Threshold estimation for stochastic processes with small noise

TL;DR

This paper introduces a filtering method to improve the stability of least squares estimators for stochastic differential equations with jump noise, showing promising asymptotic normality in numerical experiments.

Contribution

It proposes a novel filtering approach to enhance the stability and asymptotic properties of LSEs in jump noise scenarios, supported by initial theoretical insights.

Findings

Filter reduces large shocks in data

Estimator shows asymptotic normality in numerical tests

Potential for improved inference in jump processes

Abstract

Consider a process satisfying a stochastic differential equation with unknown drift parameter, and suppose that discrete observations are given. It is known that a simple least squares estimator (LSE) can be consistent, but numerically unstable in the sense of large standard deviations under finite samples when the noise process has jumps. We propose a filter to cut large shocks from data, and construct the same LSE from data selected by the filter. The proposed estimator can be asymptotically equivalent to the usual LSE, whose asymptotic distribution strongly depends on the noise process. However, in numerical study, it looked asymptotically normal in an example where filter was choosen suitably, and the noise was a L\'evy process. We will try to justify this phenomenon mathematically, under certain restricted assumptions.