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.