Approximations in actuarial and financial mathematics

Abstract

This thesis consists of three topics that are related to approximations. We first investigate the accuracy of Taylor polynomials in approximating utility functions. We show that increasing the polynomial order does not necessarily improve the approximation of the expected utility. The proofs use methods from the theory of parabolic second-order partial differential equations. In the second part of the thesis, we aim to analyse the spread process in a short time in the SIS epidemic model of computer networks. We show that the short-time asymptotics of infection probability depends on the network structure. We use concepts and methods from graph theory and defined in this chapter. In the third part of the thesis, we propose a single-network-based algorithm using deep learning techniques where neural network is used to approximate the derivatives of a function. We provide computational underpinnings for applying the replacement closeout convention in the valuation of a defaultable financial claim with counterparty credit risk. Numerical examples illustrate all results in the three parts.