# Noether Theorem in Stochastic Optimal Control Problems via Contact Symmetries

@inproceedings{DeVecchi2021NoetherTI, title={Noether Theorem in Stochastic Optimal Control Problems via Contact Symmetries}, author={Francesco C. De Vecchi and Elisa Mastrogiacomo and Mattia Turra and Stefania Ugolini}, year={2021} }

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 HamiltonJacobi-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…

## 2 Citations

Asymptotic symmetry and asymptotic solutions to Ito stochastic differential equations

- MathematicsMathematics in Engineering
- 2022

We consider several aspects of conjugating symmetry methods, including the method of invariants, with an asymptotic approach. In particular we consider how to extend to the stochastic setting several…

Some Recent Developments on Lie Symmetry Analysis of Stochastic Differential Equations

- MathematicsGeometry and Invariance in Stochastic Dynamics
- 2021

## References

SHOWING 1-10 OF 74 REFERENCES

Viscosity Solutions of Stochastic Hamilton-Jacobi-Bellman Equations

- MathematicsSIAM J. Control. Optim.
- 2018

The notion of viscosity solution is introduced, and it is proved that the value function of the optimal stochastic control problem is the maximal viscosities solution of the associated Stochastic HJB equation.

Finite dimensional solutions to SPDEs and the geometry of infinite jet bundles

- Mathematics
- 2017

Finite dimensional solutions to a class of stochastic partial differential equations are obtained extending the differential constraints method for deterministic PDE to the stochastic framework. A…

Symmetries in the stochastic calculus of variations

- Mathematics
- 1997

Summary. Given a stochastic action integral we define a notion of invariance of this action under a family of one parameter space-time transformations and a notion of prolonged transformations which…

Isovectors for the Hamilton-Jacobi-Bellman Equation, Formal Stochastic Differentials and First Integrals in Euclidean Quantum Mechanics

- Mathematics
- 2004

The study of Riemannian stochastic differential geometry is almost as old as the theory of stochastic differential equations themselves. But almost nothing is known about stochastic symplectic…

Symmetries of stochastic differential equations using Girsanov transformations

- MathematicsJournal of Physics A: Mathematical and Theoretical
- 2020

Aiming at enlarging the class of symmetries of a stochastic differential equation (SDE), we introduce a family of stochastic transformations able to change also the underlying probability measure…

Noether-Type First Integrals Associated with Autonomous Second-Order Lagrangians

- MathematicsSymmetry
- 2019

The classical Noether’s theorem and a non-standard Legendrian duality are used and a correspondence is established between the invariances under flows and the first integrals for autonomous second-order Lagrangians.

Commuting Hamiltonians and multi-time Hamilton-Jacobi equations

- Mathematics
- 2005

We prove that if a sequence of pairs of smooth commuting Hamiltonians converge in the $C^0$ topology to a pair of smooth Hamiltonians, these commute. This allows us define the notion of commuting…

Pathwise Stochastic Control Problems and Stochastic HJB Equations

- MathematicsSIAM J. Control. Optim.
- 2007

A class of pathwise stochastic control problems in which the optimality is allowed to depend on the paths of exogenous noise (or information) is studied, showing that such a control problem may not even have a “minimizing sequence,” but nevertheless the (Bellman) dynamical programming principle still holds.