Multi-Point Pade Approximation And Its Applications In Probability Theory And Finance

Abstract

This thesis proposes a computational framework for constructing and applying multi-point Padé approximations to study Stieltjes functions, distribution functions including generalized gamma convolutions (GGCs), and other special functions through their Laplace transforms. We establish algorithmic foundations for multi-point Padé approximation via both matrix-based and continued-fraction approaches, with particular emphasis on an efficient reduced-parameter continued-fraction scheme. Numerical implementations are performed using high-precision arithmetic to ensure the accuracy of results. The practical applicability of the method is demonstrated through a range of examples, including the approximation of Gaussian, lognormal, and gamma densities, as well as completely monotone functions, hockey-stick functions, and unit step functions by sums of exponentials. As a consequence, we introduce a new approach for approximating cumulative distribution functions (cdfs) through the Laplace transform of the underlying distribution. We further propose approximation methods for several well-known risk measures widely used in insurance and finance, based on the approximations of hockey-stick and unit step functions. The examples include expected shortfall, individual economic capital allocation, tail standard deviation and expectiles. The thesis also investigates dependence structures of GGCs through their lower and upper tail dependence coefficients, and finally, we present a new Laplace inversion relation for GGCs.