Tudor ‘reform’ treatises and government policy in sixteenth century Ireland

During the sixteenth century hundreds of treatises, position papers and memoranda were composed on the political state of Ireland and how best to ‘reform’, ‘conquer’ or otherwise incorporate that island into the wider Tudor kingdom. These ‘reform’ treatises attempted to identify and analyse the prevailing political, social, cultural and economic problems found in the Irish polity before positing how government policy could be altered to ameliorate these same problems. Written by a broad array of New English, Old English and Gaelic Irish authors, often serving within Irish officialdom, the military, or the Church of Ireland, these papers were generally circulated amongst senior ministers and political figures throughout the Tudor dominions. As such they were written with the express purpose of influencing the direction of government policy for Ireland. Collectively these documents are one of the most significant body of sources, not just for the study of government activity in the second Tudor kingdom, but indeed for the broader history of sixteenth century Ireland. This thesis offers the first systematic study of these texts. It does so by exploring the content of the hundreds of such works and the ‘reform’ treatise as a type of text, while the interrelationship of these documents with government policy in Tudor Ireland, and their effect thereon, is also explored. In so doing it charts the developments from origin to implementation of the principal strategies employed by Tudor Englishmen to enforce English control over the whole of Ireland. Finally, it clearly demonstrates that the ‘reform’ treatises were both central to government activity in sixteenth century Ireland and to the historical developments which occurred in that time and place.

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.