Athanasios A. Pantelous

Modeling Earthquake Risk via Extreme Value Theory and Pricing the Respective Catastrophe Bonds

The aim of the paper is twofold. Firstly, to analyze the historical data of the earthquakes in the boarder area of Greece and then to produce a reliable model for the risk dynamics of the magnitude of the earthquakes, using advanced techniques from the Extreme Value Theory. Secondly, to discuss briefly the relevant theory of incomplete markets and price earthquake catastrophe bonds, combining the model found for the earthquake risk and an appropriate model for the interest rate dynamics in an incomplete market framework. The paper ends by providing some numerical results using Monte Carlo simulation techniques and stochastic iterative equations.

Statistical linearization of nonlinear structural systems with singular matrices

AbstractA generalized statistical linearization technique is developed for determining approximately the stochastic response of nonlinear dynamic systems with singular matrices. This system modeling can arise when a greater than the minimum number of coordinates is utilized, and can be advantageous, for instance, in cases of complex multibody systems where the explicit formulation of the equations of motion can be a nontrivial task. In such cases, the introduction of additional/redundant degrees of freedom can facilitate the formulation of the equations of motion in a less labor-intensive manner. Specifically, relying on the generalized matrix inverse theory and on the Moore-Penrose (M-P) matrix inverse, a family of optimal and response-dependent equivalent linear matrices is derived. This set of equations in conjunction with a generalized excitation-response relationship for linear systems leads to an iterative determination of the system response mean vector and covariance matrix. Further, it is proved ...

A high order finite element scheme for pricing options under regime switching jump diffusion processes

This paper considers the numerical pricing of European, American and Butterfly options whose asset price dynamics follow the regime switching jump diffusion process. In an incomplete market structure and using the no-arbitrage pricing principle, the option pricing problem under the jump modulated regime switching process is formulated as a set of coupled partial integro-differential equations describing different states of a Markov chain. We develop efficient numerical algorithms to approximate the spatial terms of the option pricing equations using linear and quadratic basis polynomial approximations and solve the resulting initial value problem using exponential time integration. Various numerical examples are given to demonstrate the superiority of our computational scheme with higher level of accuracy and faster convergence compared to existing methods for pricing options under the regime switching model.

Linear Backward Stochastic Differential Equations of Descriptor Type: Regular Systems

In this article, a class of linear backward stochastic differential equations of descriptor type with time-invariant coefficients are introduced. Necessary and sufficient conditions for their solvability are obtained. It turns out that such equations may not always have a solution, and even when they do, some components of the solution could have a jump at terminal time. Exact controllability of linear descriptor stochastic control systems is also considered.

On the solution of higher order linear homogeneous complex σ–α descriptor matrix differential systems of Apostol–Kolodner type

Abstract In this paper, the solution of higher order linear homogeneous complex σ–α descriptor matrix differential systems of Apostol–Kolodner type is investigated by considering pairs of complex matrices with symmetric and skew symmetric structural properties. The results are very general, and they derive under congruence of the Thompson canonical form. The regularity (or singularity) of a matrix pencil pre-determines the number of sub-systems respectively. The special structure of these kinds of systems derives from applications in engineering, physical sciences and economics. A numerical example illustrates the main findings of the paper.

Linear Random Vibration of Structural Systems with Singular Matrices

AbstractA framework is developed for determining the stochastic response of linear multi-degree-of-freedom (MDOF) structural systems with singular matrices. This system modeling can arise when using more than the minimum number of coordinates, and can be advantageous, for instance, in cases of complex multibody systems whose dynamics satisfy a number of constraints. In such cases the explicit formulation of the equations of motion can be a nontrivial task, whereas the introduction of additional/redundant degrees of freedom can facilitate the formulation of the equations of motion in a less labor-intensive manner. Relying on the generalized matrix inverse theory and on the Moore-Penrose (M-P) matrix inverse, standard concepts, relationships, and equations of the linear random vibration theory are extended and generalized herein to account for systems with singular matrices. Adopting a state-variable formulation, equations governing the system response mean vector and covariance matrix are formed and solved...

Bonus–Malus systems with Weibull distributed claim severities

Abstract One of the pricing strategies for Bonus–Malus (BM) systems relies on the decomposition of the claims’ randomness into one part accounting for claims’ frequency and the other part for claims’ severity. By mixing an exponential with a Lévy distribution, we focus on modelling the claim severity component as a Weibull distribution. For a Negative Binomial number of claims, we employ the Bayesian approach to derive the BM premiums for Weibull severities. We then conclude by comparing our explicit formulas and numerical results with those for Pareto severities that were introduced by Frangos & Vrontos.

Discretizing LTI Descriptor (Regular) Differential Input Systems with Consistent Initial Conditions

A technique for discretizing efficiently the solution of a Linear descriptor (regular) differential input system with consistent initial conditions, and Time-Invariant coefficients (LTI) is introduced and fully discussed. Additionally, an upper bound for the error that derives from the procedure of discretization is also provided. Practically speaking, we are interested in such kind of systems, since they are inherent in many physical, economical and engineering phenomena.

The Weierstrass Canonical Form of a Regular Matrix Pencil: Numerical Issues and Computational Techniques

In the present paper, we study the derivation of the Weierstrass Canonical Form (WCF) of a regular matrix pencil. In order to compute the WCF, we use two important computational tools: a) the QZ algorithm to specify the required root range of the pencil and b) the updating technique to compute the index of annihilation. The proposed updating technique takes advantages of the already computed rank of the sequences of matrices that appears during our procedure reducing significantly the required floating-point operations. The algorithm is implemented in a numerical stable manner, giving efficient results. Error analysis and the required complexity of the algorithm are included.

Normalizability and feedback stabilization for second-order linear descriptor differential systems

ln this paper we investigate the normalizability of second-order linear descriptor differential system and simultaneously the relocation of its poles. The whole procedure is divided into three main algorithmic steps. Firstly, we normalize the descriptor system. Afterwards, we solve a linear and a multi-linear sub-problem. The proposed method computes a reduced set of quadratic Plucker relations which describes completely the specific Grassmann variety. Finally, using these relations the feedback matrices are fully determined which provide the solution to the pole assignment problem. An illustrative example of the algorithmic procedure is also discussed.