Stochastic Calculus of Variations and Stochastic Analysis with Fractal Noises

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The project aims to establish new results in three diffeent topics of stochastic analysis. First, a new approach for estimating the negative moments of the solutions to linear stochastic partial differential equations of parabolic type will be developed. These estimates will allow us to derive the regularity of the probability law of the solution at a finite number of points using the techniques of Malliavin Calculus. A second objective is to further develop the stochastic calculus with respect to the fractional Brownian motion and related processes using both Malliavin Calculus and path-wise techniques. Our third goal is to establish chaotical central limit theorems for the asymptotic behavior of different types of functionals of a Gaussian process. Examples of these functionals include the self-intersection local time of the fractional Brownian motion, and power variation and related functionals of stochastic integrals.Stochastic analysis is a modern area in mathematics which aims to study ordinary and partial differential equations perturbed by a random noise. These equations play a central role as models in many areas in physics and economics. In order to derive important properties of the solutions, like to compute the probability that the solution takes values in some interval, one needs to apply suitable mathematical techniques like the Ito Calculus and the Malliavin Calculus. We aim to make substantial contributions to the development of these thecniques and their applications to stochastic partial differential equations. On the other hand, while the classical input noise used has independent increments, motivated by some applications in hydrology, telecommunications and mathematical finance, there has been a recent interest in input noises possessing a long memory property like fractional Brownian motion. The development of a stochastic calculus with respect to these long memory processes is also one of the aims of this project.