Multitarget tracking is the process of jointly determining the number of targets present and their states from noisy sets of measurements. The difficulty of the multitarget tracking problem is that the number of targets present can change as targets appear and disappear while the sets of measurements may contain false alarms and measurements of true targets may be missed. The theory of random finite sets was proposed as a systematic, Bayesian approach to solving the multitarget tracking problem. The conceptual solution is given by Bayes filtering for the probability distribution of the set of target states, conditioned on the sets of measurements received, known as the multitarget Bayes filter. A first-moment approximation to this filter, the probability hypothesis density (PHD) filter, provides a more computationally practical, but theoretically sound, solution. The central thesis of this work is that the random finite set framework is theoretically sound, compatible with the Bayesian methodology and amenable to immediate implementation in a wide range of contexts. In advancing this thesis, new links between the PHD filter and existing Bayesian approaches for manoeuvre handling and incorporation of target amplitude information are presented. A new multitarget metric which permits incorporation of target confidence information is derived and new algorithms are developed which facilitate sequential Monte Carlo implementations of the PHD filter. Several applications of the PHD filter are presented, with a focus on applications for tracking in sonar data. Good results are presented for implementations on real active and passive sonar data. The PHD filter is also deployed in order to extract bacterial trajectories from microscopic visual data in order to aid ongoing work in understanding bacterial chemotaxis. A performance comparison between the PHD filter and conventional multitarget tracking methods using simulated data is also presented, showing favourable results for the PHD filter.

The signature of the path provides a top down description of a path in terms of its eects as a control. It is a group-like element in the tensor algebra and is an essential object in rough path theory. When the path is random, the linear independence of the signatures of different paths leads one to expect, and it has been proved in simple cases, that the expected signature would capture the complete law of this random variable. It becomes of great interest to be able to compute examples of expected signatures. In this thesis, we aim to compute the expected signature of various stochastic process solved by a PDE approach. We consider the case for an Ito diffusion process up to a fixed time, and the case for the Brownian motion up to the first exit time from a domain. We manage to derive the PDE of the expected signature for both cases, and find that this PDE system could be solved recursively. Some specific examples are included herein as well, e.g. Ornstein-Uhlenbeck (OU) processes, Brownian motion and Levy area coupled with Brownian motion.

Recent years have seen progress in the analysis of the heat kernel for certain reversible random walks in random environments. In particular the work of Barlow(2004) showed that the heat kernel for the random walk on the infinite component of supercritical bond percolation behaves in a Gaussian fashion. This heat kernel control was then used to prove a quenched functional central limit theorem. Following this work several examples have been analysed with anomalous heat kernel behaviour and, in some cases, anomalous scaling limits. We begin by generalizing the first result - looking for sufficient conditions on the geometry of the environment that ensure standard heat kernel upper bounds hold. We prove that these conditions are satisfied with probability one in the case of the random walk on continuum percolation and use the heat kernel bounds to prove an invariance principle. The random walk on dynamic environment is then considered. It is proven that if the environment evolves ergodically and is, in a certain sense, geometrically d-dimensional then standard on diagonal heat kernel bounds hold. Anomalous lower bounds on the heat kernel are also proven - in particular the random conductance model is shown to be "more anomalous" in the dynamic case than the static. Finally, the reflected random walk amongst random conductances is considered. It is shown in one dimension that under the usual scaling, this walk converges to reflected Brownian motion.

This thesis is concerned with graphical Gaussian models with equality constraints on the concentration or partial correlation matrix introduced by Højsgaard and Lauritzen (2008) as RCON and RCOR models. The models can be represented by vertex and edge coloured graphs G = (V,ε), where parameters associated with equally coloured vertices or edges are restricted to being identical. In the first part of this thesis we study the problem of estimability of a non-zero model mean μ if the covariance structure Σ is restricted to satisfy the constraints of an RCON or RCOR model but is otherwise unknown. Exploiting results in Kruskal (1968), we obtain a characterisation of suitable linear spaces Ω such that if Σ is restricted as above, the maximum likelihood estimator μ(with circumflex) and the least squares estimator μ* of μ coincide for μ ∈ Ω, thus allowing μ and Σ to be estimated independently. For the special case of Ω being specified by equality relations among the entries of μ according to a partition M of the model variables V, our characterisation translates into a necessary and sufficient regularity condition on M and (V,ε). In the second part we address model selection of RCON and RCOR models. Due to the large number of models, we study the structure of four model classes lying strictly within the sets of RCON and RCOR models, each of which is defined by desirable statistical properties corresponding to colouring regularity conditions. Two of these appear in Højsgaard and Lauritzen (2008), while the other two arise from the regularity condition ensuring equality of estimators μ(with circumflex) = μ* we find in the first part. We show each of the colouring classes to form complete lattices, which qualifies the corresponding model spaces for an Edwards-Havránek model selection procedure (Edwards and Havránek, 1987). We develop a coresponding algorithm for one of the model classes and give an algorithm for a systematic search in accordance with the Edwards-Havránek principles for a second class. Both are applied to data sets previously analysed in the literature, with very encouraging performances.

