Semi-Parametric Drift and Diffusion Estimation for Multiscale Diffusions

We consider the problem of statistical inference for the effective dynamics of multiscale diffusion processes with (at least) two widely separated characteristic time scales. More precisely, we seek to determine parameters in the effective equation describing the dynamics on the longer diffusive time scale, i.e. in a homogenization framework. We examine the case where both the drift and the diffusion coefficients in the effective dynamics are space-dependent and depend on multiple unknown parameters. It is known that classical estimators, such as Maximum Likelihood and Quadratic Variation of the Path Estimators, fail to obtain reasonable estimates for parameters in the effective dynamics when based on observations of the underlying multiscale diffusion. We propose a novel algorithm for estimating both the drift and diffusion coefficients in the effective dynamics based on a semi-parametric framework. We demonstrate by means of extensive numerical simulations of a number of selected examples that the algorithm performs well when applied to data from a multiscale diffusion. These examples also illustrate that the algorithm can be used effectively to obtain accurate and unbiased estimates.Comment: 32 pages, 10 figure

Exotic aromatic B-series for the study of long time integrators for a class of ergodic SDEs

We introduce a new algebraic framework based on a modification (called exotic) of aromatic Butcher-series for the systematic study of the accuracy of numerical integrators for the invariant measure of a class of ergodic stochastic differential equations (SDEs) with additive noise. The proposed analysis covers Runge-Kutta type schemes including the cases of partitioned methods and postprocessed methods. We also show that the introduced exotic aromatic B-series satisfy an isometric equivariance property.Comment: 33 page

Statistical and numerical methods for diffusion processes with multiple scales

In this thesis we address the problem of data-driven coarse-graining, i.e. the process of inferring simplified models, which describe the evolution of the essential characteristics of a complex system, from available data (e.g. experimental observation or simulation data). Specifically, we consider the case where the coarse-grained model can be formulated as a stochastic differential equation. The main part of this work is concerned with data-driven coarse-graining when the underlying complex system is characterised by processes occurring across two widely separated time scales. It is known that in this setting commonly used statistical techniques fail to obtain reasonable estimators for parameters in the coarse-grained model, due to the multiscale structure of the data. To enable reliable data-driven coarse-graining techniques for diffusion processes with multiple time scales, we develop a novel estimation procedure which decisively relies on combining techniques from mathematical statistics and numerical analysis. We demonstrate, both rigorously and by means of extensive simulations, that this methodology yields accurate approximations of coarse-grained SDE models. In the final part of this work, we then discuss a systematic framework to analyse and predict complex systems using observations. Specifically, we use data-driven techniques to identify simple, yet adequate, coarse-grained models, which in turn allow to study statistical properties that cannot be investigated directly from the time series. The value of this generic framework is exemplified through two seemingly unrelated data sets of real world phenomena.Open Acces

Regression-based Monte Carlo methods with optimal control variates

In der vorliegenden Dissertation werden regressionsbasierte Monte-Carlo-Verfahren für diskretisierte Diffusionsprozesse vorgestellt. Diese Verfahren beinhalten die Konstruktion von geeigneten Kontrollvariaten, die zu einer signifikanten Reduktion der Varianz führen. Dadurch kann die Komplexität des Standard-Monte-Carlo-Ansatzes (epsilon^{-3} für Schemen erster Ordnung und epsilon^{-2.5} für Schemen zweiter Ordnung) im besten Fall reduziert werden auf eine Ordnung von epsilon^{-2+delta} für ein beliebiges delta aus [0,0.25), wobei epsilon die zu erzielende Genauigkeit bezeichnet. In der Komplexitätsanalyse werden sowohl die Fehler, die auch beim Standard-Monte-Carlo-Ansatz auftreten (Diskretisierungs- und statistischer Fehler), als auch die aus der Schätzung bedingter Erwartungswerte mittels Regression resultierenden Fehler berücksichtigt. Darüber hinaus werden verschiedene Algorithmen hergeleitet, die zwar zu einer ähnlichen theoretischen Komplexität führen, jedoch numerisch gesehen bei der Regressionsschätzung unterschiedlich stabil und genau sind. Die Effektivität dieser Algorithmen wird anhand von numerischen Beispielen veranschaulicht und mit anderen bekannten Methoden verglichen. Zudem werden geeignete Kontrollvariaten für die Bewertung von Bermuda-Optionen sowie amerikanischen Optionen basierend auf einer dualen Monte-Carlo-Methode hergeleitet. Auch hierbei ergibt sich eine signifikante Komplexitätsreduktion, sofern die zugrunde liegenden Funktionen gewisse Glattheitsannahmen erfüllen

