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My current research program seeks to develop new techniques for analyzing the behaviour of systems modelling complex random phenomena. The program focuses on problems from two distinct areas: (a) stochastic partial differential equations; and (b) heavy-tailed time series.
Stochastic Partial Differential Equations (SPDEs) are mathematical objects used for modeling the behaviour of physical phenomena that evolve simultaneously in space and time, and are subject to random perturbations. Their study requires tools from stochastic analysis, such as Itô calculus or Malliavin calculus. In the classical theory, these equations are perturbed by a space-time Gaussian white noise and have random field solutions only in spatial dimension 1. My program aims to discover and study new properties of the solutions to the stochastic wave and heat equations in higher dimensions, perturbed by more general classes of noise processes (such as fractional noises), as more flexible alternatives to the white noise. These findings will offer new perspectives on the dynamical interplay between the regularity of the noise and the properties exhibited by the random field solution, leading to a deeper understanding of the effect of the noise on the behaviour of solution.
Heavy tailed time series are encountered frequently in applications in finance, insurance and environmental studies, as models for perturbations that exhibit extreme behaviour. The concept of multivariate regular variation was introduced to describe a similar behaviour in higher dimensions. When we observe processes continuously over a fixed interval of time (or a region in space), we need an infinite-dimensional theory analogous to the theory of multivariate regular variation. The goal of my program is to advance the asymptotic theory for point processes associated with various time series models with values in an infinite-dimensional space of functions, and to apply this theory for deriving new results about the partial sum or partial maximum of the variables in such series. These results will give important new insights into the extreme value theory for time series models which evolve in time and space, and could be used in a variety applications.