A Malliavin-Skorohod calculus in L0L^0 and L1L^1 for additive and Volterra-type processes

In this paper we develop a Malliavin-Skorohod type calculus for additive processes in the $L^0$ and $L^1$ settings, extending the probabilistic interpretation of the Malliavin-Skorohod operators to this context. We prove calculus rules and obtain a generalization of the Clark-Hausmann-Ocone formula for random variables in $L^1$. Our theory is then applied to extend the stochastic integration with respect to volatility modulated L\'evy-driven Volterra processes recently introduced in the literature. Our work yields to substantially weaker conditions that permit to cover integration with respect, e.g. to Volterra processes driven by $\alpha$-stable processes with $\alpha < 2$. The presentation focuses on jump type processes.Comment: 27 page

Bounded Point Derivations on Certain Function Spaces

Let be a compact subset of the complex plane and denote by () the closure of rational functions with poles off in the () norm. We show that if a point 0 admits a bounded point derivation on () for \u3e 2, then there is an approximate derivative at 0. We also prove a similar result for higher order bounded point derivations. This extends a result of Wang, which was proven for (), the uniform closure of rational functions with poles off . In addition, we show that if a point 0 admits a bounded point derivation on () and if contains an interior cone, then the bounded point derivation can be represented by the difference quotient if the limit is taken over a non-tangential ray to 0. We also extend this result to the case of higher order bounded point derivations. These results were first shown by O\u27Farrell; however, we prove them constructively by explicitly using the Cauchy integral formula

Stochastic singular control problems with state constraint

Singular control is an important and challenging class of problems in stochastic control theory. Such control problems can rarely be solved explicitly and thus numerical approximation schemes are necessary. In this work we develop approximation schemes for singular control problems with state constraints. The first problem we consider arises in problems of optimal consumption and investment under transaction costs. We use Markov chain approximations to develop a convergent numerical scheme. Proof of convergence uses techniques from the theory of weak convergence. Specific features that make the analysis nontrivial include unboundedness of state and control spaces and cost function; degeneracies in the dynamics; and presence of both singular and absolutely continuous controls. We present a computational algorithm and the results of a numerical study. Numerical schemes for singular control problems can be computationally quite intensive, and thus it is of great interest to develop less expensive schemes that exploit specific features of the underlying dynamics. To this end we investigate connections between singular control and optimal stopping problems. A key technical step in establishing such connections is proving existence of an optimal singular control. We prove such a result for a general class of singular control problems with linear dynamics and state constrained to be in a polyhedral cone. A particular example of this class of models are the so-called Brownian control problems (BCPs) and thus existence of optimal controls for BCPs follows as a consequence. Armed with this existence result, we consider a two-dimensional singular control problem that arises from queueing networks. We prove rigorously an equivalence of this iii problem with an optimal stopping problem. We exploit this connection in developing simple computational schemes for the singular control problem, and we investigate performance of the schemes in a numerical study

Well posedness and physical possibility

There is a sentiment shared among physicists that well posedness is a necessary condition for physical possibility. The arguments usually offered for well posedness have an epistemic flavor and thus they fall short of establishing the metaphysical claim that lack of well posedness implies physical impossibility. In this work we analyze the relationship of well posedness to prediction and confirmation as well as the notion of physical possibility and we devise three novel and independent argumentative strategies that may succeed where the usual epistemic arguments fail

Topological measure theory, with applications to probability

The work carried out for this thesis was motivated by a belief that the methods of topological measure theory could be more widely applied in the theory of probability. As my introduction to the subject was through the field of weak convergence, much of the thesis has developed out of the study of problems in that area; but this has often also involved consideration of the topological, measure theoretic and functional analytic background material. For example, the integral representation theory of Chapter 2 grew out of the study of relative compactness in the topology of weak convergence; the latter is the subject of Chapter 3. But the results of Chapter 2 are also of interest in their own right as they form the basis of a unified approach to Riesz type integral representation theorems. While these investigations were being carried out it became apparent that one particular concept has a most important part to play in topological measure theory. This so-called T-additivity property is examined in Chapter 4. It constitutes an intermediate stage in the progression from countable additivity to the stronger Radon measure concept. It often seems to be the minimal condition for compatibility between the topological and measure theoretic structures; this view is supported by the results of Chapter 4. The last two chapters contain some of the other applications of this research to probability theory. In Chapter 5 problems related to the existence and weak convergence of random measures on locally compact spaces are considered; and in Chapter 6 some aspects of the theory of Markov chains on topological state spaces are discussed. Weak convergence arguments are prominent in both chapters

A systematic approach for integrated product, materials, and design-process design

Designers are challenged to manage customer, technology, and socio-economic uncertainty causing dynamic, unquenchable demands on limited resources. In this context, increased concept flexibility, referring to a designer s ability to generate concepts, is crucial. Concept flexibility can be significantly increased through the integrated design of product and material concepts. Hence, the challenge is to leverage knowledge of material structure-property relations that significantly affect system concepts for function-based, systematic design of product and materials concepts in an integrated fashion. However, having selected an integrated product and material system concept, managing complexity in embodiment design-processes is important. Facing a complex network of decisions and evolving analysis models a designer needs the flexibility to systematically generate and evaluate embodiment design-process alternatives. In order to address these challenges and respond to the primary research question of how to increase a designer s concept and design-process flexibility to enhance product creation in the conceptual and early embodiment design phases, the primary hypothesis in this dissertation is embodied as a systematic approach for integrated product, materials and design-process design. The systematic approach consists of two components i) a function-based, systematic approach to the integrated design of product and material concepts from a systems perspective, and ii) a systematic strategy to design-process generation and selection based on a decision-centric perspective and a value-of-information-based Process Performance Indicator. The systematic approach is validated using the validation-square approach that consists of theoretical and empirical validation. Empirical validation of the framework is carried out using various examples including: i) design of a reactive material containment system, and ii) design of an optoelectronic communication system.Ph.D.Committee Chair: Allen, Janet K.; Committee Member: Aidun, Cyrus K.; Committee Member: Klein, Benjamin; Committee Member: McDowell, David L.; Committee Member: Mistree, Farrokh; Committee Member: Yoder, Douglas P

Development of a beam loss monitor using the Cherenkov Effect in optical fibres

The need for real-time monitoring of beam losses, including evaluation of their intensity and localization of their exact position, together with the possibility to overcome the limitations due to the reduced space for the diagnostics, makes exploitation of the Cherenkov effect in optical fibres, one of the most suitable candidates for beam loss monitoring. In this thesis, the design, development and characterization of an optical fibre beam loss monitor based on high radiation hardness pure silica fibres and silicon photomultipliers is reported. The tests were carried out at the ALICE accelerator research and development facility, Daresbury Laboratories, UK. For this purpose high numerical aperture fibres were used together with the latest generation silicon detector instead of the standard photomultiplier tubes commonly used for Cherenkov beam loss monitoring