Optimal control and Malliavin calculus applied to finance.

Summary The first part is devoted to stochastic and impulse optimal control. We propose two algorithms to numerically solve Quasi Variational inequalities that arise in a portfolio management problem with fixed and proportional transaction costs. In the second part we apply Malliavin's calculus to the calculation of sensitivities. We study pure jump processes and establish part-integration formulas using the densities of the jump amplitudes that we assume to be differentiable. Then we weaken the assumption on the densities by assuming them piecewise differentiable. Thus we use the density of jump times to establish PPI formulas. We also study models of continuous multi-factor scattering. The ellipticity of the diffusion is necessary for the classical Malliavin calculus approach. For European options we establish several PPIs independently of the ellipticity of the diffusion, using other variables that aggregate the multidimensional diffusion and reduce the dimension of the Malliavin covariance matrix. In the last chapter we study the calibration of the local volatility by minimizing the relative entropy. This involves solving a stochastic control problem. We propose improvements to existing algorithms.