We propose and investigate several new principles of rational reasoning within the framework of Pure Inductive Logic, PIL, where probability functions defined on the sentences of a first-order language are used to model an agent's beliefs. The Elephant Principle is concerned with how learning, modelled by conditioning, may be uniquely `remembered'. The Perspective Principle requires that, from a given prior, conditioning on statistically similar experiences should result in similar assignments, and is found to be a necessary condition for Reichenbach's Axiom to hold. The Abductive Inference Principle and some variations are proposed as possible formulations of a restriction of C.S. Peirce's notion of hypothesis in the context of PIL, though characterization results obtained for these principles suggest that they may be too strong. The Finite Values Property holds when a probability function takes only finitely many values when restricted to sentences containing only constant symbols from some fixed finite set. This is shown to entail a certain systematic method of assigning probabilities in terms of possible worlds, and it is considered in this light as a possible principle of inductive reasoning. Classification results are given, stating which members of certain established families of probability functions satisfy each of these new principles. Additionally, we define the theory of a principle P of PIL to be the set of those sentences which are assigned probability 1 by every probability function which satisfies P. We investigate the theory of the established principle of Spectrum Exchangeability by finding separately the theories of heterogeneous and homogeneous functions. The theory of Spectrum Exchangeability is found to be equal to the theory of finite structures. The theory of Johnson's Sufficientness Postulate is also found. Consequently, we find that Spectrum Exchangeability, Johnson's Sufficientness Postulate and the Finite Values Property are all inconsistent with the principle of Super-Regularity: that any consistent sentence should be assigned non-zero probability.

The Weibull distribution is a popular and widely used distribution in reliability and in lifetime data analysis. Since 1958, the Weibull distribution has been modified by many researchers to allow for non-monotonic hazard functions. Many modifications of the Weibull distribution have achieved the above purpose. On the other hand, the number of parameters has increased, the forms of the survival and hazard functions have become more complicated and the estimation problems have risen. This thesis provides an extensive review of some discrete and continuous versions of the modifications of the Weibull distribution, which could serve as an important reference and encourage further modifications of the Weibull distribution. Four different modifications of the Weibull distribution are proposed to address some of the above problems using different techniques. First model, with five parameters, is constructed by considering a two-component serial system with one component following a Weibull distribution and another following a modified Weibull distribution. A new method has been proposed to reduce the number of parameters of the new modified Weibull distribution from five to three parameters to simplify the distribution and address the estimation problems. The reduced version has the same desirable properties of the original distribution in spite of having two less parameters. It can be an alternative distribution for all modifications of the Weibull distribution with bathtub shaped hazard rate functions. To deal with unimodal shaped hazard rates, the third model with four parameters, named as the exponentiated reduced modified Weibull distribution is introduced. This model is flexible, has a nice physical interpretation and has the ability to capture monotonically increasing, unimodal and bathtub shaped hazard rates. It is a generalization of the reduced modified Weibull distribution. The proposed distribution gives the best fit comparing to other modifications of the Weibull distribution including those having similar properties. A three-parameter discrete distribution is introduced based on the reduced distribution. It is one of only three discrete distributions allowing for bathtub shaped hazard rate functions. Four real data sets have applied to this distribution. The new distribution is shown to outperform at least three other models including the ones allowing for bathtub shaped hazard rate functions. The new models show flexibility and can be used to model different kinds of real data sets better than other modified versions of Weibull distribution including those having the same number of parameters. The mathematical properties and statistical inferences of the new models are studied. Based on a simulation study the performances of the MLEs of each model are assessed with respect to sample size n. We find no evidence that the generalized modified Weibull distribution can provide a better fit than the exponentiated Weibull distributionfor data sets exhibiting the modified unimodal hazard function.

