On SDE systems with non-Lipschitz diffusion coefficients

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.