# Average preserving variation processes in view of optimization

## TL;DR

This paper explores least action principles for stochastic processes with filtration-preserving variations, deriving Euler-Lagrange conditions that characterize extremal laws of semi-martingales, and linking them to forward-backward McKean-Vlasov equations.

## Contribution

It introduces a novel variational framework for stochastic processes that preserves filtrations and derives new Euler-Lagrange conditions for extremal laws.

## Key findings

- Extremal processes have deterministic dynamics with martingale components.
- Certain entropy-based cost functions relate to forward-backward McKean-Vlasov equations.
- The framework characterizes laws of semi-martingales with integrable drift characteristics.

## Abstract

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.