Financial modelling with 2-EPT probability density functions

The class of all Exponential-Polynomial-Trigonometric (EPT) functions is classical and equal to the Euler-d’Alembert class of solutions of linear differential equations with constant coefficients. The class of non-negative EPT functions defined on [0;1) was discussed in Hanzon and Holland (2010) of which EPT probability density functions are an important subclass. EPT functions can be represented as ceAxb, where A is a square matrix, b a column vector and c a row vector where the triple (A; b; c) is the minimal realization of the EPT function. The minimal triple is only unique up to a basis transformation. Here the class of 2-EPT probability density functions on R is defined and shown to be closed under a variety of operations. The class is also generalised to include mixtures with the pointmass at zero. This class coincides with the class of probability density functions with rational characteristic functions. It is illustrated that the Variance Gamma density is a 2-EPT density under a parameter restriction. A discrete 2-EPT process is a process which has stochastically independent 2-EPT random variables as increments. It is shown that the distribution of the minimum and maximum of such a process is an EPT density mixed with a pointmass at zero. The Laplace Transform of these distributions correspond to the discrete time Wiener-Hopf factors of the discrete time 2-EPT process. A distribution of daily log-returns, observed over the period 1931-2011 from a prominent US index, is approximated with a 2-EPT density function. Without the non-negativity condition, it is illustrated how this problem is transformed into a discrete time rational approximation problem. The rational approximation software RARL2 is used to carry out this approximation. The non-negativity constraint is then imposed via a convex optimisation procedure after the unconstrained approximation. Sufficient and necessary conditions are derived to characterise infinitely divisible EPT and 2-EPT functions. Infinitely divisible 2-EPT density functions generate 2-EPT Lévy processes. An assets log returns can be modelled as a 2-EPT Lévy process. Closed form pricing formulae are then derived for European Options with specific times to maturity. Formulae for discretely monitored Lookback Options and 2-Period Bermudan Options are also provided. Certain Greeks, including Delta and Gamma, of these options are also computed analytically. MATLAB scripts are provided for calculations involving 2-EPT functions. Numerical option pricing examples illustrate the effectiveness of the 2-EPT approach to financial modelling.

Effect of road quality in structural health monitoring under operational conditions

The effect of unevenness in a bridge deck for the purpose of Structural Health Monitoring (SHM) under operational conditions is studied in this paper. The moving vehicle is modelled as a single degree of freedom system traversing the damaged beam at a constant speed. The bridge is modelled as an Euler-Bernoulli beam with a breathing crack, simply supported at both ends. The breathing crack is treated as a nonlinear system with bilinear stiffness characteristics related to the opening and closing of crack. The unevenness in the bridge deck considered is modelled using road classification according to ISO 8606:1995(E). Numerical simulations are conducted considering the effects of changing road surface classes from class A - very good to class E - very poor. Cumulant based statistical parameters, based on a new algorithm are computed on stochastic responses of the damaged beam due to passages of the load in order to calibrate the damage. Possibilities of damage detection and calibration under benchmarked and non-benchmarked cases are considered. The findings of this paper are important for establishing the expectations from different types of road roughness on a bridge for damage detection purposes using bridge-vehicle interaction where the bridge does not need to be closed for monitoring.