Piecewise smooth maps appear as models of various physical, economical and other systems. In such maps bifurcations can occur when a fixed point or periodic orbit crosses or collides with the border between two regions of smooth behaviour as a system parameter is varied. These bifurcations have little analogue in standard bifurcation theory for smooth maps and are often more complex. They are now known as "border collision bifurcations". The classification of border collision bifurcations is only available for one-dimensional maps. For two and higher dimensional piecewise smooth maps the study of border collision bifurcations is far from complete. In this thesis we investigate some of the bifurcation phenomena in two-dimensional continuous piecewise smooth discrete-time systems. There are a lot of studies and observations already done for piecewise smooth maps where the determinant of the Jacobian of the system has modulus less than 1, but relatively few consider models which allow area expansions. We show that the dynamics of systems with determinant greater than 1 is not necessarily trivial. Although instability of the systems often gives less useful numerical results, we show that snap-back repellers can exist in such unstable systems for appropriate parameter values, which makes it possible to predict the existence of chaotic solutions. This chaos is unstable because of the area expansion near the repeller, but it is in fact possible that this chaos can be part of a strange attractor. We use the idea of Markov partitions and a generalization of the affine locally eventually onto property to show that chaotic attractors can exist and are fully two-dimensional regions, rather than the usual fractal attractors with dimension less than two. We also study some of the local and global bifurcations of these attracting sets and attractors.Some observations are made, and we show that these sets are destroyed in boundary crises and some conditions are given.Finally we give an application to a coupled map system.

The aim of this thesis is a development of a new class of financial models with random parameters, which are computationally efficient and have the same level of performance as existing ones. In particular, this research is threefold. I have studied the evolution of storable commodity and commodity futures prices in time using a new random parameter model coupled with a Kalman filter. Such a combination allows one to forecast arbitrage-free futures prices and commodity spot prices one step ahead. Another direction of my research is a new volatility model, where the volatility is a random variable. The main advantage of this model is high calibration speed compared to the existing stochastic volatility models such as the Bates model or the Heston model. However, the performance of the new model is comparable to the latter. Comprehensive numerical studies demonstrate that the new model is a very competitive alternative to the Heston or the Bates model in terms of accuracy of matching option prices or computing hedging parameters. Finally, a new futures pricing model for electricity futures prices was developed. The new model has a random volatility parameter in its underlying process. The new model has less parameters, as compared to two-factor models for electricity commodity pricing with and without jumps. Numerical experiments with real data illustrate that it is quite competitive with the existing two-factor models in terms of pricing one step ahead futures prices, while being far simpler to calibrate. Further, a new heuristic for calibrating two-factor models was proposed. The new calibration procedure has two stages, offline and online. The offline stage calibrates parameters under a physical measure, while the online stage is used to calibrate the risk-neutrality parameters on each iteration of the particle filter. A particle filter was used to estimate the values of the underlying stochastic processes and to forecast futures prices one step ahead. The contributory material from two chapters of this thesis have been submitted to peer reviewed journals in terms of two papers: • Chapter 4: “A fast calibrating volatility model” has been submitted to the European Journal of Operational Research. • Chapter 5: “Electricity futures price models : calibration and forecasting” has been submitted to the European Journal of Operational Research.

The signature of the path provides a top down description of a path in terms of its eects as a control. It is a group-like element in the tensor algebra and is an essential object in rough path theory. When the path is random, the linear independence of the signatures of different paths leads one to expect, and it has been proved in simple cases, that the expected signature would capture the complete law of this random variable. It becomes of great interest to be able to compute examples of expected signatures. In this thesis, we aim to compute the expected signature of various stochastic process solved by a PDE approach. We consider the case for an Ito diffusion process up to a fixed time, and the case for the Brownian motion up to the first exit time from a domain. We manage to derive the PDE of the expected signature for both cases, and find that this PDE system could be solved recursively. Some specific examples are included herein as well, e.g. Ornstein-Uhlenbeck (OU) processes, Brownian motion and Levy area coupled with Brownian motion.

