On SDE systems with non-Lipschitz diffusion coefficients

Chuni, Vinayak (2020) On SDE systems with non-Lipschitz diffusion coefficients, [Dissertation thesis], Alma Mater Studiorum Università di Bologna. Dottorato di ricerca in Scienze statistiche, 32 Ciclo.
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Abstract

This thesis is a compilation of two papers. In the first paper we investigate a class of two dimensional stochastic differential equations related to susceptible-infected-susceptible epidemic models with demographic stochasticity. While preserving the key features of the model considered in \cite{Mao}, where an \emph{ad hoc} approach has been utilized to prove existence, uniqueness and non explosivity of the solution, we consider an encompassing family of models described by a stochastic differential equation with random and H\"older continuous coefficients. We prove the existence of a unique strong solution by means of a Cauchy-Euler-Peano approximation scheme which is shown to converge in the proper topologies to the unique solution. In the second paper we link a general method for modeling random phenomena using systems of stochastic differential equations to the class of affine stochastic differential equations. This general construction emphasizes the central role of the Duffie-Kan system \cite{DK} as a model for first order approximations of a wide class of nonlinear systems perturbed by noise. We also specialize to a two dimensional framework and propose a direct proof of the Duffie-Kan theorem which does not pass through the comparison with an auxiliary process. Our proof produces a scheme to obtain an explicit representation of the solution once the one dimensional square root process is assigned.

Abstract
Tipologia del documento
Tesi di dottorato
Autore
Chuni, Vinayak
Supervisore
Dottorato di ricerca
Ciclo
32
Coordinatore
Settore disciplinare
Settore concorsuale
Parole chiave
stochastic differential equations, square root process, Feller condition, two dimensional susceptible-infected-susceptible epidemic model, Brownian motion
URN:NBN
Data di discussione
2 Aprile 2020
URI

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