scholarly journals Noether Theorem in Stochastic Optimal Control Problems via Contact Symmetries

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 953
Author(s):  
Francesco C. De Vecchi ◽  
Elisa Mastrogiacomo ◽  
Mattia Turra ◽  
Stefania Ugolini
Keyword(s):  
Optimal Control ◽  
Stochastic Optimal Control ◽  
Optimal Control Problems ◽  
Local Martingale ◽  
Control Problems ◽  
Noether Theorem ◽  
Jet Bundles ◽  
Portfolio Problem ◽  
Contact Symmetry ◽  
Hamilton Jacobi Bellman

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.

scholarly journals A Novel High Dimensional Fitted Scheme for Stochastic Optimal Control Problems

2021 ◽  
Author(s):  
Christelle Dleuna Nyoumbi ◽  
Antoine Tambue
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Optimal Control ◽  
Finite Difference Method ◽  
Stochastic Optimal Control ◽  
Optimal Control Problems ◽  
Hjb Equation ◽  
High Dimensional ◽  
Control Problems ◽  
Volume Method

AbstractStochastic optimal principle leads to the resolution of a partial differential equation (PDE), namely the Hamilton–Jacobi–Bellman (HJB) equation. In general, this equation cannot be solved analytically, thus numerical algorithms are the only tools to provide accurate approximations. The aims of this paper is to introduce a novel fitted finite volume method to solve high dimensional degenerated HJB equation from stochastic optimal control problems in high dimension ($$ n\ge 3$$ n ≥ 3 ). The challenge here is due to the nature of our HJB equation which is a degenerated second-order partial differential equation coupled with an optimization problem. For such problems, standard scheme such as finite difference method losses its monotonicity and therefore the convergence toward the viscosity solution may not be guarantee. We discretize the HJB equation using the fitted finite volume method, well known to tackle degenerated PDEs, while the time discretisation is performed using the Implicit Euler scheme.. We show that matrices resulting from spatial discretization and temporal discretization are M-matrices. Numerical results in finance demonstrating the accuracy of the proposed numerical method comparing to the standard finite difference method are provided.

scholarly journals A fitted finite volume method for stochastic optimal control problems in finance

2021 ◽  
Vol 6 (4) ◽  
pp. 3053-3079
Author(s):  
Christelle Dleuna Nyoumbi ◽  
◽  
Antoine Tambue ◽  
◽  
Keyword(s):  
Optimal Control ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Stochastic Optimal Control ◽  
Optimal Control Problems ◽  
Control Problems ◽  
Volume Method
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