Infinite divisibility of interpolated gamma powers and variance

: GGC financial model

Abstract

This thesis includes three main chapters–Chapter 2,3,4. Chapter 2 is stochastic analysis. It is concerned with the distribution properties of the binomial aX + bXα, where X is a Gamma random variable. We show in particular that aX + bXα is infinitely divisible for all α ∈ [1, 2] and a, b ∈ R+, and that for α = 2 the second order polynomial aX + bX2 is a generalized Gamma convolution whose Thorin density and Wiener-Gamma integral representation are computed explicitly. As a byproduct we show that fourth order multiple Wiener integrals are in general not infinitely divisible. Chapter 3 is financial modeling. This chapter introduces the variance-GGC model which is an extension of variance-Gamma model. First we calculate the skewness and kurtosis of the variance-GGC model and get the relation of the skewness and kurtosis between GGC processes and corresponding variance-GGC processes. Then the decomposition property of variance-GGC processes is proved. At last, sensitivity analysis of this model has been conducted. In Chapter 4, our goal is to relax a sufficient condition for the exponential almost sure stability of a certain class of stochastic differential equations. Compare to the existing theory, we prove the almost sure stability, replacing Lipschitz continuity and linear growth conditions by the existence of a strong solution of the underlying stochastic differential equation. This result is extendable for the regime-switching system. An explicit example is provided for the illustration purpose.

Date of Award	2 Oct 2013
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Qingshuo SONG (Supervisor), Kam Moon Lester LIU (Supervisor) & Nicolas PRIVAULT (Supervisor)