In many physical, technological, social, and economic applications, one is commonly faced with the task of estimating statistical properties, such as mean first passage times of a temporal continuous process, from empirical data (experimental observations). Typically, however, an accurate and reliable estimation of such properties directly from the data alone is not possible as the time series is often too short, or the particular phenomenon of interest is only rarely observed. We propose here a theoretical-computational framework which provides us with a systematic and rational estimation of statistical quantities of a given temporal process, such as waiting times between subsequent bursts of activity in intermittent signals. Our framework is illustrated with applications from real-world data sets, ranging from marine biology to paleoclimatic data.

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.

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Many applications across sciences and technologies require a careful quantification of non-deterministic effects to a system output, for example when evaluating the system's reliability or when gearing it towards more robust operation conditions. At the heart of these considerations lies an accurate yet efficient characterization of uncertain system outputs. In this work we introduce and analyze novel multilevel Monte Carlo techniques for an efficient characterization of an uncertain system output's distribution. These techniques rely on accurately approximating general parametric expectations, i.e. expectations that depend on a parameter, uniformly on an interval. Applications of interest include, for example, the approximation of the characteristic function or of the cumulative distribution function of an uncertain system output. A further important consequence of the introduced approximation techniques for parametric expectations (i.e. for functions) is that they allow to construct multilevel Monte Carlo estimators for various robustness indicators, such as for quantiles (also known as value-at-risk) and for the conditional value-at-risk. These robustness indicators cannot be expressed as moments and are thus not easily accessible usually. In fact, here we provide a framework that allows to simultaneously estimate a cumulative distribution function, a quantile, and the associated conditional value-at-risk of an uncertain system output at the cost of a single multilevel Monte Carlo simulation, while each estimated quantity satisfies a prescribed tolerance goal.