Convergence in incomplete market models

The problem of pricing and hedging of contingent claims in incomplete markets has lead to the development of various valuation methodologies. This thesis examines the mean-variance and variance-optimal approaches to risk-minimisation and shows that these are robust under the convergence from discrete- to continuous-time market models. This property yields new convergence results for option prices, trading strategies and value processes in incomplete market models.Techniques from nonstandard analysis are used to develop new results for the lifting property of the minimal martingale density and risk-minimising strategies. These are applied to a number of incomplete market models:The restriction of hedging dates in a general class of discrete- and continuous-time models is studied and it is shown that the convergence of the underlying models implies the convergence of strategies and value processes.Similar results are obtained for multinomial models and approximations of the Black-Scholes model by direct observation of the price process. The concept of D 2-convergence is extended to these classes of models, including the construction of discretisation schemes. This yields new convergence results for these models as well as for option prices in a jump-diffusion model.The computational aspects of these approximations are examined and numerical results are provided in the case of European and Asian options.For ease of reference a summary of the main results from nonstandard analysis in the context of mathematical finance is given as well as a brief introduction to mean-variance hedging and variance-optimal pricing

Control strategies for road risk mitigation in kinetic traffic modelling

In this paper we present a Boltzmann-type kinetic approach to the modelling of road traffic, which includes control strategies at the level of microscopic binary interactions aimed at the mitigation of speed-dependent road risk factors. Such a description is meant to mimic a system of driver-assist vehicles, which by responding locally to the actions of their drivers can impact on the large-scale traffic dynamics, including those related to the collective road risk and safety

Time-Consistent Mean-Variance Portfolio Selection in Discrete and Continuous Time

It is well known that mean-variance portfolio selection is a time-inconsistent optimal control problem in the sense that it does not satisfy Bellman's optimality principle and therefore the usual dynamic programming approach fails. We develop a time- consistent formulation of this problem, which is based on a local notion of optimality called local mean-variance efficiency, in a general semimartingale setting. We start in discrete time, where the formulation is straightforward, and then find the natural extension to continuous time. This complements and generalises the formulation by Basak and Chabakauri (2010) and the corresponding example in Bj\"ork and Murgoci (2010), where the treatment and the notion of optimality rely on an underlying Markovian framework. We justify the continuous-time formulation by showing that it coincides with the continuous-time limit of the discrete-time formulation. The proof of this convergence is based on a global description of the locally optimal strategy in terms of the structure condition and the F\"ollmer-Schweizer decomposition of the mean-variance tradeoff. As a byproduct, this also gives new convergence results for the F\"ollmer-Schweizer decomposition, i.e. for locally risk minimising strategies

Variational Methods in Shape Space

This dissertation deals with the application of variational methods in spaces of geometric shapes. In particular, the treated topics include shape averaging, principal component analysis in shape space, computation of geodesic paths in shape space, as well as shape optimisation. Chapter 1 provides a brief overview over the employed models of shape space. Geometric shapes are identified with two- or three-dimensional, deformable objects. Deformations will be described via physical models; in particular, the objects will be interpreted as consisting of either a hyperelastic solid or a viscous liquid material. Furthermore, the description of shapes via phase fields or level sets is briefly introduced. Chapter 2 reviews different and related approaches to shape space modelling. References to related topics in image segmentation and registration are also provided. Finally, the relevant shape optimisation literature is introduced. Chapter 3 recapitulates the employed concepts from continuum mechanics and phase field modelling and states basic theoretical results needed for the later analysis. Chapter 4 addresses the computation of shape averages, based on a hyperelastic notion of shape dissimilarity: The dissimilarity between two shapes is measured as the minimum deformation energy required to deform the first into the second shape. A corresponding phase-field model is introduced, analysed, and finally implemented numerically via finite elements. A principal component analysis of shapes, which is consistent with the previously introduced average, is considered in Chapter 5. Elastic boundary stresses on the average shape are used as representatives of the input shapes in a linear vector space. On these linear representatives, a standard principal component analysis can be performed, where the employed covariance metric should be properly chosen to depend on the input shapes. Chapter 6 interprets shapes as belonging to objects made of a viscous liquid and correspondingly defines geodesic paths between shapes. The energy of a path is given as the total physical dissipation during the deformation of an object along the path. A rigid body motion invariant time discretisation is achieved by approximating the dissipation along a path segment by the deformation energy of a small solid deformation. The numerical implementation is based on level sets. Chapter 7 is concerned with the optimisation of the geometry and topology of solid structures that are subject to a mechanical load. Given the load configuration, the structure rigidity, its volume, and its surface area shall be optimally balanced. A phase field model is devised and analysed for this purpose. In this context, the use of nonlinear elasticity allows to detect buckling phenomena which would be ignored in linearised elasticity

