| dc.description.abstract | The goal of this thesis is to build on the method and result of \parencite{Espen_piecewiseconstant} to improve the bound between the optimal value function and the piece-wise constant control one in stochastic control settings by exploring the properties of semi-concavity and semi-convexity for the data. A significant part of the work was to familiarise ourselves with the fundamental aspect of optimal control theory and controlled differential equation theory. Some very basic properties are demonstrated in this way, because the author felt that these classical considerations would be helpful in understanding the whole work.
In that regard, we show how to reach the optimal error bound in two different settings before tackling the main interest. These setting are the Stochastic smooth one and the deterministic lipschitz one. We then show that while semi-concavity or semi-convexity alone are not sufficient to improve the bound, the combination of the two yields a 1/3 bound. This improves the 1/4 bound found in \parencite{Espen_piecewiseconstant} at the cost of several new stringent hypotheses. Later we present an analysis of the schemes developed in \parencite{probabilistic_error_Picarelli} and propose a new numerical experiments to show specifically the error bound induced by the time discretisation of the control. Finally some numerical experiments are conducted to show the effectiveness of the approach. | |
