A credit default swap is a contract that ensures protection against losses occurring due to a default event of an certain entity. It is crucial to know how default should be modelled for valuation or estimating of credit derivatives. In this dissertation, we first review the structural approach for modelling credit risk. The model is an approach for assessing the credit risk of a firm by typifying the firms equity as a European call option on its assets, with the strike price (or exercise price) being the promised debt repayment at the maturity. The model can be used to determine the probability that the firm will default (default probability) and the Credit Spread. We second concentrate on the valuation of credit derivatives, in particular the Credit Default Swap (CDS) when the hazard rate (or even of default) is modelled as the Vasicek-type model. The other objective is, by using South African credit spread data on defaultable bonds to estimate parameters on CIR and Vasicek-type Hazard rate models such as stochastic differential equation models of term structure. The parameters are estimated numerically by the Moment Method. / Thesis (M.Sc.)-University of KwaZulu-Natal, Durban, 2013.

This thesis focuses on matrix generalizations of Hurwitz polynomials. A real polynomial with all its roots in the open left half plane of the complex plane is called a Hurwitz polynomial. The study of these Hurwitz polynomials has a long and abundant history, which is associated with the names of Hermite, Routh, Hurwitz, Liénard, Chipart, Wall, Gantmacher et al. The direct matricial generalization of Hurwitz polynomials is naturally defined as follows: A p by p matrix polynomial F is called a Hurwitz matrix polynomial if the determinant of F is a Hurwitz polynomial. Recently, Choque Rivero followed another line of matricial extensions of the classical Hurwitz polynomial, called matrix Hurwitz type polynomials. However, the notion “matrix Hurwitz type polynomial” is still irrelative to “Hurwitz matrix polynomial” due to the totally unclear zero location of the former notion. So the main goal of this thesis is to discover the relation between the two notions “matrix Hurwitz-type polynomials” and “Hurwitz matrix polynomials' and provide some criteria to identify Hurwitz matrix polynomials. The central idea is to determine the inertia triple of matrix polynomials in terms of some related matrix sequences. Suppose that F is a p by p matrix-valued polynomial of degree n. We split F into the odd part and the even part, which allow us to introduce an essential rational matrix function of right type G. From the matrix coefficients of the Laurent series of G we construct the (n-1)-th extended sequence of right Markov parameters (SRMP) of F. Then we show that the inertia triple of F can be characterized by a combination of the inertia triples of two block Hankel matrices generated by the (n-1)-th SRMP of F and the number of zeros (counting for multiplicities) of greatest right common divisors of the even part and the odd part of F lying on the left half of the real axis. By an analogous approach we also obtain the dual results for the inertia triple of F in terms of the SLMP of F. Then we demonstrate that F is a Hurwitz matrix polynomial of degree n if and only if the (n − 1)-th SRMP (resp. SLMP) of F is a Stieltjes positive definite sequence. On this account, the two notions “Hurwitz matrix polynomials” and “matrix Hurwitz type polynomials” are equivalent. In addition, we investigate quasi-stable matrix polynomials appearing in the theory of stability, which contain Hurwitz matrix polynomials as a special case. We seek a correspondence between quasi-stable matrix polynomials, Stieltjes moment problems and multiple Nevanlinna-Pick interpolation in the Stieltjes class. Accordingly, we prove that F is a quasi-stable matrix polynomial if and only if the (n − 1)-th SRMP (resp. SLMP) of F is a Stieltjes non-negative definite extendable sequence and the zeros of right (resp. left) greatest common divisors of the even part and the odd part of F are located on the left half of the real axis.:1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Matrix polynomials and greatest common divisors. . . . . . . . . . . . . . . . . . . . . . . 7 2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Greatest common divisors of matrix polynomials . . . . . . . . . . . . . . . . . . . . . 8 3 Matrix sequences and their connection to truncated matricial moment problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4 Matrix fraction description and some related topics . . . . . . . . . . . . . . . . . . 19 4.1 Realization of Matrix fraction description from Markov parameters . . . . . . . 19 4.2 The interrelation between Hermitian transfer function matrices and monic orthogonal system of matrix polynomials . . . . . . . . . . . . . . . . . . . . . . . .27 5 The Bezoutian of matrix polynomials and the inertia problem of matrix polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33 5.2 The Anderson-Jury Bezoutian matrices in connection to special transfer function matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38 6 Para-Hermitian strictly proper transfer function matrices and their related monic Hurwitz matrix polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 7 Solution of matricial Routh-Hurwitz problems in terms of the Markov pa- rameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 8 Matrix Hurwitz type polynomials and some related topics . . . . . . . . . . . . . . 67 9 Hurwitz matrix polynomials and some related topics . . . . . . . . . . . . . . . . . . 77 9.1 Hurwitz matrix polynomials, Stieltjes positive definite sequences and matrix Hurwitz type polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 9.2 S -system of Hurwitz matrix polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 82 10 Quasi-stable matrix polynomials and some related topics . . . . . . . . . . . . 95 10.1 Particular monic quasi-stable matrix polynomials and Stieltjes moment problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .95 10.2 Particular monic quasi-stable matrix polynomials and multiple Nevanlinna- Pick interpolation in the Stieltjes class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101 10.3 General description of monic quasi-stable matrix polynomials . . . . . . . . .104 List of terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 List of notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Selbständigkeitserklärung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125