UPPSALA DISSERTATIONS IN MATHEMATICS 86 Gaussian Bridges - Modeling and Inference Maik Görgens Department of Mathematics Uppsala University UPPSALA 2014 Dissertation presented at Uppsala University to be publicly examined in Häggsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Friday, 7 November 2014 at 13:15 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Professor Mikhail Lifshits (St. Petersburg State University and Linköping University). Abstract Görgens, M. 2014. Gaussian Bridges - Modeling and Inference. Uppsala Dissertations in Mathematics 86. 32 pp. Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-506-2420-5. This thesis consists of a summary and five papers, dealing with the modeling of Gaussian bridges and membranes and inference for the α-Brownian bridge. In Paper I we study continuous Gaussian processes conditioned that certain functionals of their sample paths vanish. We deduce anticipative and non-anticipative representations for them. Generalizations to Gaussian random variables with values in separable Banach spaces are discussed. In Paper II we present a unified approach to the construction of generalized Gaussian random fields. Then we show how to extract different Gaussian processes, such as fractional Brownian motion, Gaussian bridges and their generalizations, and Gaussian membranes from them. In Paper III we study a simple decision problem on the scaling parameter in α-Brownian bridges. We generalize the Karhunen-Loève theorem and obtain the distribution of the involved likelihood ratio based on Karhunen-Loève expansions and Smirnov's formula. The presented approach is applied to a simple decision problem for Ornstein-Uhlenbeck processes as well. In Paper IV we calculate the bias of the maximum likelihood estimator for the scaling parameter and propose a bias-corrected estimator. We compare it with the maximum likelihood estimator and two alternative Bayesian estimators in a simulation study. In Paper V we solve an optimal stopping problem for the α-Brownian bridge. In particular, the limiting behavior as α tends to zero is discussed. Maik Görgens, Department of Mathematics, Analysis and Probability Theory, Box 480, Uppsala University, SE-75106 Uppsala, Sweden. © Maik Görgens 2014 ISSN 1401-2049 ISBN 978-91-506-2420-5 urn:nbn:se:uu:diva-232544 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-232544) Für Bine und Milo List of papers This thesis is based on the following papers, which are referred to in the text by their Roman numerals. I M. Görgens. Conditioning of Gaussian processes and a zero area Brownian bridge. Manuscript. II M. Görgens and I. Kaj. Gaussian processes, bridges and membranes extracted from selfsimilar random fields. Manuscript. III M. Görgens. Inference for α-Brownian bridge based on Karhunen-Loève expansions. Submitted for publication. IV M. Görgens and M. Thulin. Bias-correction of the maximum likelihood estimator for the α-Brownian bridge. Statistics and Probability Letters, 93, 78–86, 2014. V M. Görgens. Optimal stopping of an α-Brownian bridge. Submitted for publication. Reprints were made with permission from the publishers. Contents 1 Introduction .................................................................................................. 9 1.1 Gaussian processes ........................................................................... 9 1.1.1 The Brownian bridge ....................................................... 10 1.1.2 Representation of Gaussian processes ........................... 11 1.1.3 Series expansions of Gaussian processes ....................... 11 1.2 Models for Gaussian bridges and membranes .............................. 13 1.2.1 Generalized Gaussian bridges ......................................... 13 1.2.2 Gaussian selfsimilar random fields ................................. 15 1.3 Inference for α-Brownian bridges ................................................ 16 1.3.1 Estimation ........................................................................ 18 1.3.2 Hypothesis testing ........................................................... 19 1.3.3 Optimal stopping ............................................................. 20 2 Summary of Papers .................................................................................... 22 2.1 Paper I ............................................................................................. 22 2.2 Paper II ............................................................................................ 23 2.3 Paper III .......................................................................................... 24 2.4 Paper IV .......................................................................................... 