Statistical Inference for Stochastic Differential Equations with Memory

TL;DR

This paper develops a statistical inference framework for discretely observed stochastic differential equations driven by fractional Brownian motion, allowing estimation of parameters including the memory index, using data augmentation and hybrid Monte Carlo methods.

Contribution

It introduces a novel inference approach for SDEs with memory, incorporating the estimation of the Hurst index and addressing long-range dependence in the noise.

Findings

Successfully estimated the memory parameter in US interest rates.

Demonstrated the importance of careful discretization for valid inference.

Extended the methodology to other rough-path driven processes.

Abstract

In this paper we construct a framework for doing statistical inference for discretely observed stochastic differential equations (SDEs) where the driving noise has 'memory'. Classical SDE models for inference assume the driving noise to be Brownian motion, or "white noise", thus implying a Markov assumption. We focus on the case when the driving noise is a fractional Brownian motion, which is a common continuous-time modeling device for capturing long-range memory. Since the likelihood is intractable, we proceed via data augmentation, adapting a familiar discretization and missing data approach developed for the white noise case. In addition to the other SDE parameters, we take the Hurst index to be unknown and estimate it from the data. Posterior sampling is performed via a Hybrid Monte Carlo algorithm on both the parameters and the missing data simultaneously so as to improve mixing. We point out that, due to the long-range correlations of the driving noise, careful discretization of the underlying SDE is necessary for valid inference. Our approach can be adapted to other types of rough-path driving processes such as Gaussian "colored" noise. The methodology is used to estimate the evolution of the memory parameter in US short-term interest rates.