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Optimal control for estimation in partially observed elliptic and hypoelliptic linear stochastic differential equations

Lookup NU author(s): Dr Quentin Clairon

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Abstract

Multi-dimensional stochastic differential equations (SDEs) are a powerful tool to describe dynamics of phenomena that change over time. We focus on the parametric estimation of such SDEs based on partial observations when only a one-dimensional component of the system is observable. We consider two families of SDE, the elliptic family with a full-rank diffusion coefficient and the hypoelliptic family with a degenerate diffusion coefficient. Estimation for the second class is much more difficult and only few estimation methods have been proposed. Here, we adopt the framework of the optimal control theory to derive a contrast (or cost function) based on the best control sequence mimicking the (unobserved) Brownian motion. We propose a full data-driven approach to estimate the drift and diffusion coefficient parameters. Numerical simulations made on different examples (Harmonic Oscillator, FitzHugh–Nagumo, Lotka–Volterra) reveal our method produces good pointwise estimate for an acceptable computational price with, interestingly, no performance drop for hypoelliptic systems.

Publication metadata

Author(s): Clairon Q, Samson A

Publication type: Article

Publication status: Published

Journal: Statistical Inference for Stochastic Processes

Year: 2020

Volume: 23

Pages: 105-127

Online publication date: 25/06/2019

Acceptance date: 11/06/2019

Date deposited: 28/06/2019

ISSN (print): 1387-0874

ISSN (electronic): 1572-9311

Publisher: Springer Netherlands

URL: https://doi.org/10.1007/s11203-019-09199-9

DOI: 10.1007/s11203-019-09199-9

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