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<p> In 1988, Mayer published a strong form of Vaught's Conjecture for o-minimal theories. She showed Vaught's Conjecture holds, and characterized the number of countable models of an o-minimal theory <i>T</i> if <i> T</i> has fewer than continuum many countable models. Friedman and Stanley have shown that several elementary classes are Borel complete. In this thesis we address the class of countable models of an o-minimal theory <i>T </i> when <i>T</i> has continuum many countable models. Our main result gives a model theoretic dichotomy describing the Borel complexity of isomorphism on the class of countable models of <i>T</i>. The first case is if <i>T</i> has no simple types, isomorphism is Borel on the class of countable models of <i>T</i>. In the second case, <i> T</i> has a simple type over a finite set <i>A</i>, and there is a finite set <i>B</i> containing <i>A</i> such that the class of countable models of the completion of <i>T </i>over <i> B</i> is Borel complete.</p>

Graduate

This thesis studies pricing and hedging barrier and other exotic options in continuous stochastic volatility models. Classical put-call symmetry relates the price of puts and calls under a suitable dual market transform. One well-known application is the semi-static hedging of path-dependent barrier options with European options. This, however, in its classical form requires the price process to observe rather stringent and unrealistic symmetry properties. In this thesis, we provide a general self-duality theorem to develop pricing and hedging schemes for barrier options in stochastic volatility models with correlation. A decomposition formula for pricing barrier options is then derived by Ito calculus which provides an alternative approach rather than solving a partial differential equation problem. Simulation on the performance is provided. In the last part of the thesis, via a version of the reflection principle by Desire Andre, originally proved for Brownian motion, we study its application to the pricing of exotic options in a stochastic volatility context.

In this thesis, we study Parisian excursions, which are defined as excursions of Brownian motion above or below a pre-determined barrier, exceeding a certain time length. Employing a new method, a recursion formula for the densities of single barrier and double barrier Parisian stopping times are computed. This new approach allows us to obtain a semi-closed form solution for the density of the one-sided stopping times, and does not require any numerical inversions of Laplace transforms. Further, it is backed by an intuitive argument which is premised on the recursive nature of the excursions and the strong Markov property of the Brownian motion. The same method is also employed in our computation of the two-sided and the double barrier Parisian stopping times. In turn, the resultant densities are used to price Parisian options. In particular, we provide numerical expressions for down-and-in Parisian calls. Additionally, we study the tail of the distribution of the two-sided Parisian stopping time. Based on the asymptotic properties of its distribution, we propose an approximation for the option prices, alleviating the heavy computational load arising from the recursions. Finally, we use the infinitesimal generator to obtain several results on other variations of Parisian excursions. Specifically, apart from the length, we are interested in the number of excursions and the maximum height achieved during an excursion. Using the same generator, we derive the joint Laplace transform of the occupation times of the Brownian motion above and below zero, but only starting the clock each time after a certain length.

Constraint logic programming (CLP) techniques can be used in protein side-chain placement, an important sub-task in comparative modelling. In a simple formulation values for domain variables represent rotamer side-chain conformations, and constraints represent atomic clashes. These constraints can be visualised using a "rotamer contact map", and observations made with this visualisation tool have been used to develop a strategy that overcomes limitations present in CLP caused by over-constrained residues. Null rotamers provide a mechanism that can automatically identify over-constrained residues. The use of null rotamers makes possible an iterative modelling strategy where, at each iteration, a CLP program is generated automatically; each program representing successively tighter packing constraints corresponding to larger atomic radii. Different CLP enumeration heuristics have been evaluated for use with this side-chain placement method, and it has been tested with several different rotamer libraries; a backbone-dependent rotamer library, when used with first-fail enumeration heuristics, was shown to be the most successful. Side-chain conformations predicted by this CLP method compare favourably against those predicted using other side-chain placement methods. The CLP method has been applied to two modelling problems. The first involved building models of class II MHC molecules in order to increase the utility of a peptide threading program. This program uses an allele's known or modelled 3D structure with a heuristic scoring function to predict peptides that are likely to bind to it - thus using CLP to model class II MHC alleles increases the program's utility. The second application used the CLP method to build structures of ribosome inactivating proteins (RIPs). These models were built using CLP together with comparative modelling approaches, and a model of bouganin, a recently identified wild RIF protein, has been built to help design engineered therapeutic proteins.

This thesis is concerned with the project which J.N. Findlay has called absolute-theory (see his book <i>Ascent to the Absolute</i> London: George Allen and Unwin 1970, and his article 'Bradley's contribution to Absolute-theory' in <i>The Philosophy of F.H.Bradley</i> edited by Guy Stock and Anthony Manser, Oxford: Clarendon 1984). In absolute-theory one is concerned with (1) determining the abstract or formal characteristics, the form as it were, of the fundamental existent or existents on which all things depend, and (ii) evaluating the candidates that might be said to fit the abstract form. If there are a plurality of fundamental existents then the form is a universal with many instances. If there is only one fundamental existent, one primordial object, then the form itself is particular in the sense that it is not instantiable by more than one thing. The background issue is monism versus pluralism. In chapter 1 first some of the characteristics that go to make up the form are delineated. Then some candidates are briefly evaluated and rejected. Finally the kind of candidate absolute idealists offer in general is elucidated to some degree giving the authors preferred formulation. The self-differentiated Substance-Person. The rest of the thesis is concerned with elucidating this conception as a candidate and arguing for it. Through this the formal characteristics of the Absolute are also considered. The author has tried to formulate an independent and original position within the general tradition of absolute idealism. In chapter 2 the concept of substance and of ultimate substance is elucidated and argued for. In chapter 3 the substance-attribute distinction is utilized to construct an original dilemma and it is argued that the solution lies in the conception of the ultimate substance. The dilemma is this: All attributes are either essential or accidental to their substances. If interaction between substances is at the level of essential attributes loss of identity occurs. If it is at the level of accidental attributes knowledge of the real nature of other substances remains ever elusive. In chapter 4 it is argued that the ultimate substance is the source of all meaning and truth. In chapter 5 it is argued that the ultimate substance is also a self-differentiated Person. This is basically the idea that there is One Person embodied in all brain-bodies.

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