No description available.

The main contribution of the present thesis is in two aspects. The first one, which is the heart of the thesis, is to explore the fundamental relation between rough paths and their signatures. Our main goal is to give a geometric characterization of the kernel of the signature map in different situations. In Chapter Two, we start by establishing a general fact that a continuous Jordan curve on a Riemannian manifold can be arbitrarily well approximated by piecewise minimizing geodesic interpolations which are again Jordan. This result enables us to prove a generalized version of Green’s theorem for planar Jordan curves with finite p-variation 1 &le; p < 2, and to prove that two such Jordan curves have the same signature if and only if they are equal up to reparametrization. In Chapter Three, we investigate the problem for general weakly geometric rough paths. In particular, we show that a weakly geometric rough path has trivial signature if and only if it is tree-like in the sense we will define later on. In Chapter Four, we study the problem in the probabilistic setting. In particular, we show that for a class of stochastic processes, with probability one the sample paths are determined by their signatures up to reparametrization. A fundamental example is Gaussian processes including fractional Brownian motion with Hurst parameter H > 1/4, the Ornstein-Uhlenbeck process and the Brownian bridge. The second one is an application of rough path theory to the study of nonlinear diffusions on manifolds under the framework of nonlinear expectations. In Chapter Five, we begin by studying the geometric rough path nature of G-Brownian motion. This enables us to introduce rough differential equations driven by G-Brownian motion from a pathwise point of view. Next we establish the fundamental relation between rough (pathwise theory) and stochastic (L²-theory) differential equations driven by G-Brownian motion. This is a crucial point of understanding nonlinear diffusions and their generating heat flows on manifolds from an intrinsic point of view. Finally, from the pathwise point of view we construct G-Brownian motion on a compact Riemannian manifold and establish its generating heat flow for a class of G-functions under orthogonal invariance. As an independent interest, we also develop the Euler-Maruyama scheme for stochastic differential equations driven by G-Brownian motion.

This work is devoted to portfolio optimisation problem arising in the context of constrained optimisation. Despite the classical convex constraints imposed on proportion of wealth invested in the stock this work deals with the pathwise constraints. The drawdown constraint requires an investor's wealth process to dominate a given function of its up-to-date maximum. Typically, fund managers are required to post information about their maximum portfolio drawdowns as a part of the risk management procedure. One of the results of this work connects the drawdown constrained and the unconstrained asymptotic portfolio optimisation problems in an explicit manner. The main tools for achieving the connection are Azema-Yor processes which by their nature satisfy the drawdown condition. The other result deals with the constraint given as a floor process which the wealth process is required to dominate. The motivation arises from the financial market where the class of products serve as a protection from a downfall, e.g. out of the money put options. The main result provides the wealth process which dominates any fraction of a given floor and preserves the optimality. In the second part of this work we consider a problem of a lifetime utility of consumption maximisation subject to a drawdown constraint. One contribution to the existing literature consists of extending the results to incorporate a general drawdown constraint for a case of a zero interest rate market. The second result provides the first heuristic results for a problem in a presence of interest rates which differs qualitatively from a zero interest rate case. Also the last chapter concludes with the conjecture for the general case of the problem.

