Noether theorem in stochastic optimal control problems via contact symmetries

TL;DR

This paper generalizes Noether's theorem to stochastic optimal control problems using contact geometry, linking symmetries of the Hamilton-Jacobi-Bellman equation to conserved quantities in the form of local martingales, with applications to finance.

Contribution

It introduces a novel connection between symmetries and conserved quantities in stochastic control via contact geometry, extending classical results to stochastic settings.

Findings

Symmetries of the HJB equation lead to local martingales.

Application to Merton's portfolio problem reveals infinite conserved quantities.

Establishes a geometric framework for stochastic optimal control analysis.

Abstract

We establish a generalization of Noether theorem for stochastic optimal control problems. Exploiting the tools of jet bundles and contact geometry, we prove that from any (contact) symmetry of the Hamilton-Jacobi-Bellman equation associated to an optimal control problem it is possible to build a related local martingale. Moreover, we provide an application of the theoretical results to Merton's optimal portfolio problem, showing that this model admits infinitely many conserved quantities in the form of local martingales.