We propose two alternative approaches to evaluate numerically credit basket derivatives in a N-name structural model where the number of entities, N, is large, and where the names are independent and identically distributed random variables conditional on common random factors. In the first framework, we treat a N-name model as a set of N Bernoulli random variables indicating a default or a survival. We show that certain expected functionals of the proportion L_N of variables in a given state converge at rate 1/N as N [right arrow - infinity]. Based on these results, we propose a multi-level simulation algorithm using a family of sequences with increasing length, to obtain estimators for these expected functionals with a mean-square error of epsilon ² and computational complexity of order epsilon⁻², independent of N. In particular, this optimal complexity order also holds for the infinite-dimensional limit. Numerical examples are presented for tranche spreads of basket credit derivatives. In the second framework, we extend the approximation of Bush et al. [13] to a structural jump-diffusion model with discretely monitored defaults. Under this approach, a N-name model is represented as a system of particles with an absorbing boundary that is active in a discrete time set, and the loss of a portfolio is given as the function of empirical measure of the system. We show that, for the infinite system, the empirical measure has a density with respect to the Lebesgue measure that satisfies a stochastic partial differential equation. Then, we develop an algorithm to efficiently estimate CDO index and tranche spreads consistent with underlying credit default swaps, using a finite difference simulation for the resulting SPDE. We verify the validity of this approximation numerically by comparison with results obtained by direct Monte Carlo simulation of the basket constituents. A calibration exercise assesses the flexibility of the model and its extensions to match CDO spreads from precrisis and crisis periods.

New view synthesis (NVS) is the problem of generating a novel image of a scene, given a set of calibrated input images of the scene, i.e. their viewpoints, and also that of the output image, are known. The problem is generally ill-posed---a large number of scenes can generate a given set of images, therefore there may be many equally likely (given the input data) output views. Some of these views will look less natural to a human observer than others, so prior knowledge of natural scenes is required to ensure that the result is visually plausible. The aim of this thesis is to compare and improve upon the various Markov random field} and conditional random field prior models, and their associated maximum a posteriori optimization frameworks, that are currently the state of the art for NVS and stereo (itself a means to NVS). A hierarchical example-based image prior is introduced which, when combined with a multi-resolution framework, accelerates inference by an order of magnitude, whilst also improving the quality of rendering. A parametric image prior is tested using a number of novel discrete optimization algorithms. This general prior is found to be less well suited to the NVS problem than sequence-specific priors, generating two forms of undesirable artifact, which are discussed. A novel pairwise clique image prior is developed, allowing inference using powerful optimizers. The prior is shown to perform better than a range of other pairwise image priors, distinguishing as it does between natural and artificial texture discontinuities. A dense stereo algorithm with geometrical occlusion model is converted to the task of NVS. In doing so, a number of challenges are novelly addressed; in particular, the new pairwise image prior is employed to align depth discontinuities with genuine texture edges in the output image. The resulting joint prior over smoothness and texture is shown to produce cutting edge rendering performance. Finally, a powerful new inference framework for stereo that allows the tractable optimization of second order smoothness priors is introduced. The second order priors are shown to improve reconstruction over first order priors in a number of situations.

