Resumen de Applications of mallavin calculus to some problems related to gaussian and lévy processes
The main objective of this thesis is to use Malliavin calculus techniques to study some problems of Stochastic Analysis. The thesis has two parts. In the first part, we study the regularity of the density of the supremum of some Volterra type Gaussian processes. More concretely , we show that the density of the distribution of the supremum of the process is intinitely differentiable on (0,¿) for different examples. We treat the cases of subfractional, bifractional, and multifractional Brownian motion. In the second part, we obtain some results in financial modelling with infinite activity Lévy process. We study the small-time behavior of the call option price and the implied volatility of a general stochastic volatility model driven by an infinite activity Lévy process. By using the Malliavin calculus for Lévy processes, we obtain an anticipating Itô formula and a Hull-White formula. We consider only some mild Malliavin calculus assumptions on the volatility process, but no assumption on the Lévy measure. As an application, we find the small-time to maturity behaviour of European call option price for our general model. The using the asymptotic relation between the option price and the implied volatility we obtain the small-time to maturity behaviour of the implied volatility for the at-the-money and out-of-the-money cases. We also study the asymptotic behavior of the slope of the implied volatility for our general stochastic volatility model. The Malliavin calculus can be applied to other volatility models. For instance, we obtain the small-time limit for the slope of the implied volatility of the exponential Lévy model with constant volatility.
