Optimal control for parameter estimation in partially observed hypoelliptic stochastic differential equations

TL;DR

This paper introduces a novel control-based method for parameter estimation in partially observed hypoelliptic SDEs, overcoming challenges of traditional filtering and discretization techniques, and demonstrating improved accuracy and efficiency.

Contribution

It develops a new cost function and control approach that works for both elliptic and hypoelliptic SDEs, bypassing filtering and discretization issues.

Findings

Accurate parameter estimates in hypoelliptic SDEs.

Significant reduction in computational cost.

Robustness across different types of SDEs.

Abstract

We deal with the problem of parameter estimation in stochastic differential equations (SDEs) in a partially observed framework. We aim to design a method working for both elliptic and hypoelliptic SDEs, the latters being characterized by degenerate diffusion coefficients. This feature often causes the failure of contrast estimators based on Euler Maruyama discretization scheme and dramatically impairs classic stochastic filtering methods used to reconstruct the unobserved states. All of theses issues make the estimation problem in hypoelliptic SDEs difficult to solve. To overcome this, we construct a well-defined cost function no matter the elliptic nature of the SDEs. We also bypass the filtering step by considering a control theory perspective. The unobserved states are estimated by solving deterministic optimal control problems using numerical methods which do not need strong assumptions on the diffusion coefficient conditioning. Numerical simulations made on different partially observed hypoelliptic SDEs reveal our method produces accurate estimate while dramatically reducing the computational price comparing to other methods.