In this paper, we investigate specific least action principles for laws of stochastic processes within a framework which stands on filtrations preserving variations. The associated Euler–Lagrange conditions, which we obtain, exhibit a deterministic process in the dynamics aside the canonical martingale term. In particular, taking specific action functionals, extremal processes with respect to those variations encompass specific laws of continuous semi-martingales whose drift characteristic is integrable with independent increments. Then, we relate extremal processes of classical cost functions, in particular of specific entropy functions, to a class of forward-backward systems of Mckean–Vlasov stochastic differential equations.

在本文中，我们在一个基于保留变化的过滤的框架内研究随机过程定律的特定最小作用原则。我们获得的相关欧拉-拉格朗日条件在动力学中展示了一个确定性过程，除了典型的鞅项。特别是，采取特定的行动泛函，关于这些变化的极值过程包含连续半鞅的特定法则，其漂移特性是可以用独立增量积的。然后，我们将经典成本函数的极值过程，特别是特定熵函数的极值过程，与一类 Mckean-Vlasov 随机微分方程的前向-后向系统联系起来。