Stochastic control and numerical methods in mathematical finance.

Summary This thesis presents three independent research topics belonging to the field of numerical methods and stochastic control with applications in financial mathematics. In the first part, we present a non-parametric method for estimating option price sensitivities. Using a random perturbation of the parameter of interest, we represent these sensitivities as conditional expectations, which we estimate using Monte Carlo simulations and kernel regression. Using integration by parts arguments, we propose several kernel estimators of these sensitivities, which do not require knowledge of the density of the underlying, and we obtain their asymptotic properties. When the payoff function is irregular, they converge faster than the finite difference estimators, which is verified numerically. The second part focuses on the numerical solution of decoupled systems of backward progressive stochastic differential equations. For Lipschitz coefficients, we propose a discretization scheme that converges faster than $n^{-1/2+e}$, for any $e>0$, when the time step $1/n$ tends to $0$, and under stronger regularity assumptions, the scheme reaches the parametric convergence speed. The statistical error of the algorithm due to the non-parametric approximation of conditional expectations is also controlled and we present examples of numerical solution of coupled systems of semi-linear PDEs. Finally, the last part of this thesis studies the behavior of a fund manager, maximizing the intertemporal utility of his consumption, under the constraint that the value of his portfolio does not fall below a fixed fraction of its current maximum. We consider a general class of utility functions, and a financial market composed of a risky asset with black-Scholes dynamics. When the manager sets an infinite time horizon, we obtain in explicit form his optimal investment and consumption strategy, as well as the value function of the problem. In a finite horizon, we characterize the value function as the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation.