The aim of this thesis is to give a self-contained introduction to the hyperbolic geometry and the theory of discrete subgroups of PSL(2,R), and to generalize the results on equivariant functions. We show that there is a deep relation between the geometry of the group and some analytic and algebraic properties of these functions. In addition, we provide some applications of equivariant functions consisting of new results as well as providing new and simple proofs to classical results on automorphic forms.

In this thesis we investigate and compare two recently developed models of the option valuation according to the Swedish market. The first model is the Stochastic Volatility model with jumps in the stock price and the volatility (SVJJ) and the second is the Hyperbolic model. First of all we make brief introduction about the valuation of derivatives and considered models. Then we introduce methods for the estimation of parameters for each model. To solve this problem for the SVJJ model we use the Empirical Characteristic Function Estimation and for the Hyperbolic we use the Maximum Likelihood Method. Before explicit calculations (with estimated parameters) we describe the derivation of the pricing formula which is based on characteristic functions and densities. In conclusion we made numerical valuations of the call option prices for the OMXS30 index on the Swedish Stock Exchange. The main idea of this thesis is to compare 2 different models using numerical methods and the real data sets. To achieve this goal we firstly, compare the empirical characteristic function obtained from the market and the analytical ones for estimated parameters in case of both models. Secondly, we make a comparison of calculated call option prices and produce the summary.

Thesis (doctoral)--Faculté des sciences de Paris, 1913.

Thesis (Ph. D.)--University of Washington, 2002. / Vita. Includes bibliographical references (p. 165-174).

Thesis (MTech. : Mathematical Technology.)--Tshwane University of Technology, 2008.

The aim of the thesis is to study in depth a certain class of hyperbolic 3-manifolds; namely those which are the quotient of hyperbolic 3-space by an arithmetic Kleinian group. In particular we consider the distribution and characterization of arithmetic Kleinian groups in the class of all Kleinian groups of finite covolume, the Fuchsian subgroup structure and the relationship between the Fuchsian subgroups (when they exist) and the arithmetic Kleinian group. In chapter 2 a characterization of arithmetic Kleinian groups via the traces of the elements in the group is given and, appealing directly to this, in chapter 3, a set of necessary and sufficient algebraic conditions for the existence of non-elementary Fuchsian subgroups is deduced. These conditions are given an equivalent alternative description in chapter 5 from which a technique is developed making identification of the field of definition a relatively simple algebraic operation. The technique is illustrated, taking as examples the eight arithmetic tetrahedral groups of Lanner. This enables an investigation of covolumes in the commensurability class of each group. The final chapter (chapter 6) investigates geometric and topological analogues for the manifolds associated to torsion-free arithmetic Kleinian groups which contain non-elementary Fuchsian subgroups. For such manifolds we answer in the affirmative conjectures of Thurston and Waldhausen on existence of haken covers and the first betti number.

The aim of this thesis is to give a self-contained introduction to the hyperbolic geometry and the theory of discrete subgroups of PSL(2,R), and to generalize the results on equivariant functions. We show that there is a deep relation between the geometry of the group and some analytic and algebraic properties of these functions. In addition, we provide some applications of equivariant functions consisting of new results as well as providing new and simple proofs to classical results on automorphic forms.

This thesis investigates the convergence of numerical schemes for the computation of equilibrium solutions. These are solutions of evolutionary PDEs that arise from (bounded, non-decaying) boundary forcing after the dissipation of any (initial data dependent) transients. A rigorous definition of the term 'equilibrium solution' is given. Classes of evolutionary PDEs for which equilibrium solutions exist uniquely are identified. The uniform well-posedness of equilibrium problems is also investigated. Equilibrium solutions may be approximated by evolutionary initialization: that is, by finding the solution of an initial boundary value problem, with arbitrary initial data, over a period of time t ϵ [0,T]. If T is chosen large enough, the analytic transient will be small, and the analytic solution over t ϵ [T, T + T₀] will be a good approximation to the analytic equilibrium solution. However, in numerical computations, T must be chosen so that the analytical transient is small in comparison with the numerical error E_h, which depends on the fineness of the grid h. Thus T = T_h, and, in general, T_h→∞ as h→0. Convergence is required over t ϵ [T_h,T_h + T₀]. The existing Lax-Richtmyer and GKS convergence theories cannot ensure convergence over such increasing periods of time. Furthermore, neither of these theories apply when the forcing does not decay. Consequently, these theories are of little help in predicting the convergence of finite difference methods for the computation of equilibrium solutions. For these reasons, a new definition of stability - uniform stability — is proposed. Uniformly stable, consistent, finite difference schemes, for uniformly well posed problems, converge uniformly over t ≥ 0. Uniformly convergent schemes converge for bounded and nondecaying forcing. Finite difference schemes for hyperbolic PDEs may admit waves of zero group velocity, even when the underlying analytic problem does not. Such schemes may be GKS convergent, provided that the boundary conditions exclude these waves. The deficiency of the GKS theory for equilibrium computations is traced to this fact. However, uniform stability finds schemes that admit waves of zero group velocity to be (weakly) unstable, regardless of the boundary conditions. It is also shown that weak uniform instabilities are the result of time-dependent analogues of the 'spurious modes' that occur in steady-state calculations. In addition, uniform stability theory sheds new light on the phenomenon of spurious modes.

Thesis (Ph.D.)--University of Melbourne, Dept. of Mathematics and Statistics, 2005. / Typescript. Includes bibliographical references (leaves 123-126).

Thesis (Ph. D.)--Rheinische Friedrich-Wilhelms-Universität zu Bonn, 1994. / Cover title. Includes bibliographical references (p. 166-168).