Thesis Summary:

This thesis deals with the study of stochastic differential equations (SDE’s) driven by self-similar multi-parameter processes with the aim of setting a contribution to the stochastic calculus with respect to this type of processes and thus extend the set of applications of SDE’s and phenomena susceptible of being modelled by these. Specifically, three types of SDE’s driven by fractional processes were studied, analysing different characteristics and properties of these. The Wiener integral with respect to the Hermite sheet is also defined and the associated SDE’s are studied. The fractional Brownian motion (fBm) can be considered in many senses as the natural generalization of the standard Brownian motion (sBm), however, the stochastic calculus used for the sBm can not be used for the fBm, mainly because is neither a semimartingale nor a Markov process. So, the first part of this thesis it’s about a SDE with delay, driven by a fBm with self-similarity parameter H in the interval ( 1 2 , 1). By means of a numerical method a discrete time approximation for the solution of the equation is studied, the strong convergence is proved and the rate of convergence is established. The second part of this thesis is devoted to study multi-parameter processes. The fractional Ornstein-Uhlenbeck sheet (fOUs), which is defined as the solution of a Langevin equation with respect to the fractional Brownian sheet (fBs), is studied. The fBs is an anisotropic process and it is considered the case when the self-similarity parameters α and β are greater than 1 2 , that is, the long memory case. A least squares estimator for the tendency parameter of the fOUs is built, the strong consistency is proved and also that is not asymptotically normal, this last is in contrast with the one parameter case. Continuing with the study of random fields, the third part of this thesis was devoted to the study of a stochastic wave equation with additive noise, fractional in time and colored in space. Optimal bounds for the regularity of the solution were proved, in time and space, which allows to stablish the joint regularity of the solution. This along with some concepts of Potential Theory, allowed to establish upper and lower bounds for the hitting probabilities of the solution. Finally, the last part of this thesis present a contribution to the stochastic calculus with respect to the Hermite processes, which are characterized by the self-similarity parameter H and the parameter q which is associated with the numbers of integrals involved. In contrast with the processes studied previously, the Hermite processes are Gaussian only when q = 1, the case of the fractional Brownian motion. The Hermite sheet (sH) is defined as a multiple integral with respect to the standard Brownian sheet and Wiener integrals with respect to it are presented, this along with some others results presented previously in this thesis allows to analyse as an example a stochastic wave equation driven by the Hermite sheet, his solution is defined and the temporal, spatial and joint regularity are proved. Other additional results related with local time of the solution are also presented.

ISI Publications from the Thesis

Jorge CLARKE, Ciprian A. TUDOR: Wiener integrals with respect to the Hermite random field and applications to the wave equation. Collectanea Mathematica, vol. 65, 3, pp. 341-356, (2014).

Jorge CLARKE, Ciprian A. TUDOR: Least squares estimator for the parameter of the fractional Ornstein-Uhlenbeck sheet. Journal of the Korean Statistical Society, vol. 41, 3, pp. 341-350, (2012).