Malliavin Calculus for Chaos Expansions of Generalized StochasticProcesses with Applications to Some Classes of Dierential Equations

Abstract

<p>In this dissertation we study the main properties of the opera-tors of Malliavin calculus dened on a set of singular generalized stochastic<br />processes, which admit chaos expansion representation form in terms of or-thogonal polynomial basis and having values in a certain weighted space of<br />stochastic distributions in white noise framework.<br />In the rst part of the dissertation we focus on white noise spaces and<br />introduce the fractional Poissonian white noise space. All four types of white<br />noise spaces obtained (Gaussian, Poissonian, fractional Gaussian and frac-tional Poissonian) can be identied through unitary mappings.<br />As a contribution to the Malliavin dierential theory, theorems which<br />characterize the operators of Malliavin calculus, extended from the space<br />of square integrable random variables to the space of generalized stochastic<br />processes were obtained. Moreover the connections with the corresponding<br />fractional versions of these operators are emphasized and proved.<br />Several examples of stochastic dierential equations involving the&nbsp;<br />operators of the Malliavin calculus, solved by use of the chaos expansion<br />method, have found place in the last part of the dissertation. Particularly,<br />obtained results are applied to solving a generalized eigenvalue problem<br />with the Malliavin derivative and a stochastic Dirichlet problem with a<br />perturbation term driven by the Ornstein-Uhlenbeck operator.</p>

<p>Predmet istraživanja ove doktorske disertacije je&nbsp;teorijsko razmatranje glavnih svojstava operatora Maliavenovog&nbsp;računa, definisanih na klasi uop&scaron;tenih stohastičkih procesa,&nbsp;koji se mogu razviti u red po bazi, izraženoj u obliku familije ortogonalnih polinoma i sa vrednostima u nekom težinskom&nbsp;prostoru stohastičkih distribucija, na prostoru belog &scaron;uma.</p><p>Prvi deo disertacije je posvećen izučavanju raznih klasa&nbsp;prostora belog &scaron;uma i kao rezultat, uveden je prostor frakcionog&nbsp;Poasonovog belog &scaron;uma. Pokazano je da su svi &nbsp;dobijeni prostori&nbsp;belog &scaron;uma medjusobno povezani unitarnim preslikavanjima.</p><p>Kao doprinos Maliavenovoj diferencijalnoj teoriji, formulisane su teoreme koje opisuju &nbsp;aliavenove operatore na uop&scaron;tenim stohastičkim procesima. Takodje su uočene i istaknute veze ovih operatora sa odgovarajućim frakcionim Maliavenovim operatorima.</p><p>Metod haos ekspanzija, primenjen na re&scaron;avanje stohastičkih&nbsp;diferencijalnih jednačina &nbsp;u kojima figuri&scaron;u Maliavenov izvod&nbsp;i Orn&scaron;tajn-Ulembekov operator, prezentovan je u zavr&scaron;nom delu&nbsp;disertacije. Konkretno, predstavljena su re&scaron;enja uop&scaron;tenog&nbsp;problema sopstvenih vrednosti za operator Maliavenovog izvoda&nbsp;kao i Dirihleovog problema sa perturbacijama generisanim&nbsp;dejstvom Orn&scaron;tajn-Ulembekovog operatora.</p>