Bayesian Inference for partially observed SDEs Driven by Fractional Brownian Motion

We consider continuous-time diffusion models driven by fractional Brownian motion. Observations are assumed to possess a non-trivial likelihood given the latent path. Due to the non-Markovianity and high-dimensionality of the latent paths, estimating posterior expectations is a computationally challenging undertaking. We present a reparameterization framework based on the Davies and Harte method for sampling stationary Gaussian processes and use this framework to construct a Markov chain Monte Carlo algorithm that allows computationally efficient Bayesian inference. The Markov chain Monte Carlo algorithm is based on a version of hybrid Monte Carlo that delivers increased efficiency when applied on the high-dimensional latent variables arising in this context. We specify the methodology on a stochastic volatility model allowing for memory in the volatility increments through a fractional specification. The methodology is illustrated on simulated data and on the S&P500/VIX time series and is shown to be effective. Contrary to a long range dependence attribute of such models often assumed in the literature, with Hurst parameter larger than 1/2, the posterior distribution favours values smaller than 1/2, pointing towards medium range dependence

Bayesian inference for indirectly observed stochastic processes, applications to epidemic modelling

Stochastic processes are mathematical objects that offer a probabilistic representation of how some quantities evolve in time. In this thesis we focus on estimating the trajectory and parameters of dynamical systems in cases where only indirect observations of the driving stochastic process are available. We have first explored means to use weekly recorded numbers of cases of Influenza to capture how the frequency and nature of contacts made with infected individuals evolved in time. The latter was modelled with diffusions and can be used to quantify the impact of varying drivers of epidemics as holidays, climate, or prevention interventions. Following this idea, we have estimated how the frequency of condom use has evolved during the intervention of the Gates Foundation against HIV in India. In this setting, the available estimates of the proportion of individuals infected with HIV were not only indirect but also very scarce observations, leading to specific difficulties. At last, we developed a methodology for fractional Brownian motions (fBM), here a fractional stochastic volatility model, indirectly observed through market prices. The intractability of the likelihood function, requiring augmentation of the parameter space with the diffusion path, is ubiquitous in this thesis. We aimed for inference methods robust to refinements in time discretisations, made necessary to enforce accuracy of Euler schemes. The particle Marginal Metropolis Hastings (PMMH) algorithm exhibits this mesh free property. We propose the use of fast approximate filters as a pre-exploration tool to estimate the shape of the target density, for a quicker and more robust adaptation phase of the asymptotically exact algorithm. The fBM problem could not be treated with the PMMH, which required an alternative methodology based on reparameterisation and advanced Hamiltonian Monte Carlo techniques on the diffusion pathspace, that would also be applicable in the Markovian setting

Métropoles d'Amérique latine dans la mondialisation : reconfigurations territoriales, mobilité spatiale, action publique

Rapport final du projet METALThe objective of the project was to study the ongoing socio-spatial reconfigurations in the cities of Latin America since the 1980s, particularly the evolution of the settlement and of the social divisions of space. This study combines two inputs that are usually treated separately: the behaviour of inhabitants (practices of urban space and mobility systems, from international circulation to daily movements) and the various aspects of public action (migration policy, actions on housing or transportation). The comparative analysis of three cities (Bogotá, Santiago, São Paulo) was based on a consistent and uniform methodology, articulating a metropolitan-wide analysis and case studies on areas illustrative of the ongoing changes. The challenge was always to keep in perspective the global mechanisms of metropolitan areas with that of places or processes which logic can only be understood through detailed studies and production of original information, in particular regarding aspects that cannot be captured by available sources. The empirical work relied on the analysis of censuses micro-data and on the production of new data on mobility through surveys conducted in 2009 in these areas. The system of biographical surveys on mobility has combined in each city a questionnaire survey (about 1000 households) and in-depth interviews with a hundred inhabitants and with emigrants from the cities of interest living in Europe.L'objectif du projet était d'étudier les reconfigurations socio-spatiales en cours dans les métropoles d'Amérique latine depuis les années 1980, en particulier l'évolution du peuplement et des divisions sociales de l'espace. Cette étude a combiné deux entrées habituellement séparées : par le comportement des habitants (pratiques de l'espace urbain et systèmes de mobilités, depuis les circulations internationales jusqu'aux déplacements quotidiens) et par les différents registres de l'action publique (politique migratoire, action sur le logement ou le transport). L'analyse comparée de trois métropoles (Bogotá, Santiago, São Paulo) s'est appuyée sur une méthodologie commune, articulant une analyse à l'échelle métropolitaine et des études de cas sur des quartiers illustratifs des mutations en cours. L'enjeu était de garder systématiquement en perspective le fonctionnement des espaces métropolitains dans leur ensemble et celle de lieux ou de processus dont la logique ne peut être comprise que par une étude détaillée et en produisant de l'information originale sur des thèmes non saisis par les sources disponibles. Le travail empirique a reposé sur l'exploitation de recensements et sur la production de données nouvelles sur les mobilités par des enquêtes réalisées en 2009 dans ces quartiers. Le système d'enquêtes biographiques sur les mobilités a associé, dans chaque métropole, une enquête par questionnaires (près de 1000 ménages), et des entretiens approfondis auprès d'une centaine d'habitants et d'émigrants des métropoles étudiées résidant en Europe