Nonparametric estimation for SDE with sparsely sampled paths: An FDA perspective

Neda Mohammadi, Leonardo V. Santoro and Victor M. Panaretos

Stochastic Processes and their Applications, 2024, vol. 167, issue C

Abstract: We consider the problem of nonparametric estimation of the drift and diffusion coefficients of a Stochastic Differential Equation (SDE), based on n independent replicates Xi(t):t∈[0,1]1≤i≤n, observed sparsely and irregularly on the unit interval, and subject to additive noise corruption. By sparse we intend to mean that the number of measurements per path can be arbitrary (as small as two), and can remain constant with respect to n. We focus on time-inhomogeneous SDE of the form dX(t)=μ(t)X(t)αdt+σ(t)X(t)βdB(t), where α∈{0,1} and β∈{0,1/2,1}, which includes prominent examples such as Brownian motion, Ornstein–Uhlenbeck process, geometric Brownian motion and Brownian bridge. Our estimators are constructed by relating the local (drift/diffusion) parameters of the SDE to its global parameters (mean/covariance, and their derivatives) by means of an apparently novel Partial Differential Equation (PDE). This allows us to use methods inspired by functional data analysis, and pool information across the sparsely measured paths. The methodology we develop is fully non-parametric and avoids any functional form specification on the time-dependency of either the drift function or the diffusion function. We establish almost sure uniform asymptotic convergence rates of the proposed estimators as the number of observed paths n grows to infinity. Our rates are non-asymptotic in the number of measurements per path, explicitly reflecting how different sampling frequency might affect the speed of convergence. Our framework suggests possible further fruitful interactions between FDA and SDE methods in problems with replication.

Keywords: Drift and diffusion estimation; Itô diffusion process; Local linear smoothing; Nonparametric estimation; SDE; FDA; Brownian motion (search for similar items in EconPapers)
Date: 2024
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0304414923002119
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:167:y:2024:i:c:s0304414923002119

Ordering information: This journal article can be ordered from
http://http://www.elsevier.com/wps/find/supportfaq.cws_home/regional
https://shop.elsevie ... _01_ooc_1&version=01

DOI: 10.1016/j.spa.2023.104239

Access Statistics for this article

Stochastic Processes and their Applications is currently edited by T. Mikosch

More articles in Stochastic Processes and their Applications from Elsevier
Bibliographic data for series maintained by Catherine Liu ().