# Inference for stochastic differential equations driven by weighted sub-fractional Brownian motion using neural networks and the Euler approximation

arXiv: https://arxiv.org/abs/2610.00793 · HTML: https://mlsanity.com/paper/2610.00793/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 14 references that are in the map.
- First posted on arXiv 30 Sep 2026.

## Builds on (most cited first, 6 of 6)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [Learning effective stochastic differential equations from microscopic simulations: linking stochastic numerics to deep learning](https://mlsanity.com/paper/2106.09004/) (2021)
- [An end-to-end deep learning approach for extracting stochastic dynamical systems with α-stable L'evy noise](https://mlsanity.com/paper/2201.13114/) (2022)
- [Integral Fractional Ornstein-Uhlenbeck Process Model for Animal Movement](https://mlsanity.com/paper/2312.09900/) (2023)
- [Multidimensional Integral Fractional Ornstein--Uhlenbeck Process with an Application to Animal Movement](https://mlsanity.com/paper/2608.03381/) (2026)

## Abstract (arXiv)

We consider the estimation of drift, diffusion, and noise covariance from discrete observations of stochastic differential equations driven by Gaussian processes. For a fixed observation horizon T>0 and a known initial state x0∈mathbb R, we study beginequation* dXt=a(Xt),dt+σ(Xt),dZtβ,f, qquad X0=x0,quad 0≤ t≤ T. endequation* smallskipnoindent Here a:mathbb R→mathbb R is the drift coefficient, σ:mathbb R→(0,∞) is the diffusion coefficient, and Zβ,f is a centered Gaussian process from the weighted sub-fractional Brownian family, with covariance beginequation* operatornameCov(Zsβ,f,Ztβ,f) =int0swedge t f(r)qβ(s-r,t-r),dr, qquad 0≤ s,t≤ T. endequation* smallskipnoindent Here swedge t=mins,t. The temporal weight f:[0,T]→[0,∞) is measurable, bounded, and positive almost everywhere, and β∈(0,2) is the covariance exponent. For u,v≥0, the kernel is qβ(u,v)=[uβ+vβ-(u+v)β]/(1-β) when βne1. Its continuous extension at β=1 is q1(u,v)=(u+v)log(u+v)-ulog u-vlog v, with 0log0=0. Using the Euler approximation, we reconstruct the Gaussian driving increments from observed transitions and use their joint density to obtain a trajectory likelihood. Neural and radial-basis representations model the drift, diffusion, and normalized temporal weight, while a likelihood profile estimates the covariance exponent and diffusion scale. We compare the method with two neural alternatives on the same simulated trajectories in twenty coefficient settings.