Communications in Mathematical Sciences

Volume 11 (2013)

Number 4

Sub-sampling and parametric estimation for multiscale dynamics

Pages: 939 – 970

DOI: https://dx.doi.org/10.4310/CMS.2013.v11.n4.a3

Authors

Robert Azencott (Department of Mathematics, University of Houston, Houston, Texas, U.S.A.)

Arjun Beri (Mathematical Biosciences Institute, Ohio State University, Columbus, Ohio, U.S.A.)

Ankita Jain (Department of Mathematics, University of Houston, Houston, Texas, U.S.A.)

Ilya Timofeyev (Department of Mathematics, University of Houston, Houston, Texas, U.S.A.)

Abstract

We study the problem of adequate data sub-sampling for consistent parametric estimation of unobservable stochastic differential equations (SDEs), when the data are generated by multiscale dynamic systems approximating these SDEs in some suitable sense. The challenge is that the approximation accuracy is scale dependent, and degrades at very small temporal scales. Therefore, maximum likelihood parametric estimation yields inconsistent results when the sub-sampling time-step is too small. We use data from three multiscale dynamic systems—the Additive Triad, the Truncated Burgers-Hopf models, and the Model with the Fast-Oscillating Potential—to illustrate this sub-sampling problem. In addition, we also discuss an important practical question of constructing the bias-corrected estimators for a fixed but unknown value of the multiscale parameter.

Keywords

parametric estimation, stochastic differential equations, sub-sampling

2010 Mathematics Subject Classification

60F05, 60H10, 60J60, 62M05

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Published 15 June 2013