25 2.5 Paper V ........................................................................................... 25 3 Summary in Swedish ................................................................................. 27 Acknowledgements .......................................................................................... 29 References ........................................................................................................ 30 1. Introduction This is a thesis in the mathematical field of stochastics. This field is often divided into the areas probability theory, theoretical statistics, and stochastic processes. The boundaries between these areas are not sharp but intersec- tions of them exist. Paper I and Paper II contribute to the theory of stochastic modeling of Gaussian bridges and membranes and belong to the intersection of probability theory and stochastic processes, whereas in Papers III – V we study inference for a continuous time stochastic process, and those papers thus belong to the intersection of the areas theoretical statistics and stochastic pro- cesses. Moreover, throughout the thesis we make use of functional analytical tools. The term Gaussian bridges is to be understood in a broad sense. While orig- inally introduced to describe Gaussian processes which attain a certain value at a specific time almost surely1, it was later (with the prefix “generalized”) used to denote Gaussian processes conditioned on the event that one or more func- tionals of the sample paths vanish2. Paper I contributes to the theory of such generalized Gaussian bridges. In Paper II we present a general method to con- struct selfsimilar Gaussian random fields and study how to extract Gaussian processes, bridges, and membranes from them. In Papers III – V we consider another generalization of Gaussian bridges – the α-Brownian bridges – and study problems of inference for the scaling parameter α and optimal stopping of such bridges. In all five papers of this thesis the Brownian bridge occurs at least as a special case3. In this first chapter we give a short introduction to Gaussian processes in general and to the topics studied in this thesis in particular. In Chapter 2 we summarize the included papers and in Chapter 3 we give an outline of the thesis in Swedish. 1.1 Gaussian processes Among all probability distributions the normal distribution is of particular im- portance since, by the central limit theorem, sums of independent and iden- tical distributed (i.i.d.) random variables with finite variance behave roughly 1See for example [24]. 2We refer to [1] and in particular to [43]. 3A plot of the Brownian bridge is given on the cover page. 9 like normal random variables. The central limit theorem has its functional ana- =( ) = n logue as well: Random walks S Sn n∈N of the form Sn ∑k=1 Xk, where the Xk’s are i.i.d. random variables with finite variance, behave (suitably scaled) roughly like Brownian motion. The Brownian motion is the unique continu- ous stochastic process on the real line with i.i.d. and symmetric increments. It serves as a building block for many other Gaussian and non-Gaussian pro- cesses. Gaussian processes are not only of particular importance, but also very ac- cessible for investigation, since their finite dimensional distributions are solely determined by their covariance and, moreover, Gaussian random variables are independent whenever they are orthogonal in the Hilbert space spanned by them. This importance and treatability makes the class of Gaussian processes an object of intensive study. Here we just mention the monographs [12], [27], [29], and [33]. 1.1.1 The Brownian bridge If we consider standard Brownian motion W =(Ws)s∈R (i.e., Brownian motion E = E 2 = scaled to fulfill W0 0 and W1 1) and tie it down to 0 at time 1 we obtain (restricted to the interval [0,1]) the Brownian bridge. As mentioned before, this process appears in all papers included in this thesis. The Brownian bridge B =(Bs)s∈[0,1] is a continuous centered Gaussian pro- cess uniquely defined by its covariance function EBsBt = s(1 − t) for 0 ≤ s ≤ t ≤ 1. It is of particular importance in asymptotic statistics (cf. [23]): Given n i.i.d. random variables with a continuous distribution function F, con- sider their empirical distribution function Fn. By the law of large numbers, Fn(s) → F(s) almost surely as n → ∞. In 1933, Glivenko [25] and Cantelli [15] showed that this convergence is uniform on the real line. Now, by the central limit theorem, √ n(Fn(s) − F(s)) −→ d B(F(s)), as n → ∞. (1.1) Kolmogorov showed in [31] that4,asn → ∞, √ nsup|Fn(s) − F(s)|−→d sup|B(F(s))| = sup |B(s)|. (1.2) s∈R s∈R 0≤s≤1 Moreover, he proved that the law of the left hand side in (1.2) is independent of F and studied the distribution of the right hand side in (1.2)