Huiling Le’s research while affiliated with University of Nottingham and other places

This paper investigates optimal investment-consumption strategies that maximize the expected utility of consumption and/or terminal wealth under jump-diffusion models using a martingale method. We characterize the optimal trading strategy and the optimal martingale measure in terms of a system of equations and obtain explicit solutions for the power and logarithmic utility case.

We prove weak laws of large numbers and central limit theorems of Lindeberg type for empirical centres of mass (empirical Fr\'echet means) of independent non-identically distributed random variables taking values in Riemannian manifolds. In order to prove these theorems we describe and prove a simple kind of Lindeberg-Feller central approximation theorem for vector-valued random variables, which may be of independent interest and is therefore the subject of a self-contained section. This vector-valued result allows us to clarify the number of conditions required for the central limit theorem for empirical Fr\'echet means, while extending its scope.

We consider inference for functions of the marginal covariance matrix under a class of stationary vector time series models, referred to as time-orthogonal principal components models. The main application which motivated this work involves the estimation of configurational entropy from molecular dynamics simulations in computational chemistry, where current methods of entropy estimation involve calculations based on the sample covariance matrix. The theoretical results we obtain provide a basis for approximate inference procedures, including confidence interval calculations for scalar quantities of interest; these results are applied to the molecular dynamics application, and some further applications are discussed briefly. KeywordsAutoregressive-Central limit theorem-Configurational entropy-Principal components-Procrustes-Sample covariance-Shape-Size-and-shape

A family of shape curves is introduced that is useful for modelling the changes in shape in a series of geometrical objects. The relationship between the preshape sphere and the shape space is used to define a general family of curves based on horizontal geodesics on the preshape sphere. Methods for fitting geodesics and more general curves in the non-Euclidean shape space of point sets are discussed, based on minimizing sums of squares of Procrustes distances. Likelihood-based inference is considered. We illustrate the ideas by carrying out statistical analysis of two-dimensional landmarks on rats’ skulls at various times in their development and three-dimensional landmarks on lumbar vertebrae from three primate species.

Using the geometry of the Kendall shape space, in this paper we study the shape, as well as the size-and-shape, of the projection of a configuration after it has been rotated and, when the given configuration lies in a Euclidean space of an arbitrary dimension, we obtain expressions for the induced distributions of such shapes when the rotation is uniformly distributed.

Using the geometry of the Kendall shape space, in this paper we study the shape, as well as the size-and-shape, of the projection of a configuration after it has been rotated and, when the given configuration lies in a Euclidean space of an arbitrary dimension, we obtain expressions for the induced distributions of such shapes when the rotation is uniformly distributed.

There are many variations on what one may regard as statistical shape, depending on the application in mind. The focus of this chapter is the statistical analysis of the shapes determined by finite sequences of points in a Euclidean space. We shall draw together a range of ideas from statistical shape theory, including distributions, diffusions, estimations and computations, emphasizing the role played by the underlying geometry. Applications in selected areas of current interest will be discussed.

We propose an alternative to Kendall's shape space for reflection shapes of configurations in with k labelled vertices, where reflection shape consists of all the geometric information that is invariant under compositions of similarity and reflection transformations. The proposed approach embeds the space of such shapes into the space of (k − 1) × (k − 1) real symmetric positive semidefinite matrices, which is the closure of an open subset of a Euclidean space, and defines mean shape as the natural projection of Euclidean means in on to the embedded copy of the shape space. This approach has strong connections with multi-dimensional scaling, and the mean shape so defined gives good approximations to other commonly used definitions of mean shape. We also use standard perturbation arguments for eigenvalues and eigenvectors to obtain a central limit theorem which then enables the application of standard statistical techniques to shape analysis in two or more dimensions.

We generalise, to complete, connected and locally symmetric Riemannian manifolds, the construction of coupled semimartingales X and Y given in Le and Barden (J Lond Math Soc 75:522–544, 2007). When such a manifold has non-negative curvature, this makes it possible for the stochastic anti-development of the corresponding semimartingale expXt (aexp-1Xt(Yt)){\rm exp}_{X_t} \big(\alpha\,{\rm exp}^{-1}_{X_t}(Y_t)\big) to be a time-changed Brownian motion with drift when X and Y are. As an application, we use the latter result to strengthen, and extend to locally symmetric spaces, the results of Le and Barden (J Lond Math Soc 75:522–544, 2007) concerning an inequality involving the solutions of the parabolic equation \frac¶y ¶t = \frac12Dy- hy\frac{\partial\psi} {\partial t} = \frac{1}{2}\Delta\psi - h\,\psi with Dirichlet boundary condition and an inequality involving the first eigenvalues of the Laplacian, both on three related convex sets.

The problemWhen is a space familiarCW complexesA cellular decomposition of the unit sphereThe cellular decomposition of shape spacesInclusions and isometriesSimple connectivity and higher homotopy groupsThe mapping cone decompositionHomotopy type and Casson's theorem