Abstract
We establish a generalization of the 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 with 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.
Highlights
In [29], a generalization of the Noether theorem has been proved: to any oneparameter symmetry of a variational problem it is possible to associate a martingale that is independent both from the initial and final condition of the system. This first step was quite important since it stressed that the suitable generalization of conserved quantities in a stochastic setting is not a function that remains constant during the time evolution of a stochastic system, but a function that is constant in mean. Another remarkable advance in the study of variational symmetries was achieved in [24,25,30], where it was noted that the symmetries of the Hamilton–Jacobi–Bellman (HJB) equation of the considered variational problem are the correct objects to be associated to the aforementioned martingales and the contact geometry is a good framework in which a stochastic version of the Noether theorem can be formulated
We proposed a generalization of the Noether theorem to a generic stochastic optimal control problem exploiting the tools of contact geometry and contact transformations
The results are formulated in Theorems 12 and 13 and Corollary 1, and they establish a relation between any contact symmetry of the HJB equation associated with an optimal control problem and a martingale given by the generator of the contact symmetry
Summary
The concept of symmetry of ordinary or partial differential equations (ODEs and PDEs) was introduced by Sophus Lie at the end of the 19th century with the aim of extending the Galois theory from polynomial to differential equations. This first step was quite important since it stressed that the suitable generalization of conserved quantities in a stochastic setting is not a function that remains constant during the time evolution of a stochastic system, but a function that is constant in mean Another remarkable advance in the study of variational symmetries was achieved in [24,25,30], where it was noted that the symmetries of the Hamilton–Jacobi–Bellman (HJB) equation of the considered variational problem are the correct objects to be associated to the aforementioned martingales and the contact geometry is a good framework in which a stochastic version of the Noether theorem can be formulated.
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call