Made available in DSpace on 2014-06-11T19:25:30Z (GMT). No. of bitstreams: 0 Previous issue date: 2005-04Bitstream added on 2014-06-13T20:48:05Z : No. of bitstreams: 1 bessada_dfa_me_ift.pdf: 3052096 bytes, checksum: bfe2b25d2283cf5ec06ca7dc7407c70c (MD5) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Realizamos neste trabalho uma exposição geral da Teoria do Movimento Browniano, desde suas primeiras observações, feitas no âmbito da Biologia, até sua completa descrição seundo as leis da Mecânica estatística, formulação esta efetuada por Einstein em 1905. Com base nestes princípios físicos analisamos a Teoria do Movimento Browniano de Einstein como sendo um processo estocástico, o que permite sua generalização para um processo de Lévy. Fazemos uma exposição da Teoria de Lévy, e aplicamo-la em seguida na análise de dados provenientes do índice IBOVESPA. Camparamos os resultados com as distribuições empíricas e a modelada via distribuição gaussiana, demonstrando efetivamente que a série financeira analisada apresenta um comportamento não-gaussiano. / Abstracts: We review in this work the foundations of the Theory of Brownian Motion, from the first observations made in Biology to its complete description according to the laws of Statistical Mechanics performed by einstein in 1905. Afterwards we discuss the Einstein's Theory of Brownian Motion as a stochastic process, since this connection allows its generalization to a Lévy process. After a brief review of Lévy Theory we analyse IBOVESPA data within this framework. We compare the outcomes with the empirical and gaussian distributions, showing effectively that the analyzed financial series behaves exactly as a non-gaussian stochastic process.

The main objects of interest in this thesis are relatively hyperbolic groups. We will study some of their geometric properties, and we will be especially concerned with geometric properties of their boundaries, like linear connectedness, avoidability of parabolic points, etc. Exploiting such properties will allow us to construct, under suitable hypotheses, quasi-isometric embeddings of hyperbolic planes into relatively hyperbolic groups and quasi-isometric embeddings of relatively hyperbolic groups into products of trees. Both results have applications to fundamental groups of 3-manifolds. We will also study probabilistic properties of relatively hyperbolic groups and of groups containing ``hyperbolic directions'' despite not being relatively hyperbolic, like mapping class groups, Out(F<sub>n</sub>), CAT(0) groups and subgroups of the above. In particular, we will show that the elements that generate the ``hyperbolic directions'' (hyperbolic elements in relatively hyperbolic groups, pseudo-Anosovs in mapping class groups, fully irreducible elements in Out(F_n) and rank one elements in CAT(0) groups) are generic in the corresponding groups (provided at least one exists, in the case of CAT(0) groups, or of proper subgroups). We also study how far a random path can stray from a geodesic in the context of relatively hyperbolic groups and mapping class groups, but also of groups acting on a relatively hyperbolic space. We will apply this, for example, to show properties of random triangles.

The purpose of this thesis is to develop techniques for analysing interacting particle systems on the half-line. When the number of particles becomes large, stochastic partial differential equations (SPDEs) with Dirichlet boundary conditions will be the natural objects for describing the dynamics of the population's empirical measure. As a source of motivation, we consider systems that arise naturally as models for the pricing of portfolio credit derivatives, although similar applications are found in mathematical neuroscience, stochastic filtering and mean-field games. We will focus on a stochastic McKean--Vlasov system in which a collection of Brownian motions interact through a correlation which is a function of the proportion of particles that have been absorbed at level zero. We prove a law of large numbers where the limiting object is the unique solution to (the weak formulation of) the loss-dependent SPDE: dV_t(x) = 1/2 &part;_xxV_t(x)dt - p(L_t)&part;_xV_t(x)dW_t, V_t(0)=0, where L_t = 1-&lmoust;^&infin;<sub style='position: relative; left: -.8em;'>t</sub></sup>V_t(x)dx, V is a density process on the half-line and W is a Brownian motion. The correlation function is assumed to be piecewise Lipschitz, which encompasses a natural class of credit models. The first of our theoretical developments is to introduce the kernel smoothing method in the dual of the first Sobolev space, H^-1, with the aim of proving uniqueness results for SPDEs. A benefit of this approach is that only first order moment estimates of solutions are required, and in the particle setting this translates into studying the particles at an individual level rather than as a correlated collection. The second idea is to extend Skorokhod's M₁ topology to the space of processes that take values in the tempered distributions. The benefit we gain is that monotone functions have zero modulus of continuity under this topology, so the loss process, L, is easy to control. As a final example, we consider the fluctuations in the convergence of a basic particle system with constant correlation. This gives rise to a central limit theorem, for which the limiting object is a solution to an SPDE with random transport and an additive idiosyncratic driver acting on the first derivative terms. Conditional on the systemic random variables, this driver is a space-time white noise with intensity controlled by the empirical measure of the underlying system. The SPDE has insufficient regularity for us to work in any Sobolev space higher than H^-1, hence we have an example of where our extension to the kernel smoothing method is necessary.