Semiparametric Drift and Diffusion Estimation for Multiscale Diffusions

Abstract

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.

Nonlinear Forecasting of the Generalized Kuramoto–Sivashinsky Equation

“…For instance, how can one use elements from stochastic processes (e.g. in [Krumscheid et al, 2013]) to appropriately modify the nonlinear forecasting method when the gKS equation is postulated for a nonlinear process for which the precise underlying model is not available. Another interesting problem would be the extension of the methodology proposed here to more involved equations, such as those describing the dynamics of the falling film away from criticality.…”

Perturbation-based inference for diffusion processes: Obtaining effective models from multiscale data

“…Here we introduce a procedure for the parametric inference problem of diffusion processes which is motivated by the recent computational results in [26,31]. In fact, we extend and generalise the introduced procedure further to make it more amenable to a theoretical treatment.…”

“…In fact, for t ≥ 0.1 the relative error drops well below 5% with only minor fluctuations. Since bound (31) is not guaranteed to be valid in this case, the constants in front of the rates might depends on other parameters (see discussion in Section 4.3). Therefore we chose a rather long time series to focus solely on ε-stability, that is on the influence of the perturbation of the input, and to illustrate the convergent behaviour of the estimation procedure.…”

Data-driven coarse graining in action: Modeling and prediction of complex systems

“…It is worth emphasizing that while the MLE approach works well for data sets (time series) with a single characteristic time scale, it becomes asymptotically biased when applied to data coming from multiscale stochastic systems. For such systems, statistical inference methodologies that take into account the multiscale nature of the data set have to be used [11,12]. We emphasize though, that the general framework as illustrated in Fig.…”