Thesis (doctoral)--Grossherzoglich Hessische Ludwigs-Universität zu Giessen, 1910. / Vita.

Thèse - Genève.

Thèse : Mathématiques : Université, Faculté des sciences de Paris : 1843. / Titre provenant de l'écran-titre.

Thesis (doctoral)--Georg-August-Universität zu Göttingen, 1913. / Cover title. Vita.

Thesis (doctoral)--Friedrich-Wilhelms-Universität Berlin, 1863. / Vita. On t.p., the expression in parentheses after e and [square root of minus one] are superscript and [m̳u̳], [n̳u̳] and n̳ are subscript.

Thesis (doctoral)--Universität Bonn, 1989. / Bibliography: p. 114-116.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1966. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

There are many problems in mathematical finance which require the evaluation of a multivariate integral. Since these problems typically involve the discretisation of a continuous random variable, the dimension of the integrand can be in the thousands, tens of thousands or even more. For such problems the Monte Carlo method has been a powerful and popular technique. This is largely related to the fact that the performance of the method is independent of the number of dimensions. Traditional quasi-Monte Carlo techniques are typically not independent of the dimension and as such have not been suitable for high-dimensional problems. However, recent work has developed new types of quasi-Monte Carlo point sets which can be used in practically limitless dimension. Among these types of point sets are Sobol' sequences, Faure sequences, Niederreiter-Xing sequences, digital nets and lattice rules. In this thesis, we will concentrate on results concerning lattice rules. The typical setting for analysis of these new quasi-Monte Carlo point sets is the worst-case error in a weighted function space. There has been much work on constructing point sets with small worst-case errors in the weighted Korobov and Sobolev spaces. However, many of the integrands which arise in the area of mathematical finance do not lie in either of these spaces. One common problem is that the integrands are unbounded on the boundaries of the unit cube. In this thesis we construct function spaces which admit such integrands and present algorithms to construct lattice rules where the worst-case error in this new function space is small. Lattice rules differ from other quasi-Monte Carlo techniques in that the points can not be used sequentially. That is, the entire lattice is needed to keep the worst-case error small. It has been shown that there exist generating vectors for lattice rules which are good for many different numbers of points. This is a desirable property for a practitioner, as it allows them to keep increasing the number of points until some error criterion is met. In this thesis, we will develop fast algorithms to construct such generating vectors. Finally, we apply a similar technique to show how a particular type of generating vector known as the Korobov form can be made extensible in dimension.

In both physical and social sciences, we usually use controlled differential equation to model various continuous evolving system; describing how a response y relates to another process x called control. For regular controls x, the unique existence of the response y is guaranteed while it would never be the case for non-smooth controls via the classical approach. Besides, uniform closeness of controls may not imply closeness of their corresponding responses. Theory of rough paths provides a solution to both concerns. Since the creation of rough path theory, it enjoys a fruitful development and finds wide applications in stochastic analysis. In particular, rough path theory provides an effective method to study irregularity of curves and its geometric consequences in relation to integration of differential forms. In the chapter 2, we demonstrate the power of rough path theory in classical complex analysis by showing the rough path nature of the boundaries of a class of Holder's domains; as an immediate application, we extend the classical Gauss-Green's theorem. Until recently, there has been only limited research on applications of theory of rough paths to high dimensional geometry. It is clear to us that many geometric objects, in some senses appearing as solids, are actually comprised of filaments. In the chapter 3, two basic results in the theory of rough paths which will motivate later development of my thesis has been included. In the chapters 4 and 5, we identify a sensible way to do geometric calculus via those filaments (more precisely, space-filling rough paths) in dimension 3. In a recent joint work of Hambly and Lyons, they have shown that every rectifiable path can be completely characterized, up to tree-like deformation, by an algebraic object called the signature, tensor of all iterated integrals, of the path. It is clear that all tree-like deformation of the path would not change its topological features. For instance, the number of times a planar loop of finite length winds around a point (not lying on the path) is unaltered if one deforms the path in tree-like ways. Therefore, it should be plausible to extract this topological information out from the signature of the loop since the signature is a complete algebraic invariant. In the chapter 6, we express the winding number of a nice loop (respectively linking number of a pair of nice loops) as a linear functional of the signature of the loop (respectively signatures of the pair of loops).

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