Optimal trajectory design for interception and deflection of Near Earth Objects

Many asteroids and comets orbit the inner solar system; among them Near Earth Objects (NEOs) are those celestial bodies for which the orbit lies close, and sometimes crosses, the Earth's orbit. Over the last decades the impact hazard they pose to the Earth has generated heated discussions on the required measures to react to such a scenario. The aim of the research presented in this dissertation is to develop methodologies for the trajectory design of interception and deflection missions to Near Earth Objects. The displacement, following a deflection manoeuvre, of the asteroid at the minimum orbit intersection distance with the Earth is expressed by means of a simple and general formulation, which exploits the relative motion equations and Gauss' equations. The variation of the orbital elements achieved by any impulsive or low-thrust action on the threatening body is derived through a semi-analytical approach, whose accuracy is extensively shown. This formulation allows the analysis of the optimal direction of the deflection manoeuvre to maximise the achievable deviation. The search for optimal opportunities for mitigation missions is done through a global optimisation approach. The transfer trajectory, modelled through preliminary design techniques, is integrated with the deflection model. In this way, the mission planning can be performed by optimising different contrasting criteria, such as the mass at launch, the warning time, and the total deflection. A set of Pareto fronts is computed for different deflection strategies and considering various asteroid mitigation scenarios. Each Pareto set represents a number of mission opportunities, over a wide domain of launch windows and design parameters. A first set of results focuses on impulsive deflection missions, to a selected group of potentially hazardous asteroids; the analysis shows that the ideal optimal direction of the deflection manoeuvre cannot always be achieved when the transfer trajectory is integrated with the deflection phase. A second set of results includes solutions for the deviation of some selected NEOs by means of a solar collector strategy. The semi-analytical formulation derived allows the reduction of the computational time, hence the generation of a large number of solutions. Moreover, sets of Pareto fronts for asteroid mitigation are computed through the more feasible deflection schemes proposed in literature: kinetic impactor, nuclear interceptor, mass driver device, low-thrust attached propulsion, solar collector, and gravity tug. A dominance criterion is used to perform a comparative assessment of these mitigation strategies, while also considering the required technological development through a technology readiness factor. The global search of solutions through a multi-criteria optimisation approach represents the first stage of the mission planning, in which preliminary design techniques are used for the trajectory model. At a second stage, a selected number of trajectories can be optimised, using a refined model of the dynamics. For this purpose, the use of Differential Dynamic Programming (DDP) is investigated for the solution of the optimal control problem associated to the design of low-thrust trajectories. The stage-wise approach of DDP is exploited to integrate an adaptive step discretisation scheme within the optimisation process. The discretisation mesh is adjusted at each iteration, to assure high accuracy of the solution trajectory and hence fully exploit the dynamics of the problem within the optimisation process. The feedback nature of the control law is preserved, through a particular interpolation technique that improves the robustness against some approximation errors. The modified DDP-method is presented and applied to the design of transfer trajectories to the fly-by or rendezvous of NEOs, including the escape phase at the Earth. The DDP approach allows the optimisation of the trajectory as a whole, without recurring to the patched conic approach. The results show how the proposed method is capable of fully exploiting the multi-body dynamics of the problem; in fact, in one of the study cases, a fly-by of the Earth is scheduled, which was not included in the first guess solution

On Optimal Regularization Parameters via Bilevel Learning

Variational regularization is commonly used to solve linear inverse problems, and involves augmenting a data fidelity by a regularizer. The regularizer is used to promote a priori information, and is weighted by a regularization parameter. Selection of an appropriate regularization parameter is critical, with various choices leading to very different reconstructions. Existing strategies such as the discrepancy principle and L-curve can be used to determine a suitable parameter value, but in recent years a supervised machine learning approach called bilevel learning has been employed. Bilevel learning is a powerful framework to determine optimal parameters, and involves solving a nested optimisation problem. While previous strategies enjoy various theoretical results, the well-posedness of bilevel learning in this setting is still a developing field. One necessary property is positivity of the determined regularization parameter. In this work, we provide a new condition that better characterises positivity of optimal regularization parameters than the existing theory. Numerical results verify and explore this new condition for both small and large dimensional problems.Comment: 26 pages, 6 figure