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Section: New Results
Stochastic Control
On the time discretization of stochastic optimal control problems: the dynamic programming approach
Participant : Frederic Bonnans.
With Justina Gianatti (U. Rosario) and Francisco J. Silva (U. Limoges) In this work we consider the time discretization of stochastic optimal control problems. Under general assumptions on the data, we prove the convergence of the value functions associated with the discrete time problems to the value function of the original problem. Moreover, we prove that any sequence of optimal solutions of discrete problems is minimizing for the continuous one. As a consequence of the Dynamic Programming Principle for the discrete problems, the minimizing sequence can be taken in discrete time feedback form. See [17].
Variational analysis for options with stochastic volatility and multiple factors
Participants : Frederic Bonnans, Axel Kroner.
We perform a variational analysis for a class of European or American options with stochastic volatility models, including those of Heston and Achdou-Tchou. Taking into account partial correlations and the presence of multiple factors, we obtain the well-posedness of the related partial differential equations, in some weighted Sobolev spaces. This involves a generalization of the commutator analysis introduced by Achdou and Tchou. See [18].
Infinite Horizon Stochastic Optimal Control Problems with Running Maximum Cost
Participant : Axel Kroner.
With Athena Picarelli (U. Oxford) and Hasna Zidani (ENSTA).
An infinite horizon stochastic optimal control problem with running maximum cost is considered. The value function is characterized as the viscosity solution of a second-order HJB equation with mixed boundary condition. A general numerical scheme is proposed and convergence is established under the assumptions of consistency, monotonicity and stability of the scheme. A convergent semi-Lagrangian scheme is presented in detail. See [19].