Average preserving variation processes in view of optimization

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, obtain, exhibit deterministic process in the dynamics aside canonical martingale term. particular, taking functionals, extremal with respect to those variations encompass continuous semi-martingales whose drift characteristic is integrable independent increments. Then, relate classical cost functions, particular entropy class forward-backward systems Mckean–Vlasov differential equations.