On stochastic variational principles

The study of stochastic variational principles involves the problem of constructing fixed-endpoint and adapted variations of semimartingales. We provide a detailed construction of variations of semimartingales that are not only fixed at deterministic endpoints, but also fixed at first entry times and first exit times for charts in a manifold. We prove a stochastic version of the fundamental lemma of calculus of variations in the context of these variations. Using this framework, we provide a generalization of the stochastic Hamilton–Pontryagin principle in local coordinates to arbitrary noise semimartingales. We also formulate a stochastic analogue of Noether’s theorem in this context. For the corresponding global form of the stochastic Hamilton–Pontryagin principle, we introduce a novel approach to global variational principles by restricting to semimartingales obtained as solutions of Stratonovich equations.