The main contribution of the present thesis is in two aspects. The first one, which is the heart of the thesis, is to explore the fundamental relation between rough paths and their signatures. Our main goal is to give a geometric characterization of the kernel of the signature map in different situations. In Chapter Two, we start by establishing a general fact that a continuous Jordan curve on a Riemannian manifold can be arbitrarily well approximated by piecewise minimizing geodesic interpolations which are again Jordan. This result enables us to prove a generalized version of Green’s theorem for planar Jordan curves with finite p-variation 1 &le; p < 2, and to prove that two such Jordan curves have the same signature if and only if they are equal up to reparametrization. In Chapter Three, we investigate the problem for general weakly geometric rough paths. In particular, we show that a weakly geometric rough path has trivial signature if and only if it is tree-like in the sense we will define later on. In Chapter Four, we study the problem in the probabilistic setting. In particular, we show that for a class of stochastic processes, with probability one the sample paths are determined by their signatures up to reparametrization. A fundamental example is Gaussian processes including fractional Brownian motion with Hurst parameter H > 1/4, the Ornstein-Uhlenbeck process and the Brownian bridge. The second one is an application of rough path theory to the study of nonlinear diffusions on manifolds under the framework of nonlinear expectations. In Chapter Five, we begin by studying the geometric rough path nature of G-Brownian motion. This enables us to introduce rough differential equations driven by G-Brownian motion from a pathwise point of view. Next we establish the fundamental relation between rough (pathwise theory) and stochastic (L²-theory) differential equations driven by G-Brownian motion. This is a crucial point of understanding nonlinear diffusions and their generating heat flows on manifolds from an intrinsic point of view. Finally, from the pathwise point of view we construct G-Brownian motion on a compact Riemannian manifold and establish its generating heat flow for a class of G-functions under orthogonal invariance. As an independent interest, we also develop the Euler-Maruyama scheme for stochastic differential equations driven by G-Brownian motion.

In this work we introduce a novel queueing model with two classes of users in which, instead of accessing a resource, users wait in the system to match with a candidate from the other class. The users are selective and the matchings occur probabilistically. This new model is useful for analysing the traffic in web portals that match people who provide a service with people who demand the same service, e.g. employment portals, matrimonial and dating sites and rental portals. We first provide a Markov chain model for these systems and derive the probability distribution of the number of matches up to some finite time given the number of arrivals. We then prove that if no control mechanism is employed these systems are unstable for any set of parameters. We suggest four different classes of control policies to assure stability and conduct analysis on performance measures under the control policies. Contrary to the intuition that the rejection rate should decrease as the users become more likely to be matched, we show that for certain control policies the rejection rate is insensitive to the matching probability. Even more surprisingly, we show that for reasonable policies the rejection rate may be an increasing function of the matching probability. We also prove insensitivity results related to the average queue lengths and waiting times. Further, to gain more insight into the behaviour of probabilistic matching systems, we propose approximation methods based on fluid and diffusion limits using different scalings. We analyse the basic properties of these approximations and show that some performance measures are insensitive to the matching probability agreeing with the results found by the exact analysis. Finally we study the optimal control and revenue management for the systems with the objective of profit maximization. We formulate mathematical models for both unobservable and observable systems. For an unobservable system we suggest a deterministic optimal control, while for an observable system we develop an optimal myopic state dependent pricing.

In this thesis we investigate the role of odd Poisson brackets in related areas of supergeometry. In particular we study three different cases of their appearance: Courant algebroids and their homotopy analogues, weak Poisson structures and their relation to foliated manifolds, and the structure of odd Poisson manifolds and their modular class. In chapter 2 we introduce the notion of a homotopy Courant algebroid, a subclass of which is suggested to stand as the double objects to L-bialgebroids. We provide explicit formula for the higher homotopy Dorfman brackets introduced in this case, and the higher relations between these and the anchor maps. The homotopy Loday structure is investigated, and we begin a discussion of what other constructions in the theory of Courant algebroids can be carried out in this homotopy setting. Chapter 3 is devoted to lifting a weak Poisson structure corresponding to a local foliation of a submanifold to a weak Koszul bracket, and interpreting the results in terms of the cohomology of an associated differential. This bracket is shown to produce a bracket on co-exact differential forms. In chapter 5 studies classes of second order differential operators acting on semidensities on an arbitrary supermanifold. In particular, when the supermanifold is odd Poisson, we given an explicit description of the modular class of the odd Poisson manifold, and provide the first non-trivial examples of such a class. We also introduce the potential field of a general odd Laplacian, and discuss its relation to the geometry of the odd Poisson manifold and its status as a connection-like object.