Topics in multiscale modeling: numerical analysis and applications

We explore several topics in multiscale modeling, with an emphasis on numerical analysis and applications. Throughout Chapters 2 to 4, our investigation is guided by asymptotic calculations and numerical experiments based on spectral methods. In Chapter 2, we present a new method for the solution of multiscale stochastic differential equations at the diffusive time scale. In contrast to averaging-based methods, the numerical methodology that we present is based on a spectral method. We use an expansion in Hermite functions to approximate the solution of an appropriate Poisson equation, which is used in order to calculate the coefficients in the homogenized equation. Extensions of this method are presented in Chapter 3 and 4, where they are employed for the investigation of the Desai—Zwanzig mean-field model with colored noise and the generalized Langevin dynamics in a periodic potential, respectively. In Chapter 3, we study in particular the effect of colored noise on bifurcations and phase transitions induced by variations of the temperature. In Chapter 4, we investigate the dependence of the effective diffusion coefficient associated with the generalized Langevin equation on the parameters of the equation. In Chapter 5, which is independent from the rest of this thesis, we introduce a novel numerical method for phase-field models with wetting. More specifically, we consider the Cahn—Hilliard equation with a nonlinear wetting boundary condition, and we propose a class of linear, semi-implicit time-stepping schemes for its solution.Open Acces

Recursive Bayesian inference on stochastic differential equations

This thesis is concerned with recursive Bayesian estimation of non-linear dynamical systems, which can be modeled as discretely observed stochastic differential equations. The recursive real-time estimation algorithms for these continuous-discrete filtering problems are traditionally called optimal filters and the algorithms for recursively computing the estimates based on batches of observations are called optimal smoothers. In this thesis, new practical algorithms for approximate and asymptotically optimal continuous-discrete filtering and smoothing are presented. The mathematical approach of this thesis is probabilistic and the estimation algorithms are formulated in terms of Bayesian inference. This means that the unknown parameters, the unknown functions and the physical noise processes are treated as random processes in the same joint probability space. The Bayesian approach provides a consistent way of computing the optimal filtering and smoothing estimates, which are optimal given the model assumptions and a consistent way of analyzing their uncertainties. The formal equations of the optimal Bayesian continuous-discrete filtering and smoothing solutions are well known, but the exact analytical solutions are available only for linear Gaussian models and for a few other restricted special cases. The main contributions of this thesis are to show how the recently developed discrete-time unscented Kalman filter, particle filter, and the corresponding smoothers can be applied in the continuous-discrete setting. The equations for the continuous-time unscented Kalman-Bucy filter are also derived. The estimation performance of the new filters and smoothers is tested using simulated data. Continuous-discrete filtering based solutions are also presented to the problems of tracking an unknown number of targets, estimating the spread of an infectious disease and to prediction of an unknown time series.reviewe

Weak and strong approximation of the Log-Heston model by Euler-Type methods and related topics

This thesis deals with the weak and strong numerical approximation of so-called stochastic volatility models. In particular, the focus is on the log-Heston model and its associated Euler methods, for which there have been only a few convergence results with a polynomial rate in the literature so far. The biggest challenge here is the approximation of the CIR process, which models the stochastic variance and whose diffusion coefficient is not Lipschitz continuous. We first study the weak order of convergence of two Euler methods that keep the approximation of the CIR process positive. When the Feller index ν of the CIR process is greater than one, weak convergence of order one is obtained as under standard assumptions. For ν ≤ 1 we obtain a weak order of convergence of ν − ε for ε > 0 arbitrarily small. For the L1-error for a large class of Euler methods, we can recover the order 1/2 obtained under standard assumptions under the condition ν > 1. Moreover, we prove that this is already the optimal L1-convergence order for the log-Heston model. Finally, in the last part of this dissertation we deal with the optimal L2 approximation of more general stochastic volatility models

Doctor of Philosophy

dissertationActive transport of cargoes is critical for cellular function. To accomplish this, networks of cytoskeletal filaments form highways along which small teams of mechanochemical enzymes (molecular motors) take steps to pull associated cargoes. The robustness of this transport system is juxtaposed by the stochasticity that exists at several spatial and temporal scales. For instance, individual motors stochastically step, bind, and unbind while the cargo undergoes nonnegligible thermal fluctuations. Experimental advances have produced rich quantitative measurements of each of these stochastic elements, but the interaction between them remains elusive. In this thesis, we explore the roles of stochasticity in motor-mediated transport with four specific projects at different scales. We first construct a mean-field model of a cargo transported by two teams of opposing motors. This system is known to display bidirectionality: switching between phases of transport in opposite directions. We hypothesize that thermal fluctuations of the cargo drive the switching. From our model, we predict how cargo size influences the switching time, an experimentally measurable quantity to verify the hypothesis. In the second work, we investigate the force dependence of motor stepping, formulated as a state-dependent jump-diffusion model. We prove general results regarding the computation of the statistics of this process. From this framework, we find that thermal fluctuations may provide a nonmonotonic influence on the stepping rate of motors. The remaining projects investigate the behavior of nonprocessive motors, which take few steps before detaching. In collaboration with experimentalists, we study seemingly diffusive data of motor-mediated transport. Using a jump-diffusion model, the active and passive portions of the diffusivity are disentangled, and curious higher order statistics are explained as a sampling issue. Lastly, we construct a model of cooperative transport by nonprocessive motors, which we study using reward-renewal theory. The theory provides predictions about measured quantities such as run length, which suggest that geometric effects have a large influence on the transport ability of these motors