Risk comparison theorems and application to deep learning of diffusion coefficients
Abstract
We investigate the nonparametric estimation of the diffusion matrix in stochastic differential equations featuring multidimensional, strong mixing covariate processes. We propose a flexible statistical framework based on general function classes that does not require a linear basis representation, rendering our results directly applicable to deep neural network estimators. Our approach employs a two-step estimation procedure: constructing a preliminary nonparametric quasi-likelihood estimator and subsequently regularizing it via a β-Hölder class approximation. We establish general risk comparison theorems between empirical and generalization risks for arbitrary estimators without relying on a specific probabilistic structure of the underlying process. In diffusion matrix learning based on n + 1 observations over the time interval [0,T], the derived upper bounds capture the intrinsic interplay between the T-rate, associated with the mixing behavior of the covariate process, and the intrinsic n-rate governing the volatility estimation.
Related
- Gaps theorems for linear forms with cubic coefficients
- Strong solutions of SDEs with critical discontinuities in diffusion coefficients
- Non-uniqueness for McKean--Vlasov SDEs with Hölder diffusion coefficients
- Eigenvalue comparison theorems for the Witten-Laplacian and the weighted p-Laplacian on complete manifolds with a modified Ricci curvature bounded from below
- Model-based clustering of diffusion processes with constant volatility coefficients using high-frequency data
BibTeX
@misc{gloter2026risk,
title = {{Risk comparison theorems and application to deep learning of diffusion coefficients}},
author = {Arnaud Gloter and Nakahiro Yoshida},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13898},
eprint = {2609.13898},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13898},
}
RIS
TY - UNPB
TI - Risk comparison theorems and application to deep learning of diffusion coefficients
AU - Arnaud Gloter
AU - Nakahiro Yoshida
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.13898
UR - https://arxiv.org/abs/2609.13898
AB - We investigate the nonparametric estimation of the diffusion matrix in stochastic differential equations featuring multidimensional, strong mixing covariate processes. We propose a flexible statistical framework based on general function classes that does not require a linear basis representation, rendering our results directly applicable to deep neural network estimators. Our approach employs a two-step estimation procedure: constructing a preliminary nonparametric quasi-likelihood estimator and subsequently regularizing it via a β-Hölder class approximation. We establish general risk comparison theorems between empirical and generalization risks for arbitrary estimators without relying on a specific probabilistic structure of the underlying process. In diffusion matrix learning based on n + 1 observations over the time interval [0,T], the derived upper bounds capture the intrinsic interplay between the T-rate, associated with the mixing behavior of the covariate process, and the intrinsic n-rate governing the volatility estimation.
ER -