注于二维框架，并提出了 Duffie-Kan 定理的直接证明，它不通过与辅助过程的比较。一旦分配了一维平方根过程，我们的证明产生了一种方案来获得解决方案的显式表示。