Averaged stochastic processes and Kolmogorov operators

In this thesis we study a class of multidimensional stochastic processes in which a component is the time integral of another. Our interest stems both from the great variety of applications and the challenging structure of the related Kolmogorov backward operators. In fact, while such processes are widely used in physics and finance, the natural geometric framework to study them is considerably far from the standard Euclidean one and still vague. We wish to clarify it developing a new notion of Hölder spaces of any order and proving a Taylor type formula for functions on them. As applications, we prove an error estimate for an asymptotic expansion arising in studying Asian financial options and we also present and analytically investigate a new model for mine valuation.