Inference for stochastic differential equations driven by weighted sub-fractional Brownian motion using neural networks and the Euler approximation
Abstract
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∈ℝ, we study dXt=a(Xt) dt+σ(Xt) dZtβ,f, X0=x0, 0≤ t≤ T. \smallskip\noindent Here a:ℝ→ℝ is the drift coefficient, σ:ℝ→(0,∞) is the diffusion coefficient, and Zβ,f is a centered Gaussian process from the weighted sub-fractional Brownian family, with covariance Cov(Zsβ,f,Ztβ,f) =∫0s∧ t f(r)qβ(s−r,t−r) dr, 0≤ s,t≤ T. \smallskip\noindent Here s∧ t=min{s,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 β≠1. 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.
Related
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BibTeX
@misc{ramirezgonzalez2026inference,
title = {{Inference for stochastic differential equations driven by weighted sub-fractional Brownian motion using neural networks and the Euler approximation}},
author = {J. H. Ramirez-Gonzalez},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00793},
eprint = {2610.00793},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00793},
note = {CC0},
}
RIS
TY - UNPB
TI - Inference for stochastic differential equations driven by weighted sub-fractional Brownian motion using neural networks and the Euler approximation
AU - J. H. Ramirez-Gonzalez
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00793
UR - https://arxiv.org/abs/2610.00793
AB - 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 x_0∈ℝ, we study dX_t=a(X_t) dt+σ(X_t) dZ_t^β,f, X_0=x_0, 0≤ t≤ T. \smallskip\noindent Here a:ℝ→ℝ is the drift coefficient, σ:ℝ→(0,∞) is the diffusion coefficient, and Z^β,f is a centered Gaussian process from the weighted sub-fractional Brownian family, with covariance Cov(Z_s^β,f,Z_t^β,f) =∫_0^s∧ t f(r)q_β(s−r,t−r) dr, 0≤ s,t≤ T. \smallskip\noindent Here s∧ t=min{s,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 q_β(u,v)=[u^β+v^β−(u+v)^β]/(1−β) when β≠1. Its continuous extension at β=1 is q_1(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.
ER -