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Contrôle stochastique et méthodes numériques en finance mathématique

Abstract : This PhD dissertation presents three independent research topics in the fields of numerical methods and stochastic control with applications to financial mathematics.

The first part of this thesis is dedicated to the estimation of the sensitivities of option prices, by means of non-parametric techniques. When the density of the underlying is unknown, we propose several non-parametric estimators of the so called Greeks, based on the randomization of the parameter of interest combined with Monte Carlo simulations and Kernel regression techniques. We provide an asymptotic analysis of the mean squared error of these estimators, as well as their asymptotic distributions. For a discontinuous payoff function, the kernel estimators outperforms the classical finite differences one in terms of the asymptotic rate of convergence. This result is confirmed by our numerical experiments.

The second part of this dissertation deals with the numerical resolution of systems of decoupled forward-backward stochastic differential equations with jumps. Assuming that the coefficients are Lipschitz-continuous, we propose a convergent discrete-time scheme whose rate of convergence is at least $n^{-1/2+e}$, for any $e>0$, when the number of time steps $n$ goes to infinity. Under additional regularity assumption the scheme achieves the optimal parametric convergence rate. The statistical error due to the non parametric approximation of conditional expectations is controlled and we provide the numerical solution of systems of coupled semilinear parabolic PDE's.

The third part of this thesis is concerned with the resolution of the optimal consumption-investment problem under a drawdown constraint, i.e. the wealth process never falls below a fixed fraction of its running maximum. We assume that the risky asset is driven by the constant coefficients Black and Scholes model and we consider a general class of utility functions. On an infinite time horizon, we provide the value function in explicit form, and we derive closed-form expressions for the optimal consumption and investment strategy. On a finite time horizon, we interpret the value function as the unique viscosity solution of its corresponding Hamilton-Jacobi-Bellman equation. This leads to a consistent numerical scheme of approximation and allows for a comparison with the explicit solution in infinite horizon.
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Contributor : Romuald Elie Connect in order to contact the contributor
Submitted on : Friday, January 5, 2007 - 1:25:41 PM
Last modification on : Friday, August 5, 2022 - 2:49:41 PM
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  • HAL Id : tel-00122883, version 1



Romuald Elie. Contrôle stochastique et méthodes numériques en finance mathématique. Mathématiques [math]. ENSAE ParisTech, 2006. Français. ⟨tel-00122883⟩



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