In this thesis we investigate the convergence of various quantum random walks to quantum stochastic cocycles defined on a Bosonic Fock space. We prove a quantum analogue of the Donsker invariance principle by invoking the so-called semigroup representation of quantum stochastic cocycles. In contrast to similar results by other authors our proof is relatively elementary. We also show convergence of products of ampliated random walks with different system algebras; in particular, we give a sufficient condition to obtain a cocycle via products of cocycles. The CCR algebra, its quasifree representations and the corresponding quasifree stochastic calculus are also described. In particular, we study in detail gauge-invariant and squeezed quasifree states. We describe repeated quantum interactions between a `small' quantum system and an environment consisting of an infinite chain of particles. We study different cases of interaction, in particular those which occur in weak coupling limits and low density limits. Under different choices of scaling of the interaction part we show that random walks, which are generated by the associated unitary evolutions of a repeated interaction system, strongly converge to unitary quantum stochastic cocycles. We provide necessary and sufficient conditions for such convergence. Furthermore, under repeated quantum interactions, we consider the situation of an infinite chain of identical particles where each particle is in an arbitrary faithful normal state. This includes the case of thermal Gibbs states. We show that the corresponding random walks converge strongly to unitary cocycles for which the driving noises depend on the state of the incoming particles. We also use conditional expectations to obtain a simple condition, at the level of generators, which suffices for the convergence of the associated random walks. Limit cocycles, for which noises depend on the state of the incoming particles, are also obtained by investigating what we refer to as `compressed' random walks. Lastly, we show that the cocycles obtained via the procedure of repeated quantum interactions are quasifree, thus the driving noises form a representation of the relevant CCR algebra. Both gauge-invariant and squeezed representations are shown to occur.

This thesis addresses several challenging problems in low-dimensional systems, which have rarely or never been studied using quantum Monte Carlo methods. It begins with an investigation into weak van der Waals-like interactions in bilayer graphene and extends to graphene placed on top of boron nitride at four different stacking configurations. The in-plane optical phonon frequencies for the latter heterostructure as well as the out-of-plane phonon frequencies for both structures are calculated. We find that the binding energies (BEs) of these structures are almost within the same range and are less than 20 meV/atom. Although the phonon vibrations are comparable within both the diffusion quantum Monte Carlo (DMC) method and density functional theory (DFT), DFT gives quantitatively wrong BEs for vdW structures. Next, the BEs of 2D biexcitons are studied at different mass ratios and a variety of screening lengths. Our exact DMC results show that the BEs of biexcitons in different kinds of transition-metal dichalcogenides are in the range 15 − 30 meV bound at room temperature. Besides 2D systems, the electronic properties of 1D hydrogen-terminated oligoynes and polyyne are studied by calculating their DMC quasiparticle and excitonic gaps. By minimising the DMC energy of free-standing polyyne with respect to the lattice constant and the bond-length alternation, DMC predicts geometry in agreement with that obtained by accurate quantum chemistry methods. The DMC longitudinal optical phonon is within the range of experimental values. Our results confirm that DMC is capable of accurately describing Peierls-distorted materials.

In this thesis, Mean-Variance Asset-Liability management is studied ina multi-period setting. An investor aims at nding an optimal investmentstrategy in order to maximise the mean-variance objective. The prices ofassets and liabilities are formulated as geometric Brownian motions and wefurther extend them to exponential Levy process. By the Bellman principle,the explicit optimal solution is obtained under backward induction.

In this thesis, we consider radial basis function interpolations in d-dimensional Euclidean space Hd and the unit sphere 5d_1, where the data is generated not only by point-evaluations, but also by the derivatives, or differential/pseudo-differential operators. Some sufficient and necessary conditions for the well-posedness of the interpolations are given. The results on sensitivity and sta bility of the interpolation systems are obtained. The optimal properties of the interpolants are analysed through the variational framework and reproducing kernel Hilbert space property, the error bounds and convergence rates of the interpolants are derived. The admissible reproducing kernel Hilbert spaces are also characterised.

This thesis aims at investigating hedging and portfolio optimisation problems under weak stochastic target constraints. Our first contribution consists in the representation of the hedging price of some contingent claims under both probabilistic and expected shortfall ("weak") constraints holding on a set of dates. We consider a Markovian and complete market framework and favour a dual approach. This work is an extension to Föllmer and Leukert (1999,2000). We then extend the previous results to the case where the wealth process diffusion is semi-linear in the control/strategy variable. The previous convex duality machinery does not apply anymore and we rely on PDE arguments. Bouchard, Elie and Touzi (2009) already proved the PDE characterisation of such price functions but a comparison result, necessary to build a convergent numerical scheme, is still missing in the literature. We will prove that such a result actually holds. The main difficulty arises from the discontinuity of the operators involved in the PDE characterisation of the price function. An application to the quantile hedging of Bermudan options is provided. Our third contribution relies on the PDE characterisation of the problem of portfolio optimisation under a European quantile hedging constraint. We extend the results of Bouchard, Elie and Imbert (2010) to the case where the constraint holds in a weaker sense. The study is based on a reformulation of the initial constraint into an obstacle and almost-sure stochastic target one. This reduction is done by the introduction of an additional controlled state variable coming from the diffusion of the probability of reaching the target (see Bouchard, Elie and Touzi (2009)) and by means of the Geometric Dynamic Programming principle of Soner and Touzi (2002). However this additional controlled state variable raises non-trivial boundary conditions that have to be characterised. We also have to handle the discontinuity of the operators involved in the characterisation.

No description available.

No description available.

This thesis deals with the problem of multimodal dimensionality reduction (DR), which arises when the input objects, to be mapped on a low-dimensional space, consist of multiple vectorial representations, instead of a single one. Herein, the problem is addressed in two alternative manners. One is based on the traditional notion of modality fusion, but using a novel approach to determine the fusion weights. In order to optimally fuse the modalities, the known graph embedding DR framework is extended to multiple modalities by considering a weighted sum of the involved affinity matrices. The weights of the sum are automatically calculated by minimizing an introduced notion of inconsistency of the resulting multimodal affinity matrix. The other manner for dealing with the problem is an approach to consider all modalities simultaneously, without fusing them, which has the advantage of minimal information loss due to fusion. In order to avoid fusion, the problem is viewed as a multi-objective optimization problem. The multiple objective functions are defined based on graph representations of the data, so that their individual minimization leads to dimensionality reduction for each modality separately. The aim is to combine the multiple modalities without the need to assign importance weights to them, or at least postpone such an assignment as a last step. The proposed approaches were experimentally tested in mapping multimedia data on low-dimensional spaces for purposes of visualization, classification and clustering. The no-fusion approach, namely Multi-objective DR, was able to discover mappings revealing the structure of all modalities simultaneously, which cannot be discovered by weight-based fusion methods. However, it results in a set of optimal trade-offs, from which one needs to be selected, which is not trivial. The optimal-fusion approach, namely Multimodal Graph Embedding DR, is able to easily extend unimodal DR methods to multiple modalities, but depends on the limitations of the unimodal DR method used. Both the no-fusion and the optimal-fusion approaches were compared to state-of-the-art multimodal dimensionality reduction methods and the comparison showed performance improvement in visualization, classification and clustering tasks. The proposed approaches were also evaluated for different types of problems and data, in two diverse application fields, a visual-accessibility-enhanced search engine and a visualization tool for mobile network security data. The results verified their applicability in different domains and suggested promising directions for future advancements.

No description available.

No description available.