# Statistical Inference for Stochastic Differential Equations with Memory

arXiv: https://arxiv.org/abs/1307.1164 · HTML: https://mlsanity.com/paper/1307.1164/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Jul 2013.

## Built on by (most cited first, 3 of 3)

- [Variational Inference for SDEs Driven by Fractional Noise](https://mlsanity.com/paper/2310.12975/) (2023)
- [Efficient Training of Neural SDEs Using Stochastic Optimal Control](https://mlsanity.com/paper/2505.17150/) (2025)
- [Meta-Posterior Consistency for the Bayesian Inference of Metastable System](https://mlsanity.com/paper/2408.01868/) (2024)

## Abstract (arXiv)

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.