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.

Characterization of TRAF6 mediated ubiquitination of presenilins and γ-secretase substrates

Post-translational modification of the γ-secretase protease complexes and their substrates has an important role in controlling receptor-initiated signalling events, which are critically important in the pathogenesis of cancer, inflammatory and Alzheimer’s disease. Our lab has previously characterised an interaction between TRAF6 and presenilin-1, which lead to the identification of interleukin-1 (IL-1) receptor type 1 (IL-1R1) and Toll-like receptor-4 (TLR4) as novel γ-secretase substrates. Subsequently our group showed that TRAF6 promoted ubiquitination and γ-secretase cleavage of IL-1R1. The aim of this project is to study the association between TRAF6 and the presenilins, the critical γ-secretase complex components, and to determine the functional importance of TRAF6-mediated ubiquitination of γ-secretase substrates. Firstly, we show that the full-length presenilins are novel substrates of TRAF6-mediated Lysine-63-linked polyubiquitination. Secondly, we show that co-expression of TRAF6 and the presenilins increases the stability and alters the turnover of the presenilins. Thirdly, we reveal that TRAF6-mediated ubiquitination of presenilin does not affect γ-secretase enzyme activity, but may regulate the full-length presenilin functions such as ER Ca2+ signalling. Previously, we have reported IL-1R1 as a novel substrate of TRAF6-mediated ubiquitination. In this study, we identified five lysine residues in the IL-1R1 intracellular domain targeted by TRAF6-mediated polyubiquitination. Furthermore, mutagenesis of these five lysine residues led to decreased IL-1R1 cell surface expression, precluded the ectodomain shedding and attenuated the responsiveness to IL-1β stimulation, demonstrating the critical role of TRAF6 in IL-1R1 trafficking.

Analysing energy detector diversity receivers for spectrum sensing

The analysis of energy detector systems is a well studied topic in the literature: numerous models have been derived describing the behaviour of single and multiple antenna architectures operating in a variety of radio environments. However, in many cases of interest, these models are not in a closed form and so their evaluation requires the use of numerical methods. In general, these are computationally expensive, which can cause difficulties in certain scenarios, such as in the optimisation of device parameters on low cost hardware. The problem becomes acute in situations where the signal to noise ratio is small and reliable detection is to be ensured or where the number of samples of the received signal is large. Furthermore, due to the analytic complexity of the models, further insight into the behaviour of various system parameters of interest is not readily apparent. In this thesis, an approximation based approach is taken towards the analysis of such systems. By focusing on the situations where exact analyses become complicated, and making a small number of astute simplifications to the underlying mathematical models, it is possible to derive novel, accurate and compact descriptions of system behaviour. Approximations are derived for the analysis of energy detectors with single and multiple antennae operating on additive white Gaussian noise (AWGN) and independent and identically distributed Rayleigh, Nakagami-m and Rice channels; in the multiple antenna case, approximations are derived for systems with maximal ratio combiner (MRC), equal gain combiner (EGC) and square law combiner (SLC) diversity. In each case, error bounds are derived describing the maximum error resulting from the use of the approximations. In addition, it is demonstrated that the derived approximations require fewer computations of simple functions than any of the exact models available in the literature. Consequently, the regions of applicability of the approximations directly complement the regions of applicability of the available exact models. Further novel approximations for other system parameters of interest, such as sample complexity, minimum detectable signal to noise ratio and diversity gain, are also derived. In the course of the analysis, a novel theorem describing the convergence of the chi square, noncentral chi square and gamma distributions towards the normal distribution is derived. The theorem describes a tight upper bound on the error resulting from the application of the central limit theorem to random variables of the aforementioned distributions and gives a much better description of the resulting error than existing Berry-Esseen type bounds. A second novel theorem, providing an upper bound on the maximum error resulting from the use of the central limit theorem to approximate the noncentral chi square distribution where the noncentrality parameter is a multiple of the number of degrees of freedom, is also derived.

Molecular mechanisms underpinning IFN-gamma and TNF-alpha synergy in intestinal epithelial cells

Cytokine-driven signalling shapes immune homeostasis and guides inflammatory responses mainly through induction of specific gene expression programmes both within and outside the immune cell compartment. These transcriptional outputs are often amplified via cytokine synergy, which sets a stimulatory threshold that safeguards from exacerbated inflammation and immunopathology. In this study, we investigated the molecular mechanisms underpinning synergy between two pivotal Th1 cytokines, IFN-γ and TNF-α, in human intestinal epithelial cells. These two proinflammatory mediators induce a unique state of signalling and transcriptional synergy implicated in processes such as antiviral and antitumour immunity, intestinal barrier and pancreatic β-cell dysfunction. Since its discovery more than 30 years ago, this biological phenomenon remains, however, only partially defined. Here, using a functional genomics approach including RNAi perturbation screens and small-molecule inhibitors, we identified two new regulators of IFN-γ/TNF-α-induced chemokine and antiviral gene and protein expression, a Bcl-2 protein BCL-G and a histone demethylase UTX. We also discovered that IFN-γ/TNF-α synergise to trigger a coordinated shutdown of major receptor tyrosine kinases expression in colon cancer cells. Together, these findings extend our current understanding of how IFN-γ/TNF-α synergy elicits qualitatively and quantitatively distinct outputs in the intestinal epithelium. Given the well-documented role of this synergistic state in immunopathology of various disorders, our results may help to inform the identification of high quality and biologically relevant druggable targets for diseases characterised by an IFN-γ/TNF-α high immune signature