We consider a linear stochastic differential equation with stochastic drift. We study the problem of approximating the solution of such equation through an Ornstein-Uhlenbeck type process, by using direct methods of calculus of variations. We show that general power cost functionals satisfy the conditions for existence and uniqueness of the approximation. We provide some examples of general interest and we give bounds on the goodness of the corresponding approximations. Finally, we focus on a model of a neuron embedded in a simple network and we study the approximation of its activity, by exploiting the aforementioned results.

中文翻译：

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具有随机漂移和应用的过程的最优高斯-马尔可夫近似

我们考虑具有随机漂移的线性随机微分方程。我们通过使用变分法的直接方法，研究通过 Ornstein-Uhlenbeck 类型的过程来逼近此类方程的解的问题。我们表明一般电力成本泛函满足近似的存在性和唯一性的条件。我们提供了一些普遍感兴趣的例子，并给出了相应近似值的边界。最后，我们关注嵌入在简单网络中的神经元模型，并通过利用上述结果研究其活动的近似值。