LIE SYMMETRY ANALYSIS AND GEOMETRICAL METHODS FOR FINITE AND INFINITE DIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS

The main aim of the thesis is a systematic application (via suitable generalizations) of Lie symmetry analysis, or more generally, of the various geometric techniques for differential equations, to the study of finite and infinite dimensional stochastic differential equations (SDEs). The work can be divided in three main parts. In the first part a new geometric approach to finite dimensional SDEs driven by a multidimensional Brownian motion is proposed, which is based on a new notion of random transformations of a stochastic process called stochastic transformations. After having studied the probabilistic and geometric properties of stochastic transformations, we provide a useful generalization of the well-known results of reduction and reconstruction of symmetric ODEs to the stochastic setting. We give many applications of previous results to some interesting SDEs among which the two dimensional Brownian motion, the Kolmogorov-Pearson equation, a generalized Langevin equation and the SABR model. Finally, using the previous theorems, we propose a symmetry-adapted numerical scheme whose effectiveness is verified through both theoretical estimates and numerical simulations. The second part contains an extension of the results obtained in the first part to finite dimensional SDEs driven by a general semimartingale taking values in a Lie group. In order to provide such an extension we use the notion of geometrical SDEs introduced by Serge Choen, and we introduce some new notions of stochastic invariance for semimartingales called gauge and time symmetries of a semimartingale. Using these mathematical tools we generalize the notion of stochastic transformations in this setting and we propose the natural definition of symmetry based on this group of transformations. The formulated theory allows us to analyze in detail an important class of SDEs with possible relevant applications to iterated random maps theory. In the third part we take advantage of the geometry of the infinite jets bundle to develop a convenient algorithm for the explicit determination of finite dimensional solutions to stochastic partial differential equations (SPDEs). In this setting we are able to propose a generalization of Frobenius theorem in the infinite jet bundles setting, which, exploiting the classical notion of characteristics of a PDE, allows us to find some sufficient conditions for the existence of finite dimensional solutions to an SPDE and then to explicitly reduce the SPDE to a finite dimensional SDE. Our techniques permits to individuate new finite dimensional solutions to interesting SPDEs among which the proportional volatility equation in Heath-Jarrow-Morton framework, a stochastic perturbation of Hunter-Saxton equation and a filtering problem related to affine type processes.