Drift Estimation of Multiscale Diffusions Based on Filtered Data

Abstract

We study the problem of drift estimation for two-scale continuous time series. We set ourselves in the framework of overdamped Langevin equations, for which a single-scale surrogate homogenized equation exists. In this setting, estimating the drift coefficient of the homogenized equation requires pre-processing of the data, often in the form of subsampling; this is because the two-scale equation and the homogenized single-scale equation are incompatible at small scales, generating mutually singular measures on the path space. We avoid subsampling and work instead with filtered data, found by application of an appropriate kernel function, and compute maximum likelihood estimators based on the filtered process. We show that the estimators we propose are asymptotically unbiased and demonstrate numerically the advantages of our method with respect to subsampling. Finally, we show how our filtered data methodology can be combined with Bayesian techniques and provide a full uncertainty quantification of the inference procedure.

Frequentist perspective on robust parameter estimation using the ensemble Kalman filter

“…where u t ∈ R d denotes the vector of grid approximations at time t, D ∈ R d×d a finite difference approximation of the spatial derivative ∂ y , and W t standard d-dimensional Brownian motion. We can now set x = u, γ = ∆y −1 and identify either θ = U or θ = ρ as the unknown parameter in order to obtain a SDE of the form (1).…”

“…In this section, we investigate the impact of a possible discrepancy between the SDE model (1), for which we aim to estimate the parameter θ, and the data generating SDE (2). We therefore replace (2) by the the following two-scale SDE [16]:…”

“…The frequentist perspective also suggests a rather natural approach to the estimation of the required correction term in case an EnKBF is implemented without subsampling. We end this introductory paragraph with a reference to [1], which includes a broad survey on alternative estimation techniques. We also point to [8] for a more in-depth discussion of rough path theory in connection with filtering and parameter estimation.…”

Frequentist Perspective on Robust Parameter Estimation Using the Ensemble Kalman Filter

“…where u t ∈ R d denotes the vector of grid approximations at time t, D ∈ R d×d a finite difference approximation of the spatial derivative ∂ y , and W t the standard d-dimensional Brownian motion. We can now set X t = u t , γ = Δy −1 and identify either θ = U or θ = ρ as the unknown parameter in order to obtain an SDE of the form (1).…”

“…We end this introductory paragraph with a reference to [1], which includes a broad survey on alternative estimation techniques. We also point to [9] for an indepth discussion of rough path theory in connection to filtering and parameter estimation.…”