Nonparametric Estimation in Delayed Equations Driven by a Differentiable Approximation of the Brownian Motion
Abstract
This paper deals with a nonparametric estimator of the drift function b in a delayed differential equation perturbed by a differentiable approximation Wε (ε > 0) of the Brownian motion. Unlike diffusion processes, the solution Yε of this equation has continuously differentiable paths and a memory controlled by ε. For these reasons, Yε is well-suited for applications in pharmacokinetics or mathematical finance. The first part of the paper deals with probabilistic properties of the delayed equation, which are crucial for our statistical purposes. In particular, the absolute continuity of the probability distribution of Ytε (t∈ (0,T]) with respect to Lebesgue's measure on ℝ is investigated via the Malliavin calculus. Thanks to these probabilistic preliminaries, the second part of the paper deals with a (nonparametric) projection least squares estimator b̂m of b computed from N independent copies of Yε (0 < mleqslant N). Precisely, a risk bound is established on b̂m, and then a model selection procedure is investigated.
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BibTeX
@misc{marie2026nonparametric,
title = {{Nonparametric Estimation in Delayed Equations Driven by a Differentiable Approximation of the Brownian Motion}},
author = {Nicolas Marie},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.science/hal-05712907v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Nonparametric Estimation in Delayed Equations Driven by a Differentiable Approximation of the Brownian Motion
AU - Nicolas Marie
PY - 2026
DA - 2026/10/27
PB - HAL
UR - https://hal.science/hal-05712907v2
AB - This paper deals with a nonparametric estimator of the drift function b in a delayed differential equation perturbed by a differentiable approximation W^ε (ε > 0) of the Brownian motion. Unlike diffusion processes, the solution Y^ε of this equation has continuously differentiable paths and a memory controlled by ε. For these reasons, Y^ε is well-suited for applications in pharmacokinetics or mathematical finance. The first part of the paper deals with probabilistic properties of the delayed equation, which are crucial for our statistical purposes. In particular, the absolute continuity of the probability distribution of Y_t^ε (t∈ (0,T]) with respect to Lebesgue's measure on ℝ is investigated via the Malliavin calculus. Thanks to these probabilistic preliminaries, the second part of the paper deals with a (nonparametric) projection least squares estimator b̂_m of b computed from N independent copies of Y^ε (0 < mleqslant N). Precisely, a risk bound is established on b̂_m, and then a model selection procedure is investigated.
ER -