Stochastic Analysis of Gaussian Fractional Noises

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The project aims to establish new results in different topics of stochastic analysis. First a new approach for proving the smoothness of the density for solutions to stochastic partial differential equations will be introduced. This method is based on a stochastic version of Feynman-Kac's formula for the Malliavin derivative of the solution. Moreover, the techniques of Malliavin calculus will be applied to derive upper and lower Gaussian estimates for the density of the solution. A second objective of the project is to obtain results on the rate of convergence of Euler-type numerical approximation schemes for stochastic differential equations driven by a fractional Brownian motion. The application of the techniques of Malliavin calculus to analyze numerical approximation schemes for backward stochastic differential equation is also one of the topics of the project. Another research direction deals with the proof of central and noncentral limit theorems for a large variety of functionals of a Gaussian process, including multiple stochastic integrals, and weighted power variations. The project also aims to establish Feynman-Kac's formulas for the one-dimensional stochastic heat equation driven by a fractional multiplicative Gaussian noise.Stochastic analysis is an active area in mathematics which is motivated by the study of ordinary and partial differential equations perturbed by a random noise. These equations play a central role as models in many areas of physics and economics. The application of these equations in concrete problems requires suitable numerical approximation schemes, and convenient estimates for the probability distribution of the solution. This project aims to make new relevant contributions to these problems, by developing and applying powerful mathematical techniques such as the Malliavin calculus. On the other hand, motivated by some applications in hydrology, telecommunications and mathematical finance, there has been a recent interest in input noises possessing a long memory property such as the 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.