Nonparametric Estimation for SDE with Sparsely Sampled Paths: an FDA Perspective

10/27/2021
by   Neda Mohammadi, et al.
0

We consider the problem of nonparametric estimation of the drift and diffusion coefficients of a Stochastic Differential Equation (SDE), based on n independent replicates {X_i(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 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^βdW_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 diffusion to their global parameters (mean/covariance, and their derivatives) by means of an apparently novel 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 curves 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.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
07/08/2023

Nonparametric estimation of the diffusion coefficient from S.D.E. paths

Consider a diffusion process X=(X_t), with t in [0,1], observed at discr...
research
03/14/2018

Maximum likelihood drift estimation for a threshold diffusion

We study the maximum likelihood estimator of the drift parameters of a s...
research
04/04/2020

Nonparametric Estimation for I.I.D. Paths of Fractional SDE

This paper deals with nonparametric projection estimators of the drift f...
research
10/24/2022

Nonparametric Drift Estimation from Diffusions with Correlated Brownian Motions

In the present paper, we consider that N diffusion processes X^1,…,X^N a...
research
11/15/2018

State-dependent jump activity estimation for Markovian semimartingales

The jump behavior of an infinitely active Itô semimartingale can be conv...
research
05/25/2021

Functional Data Analysis with Rough Sampled Paths?

Functional data are typically modeled as sampled paths of smooth stochas...
research
06/01/2023

From sparse to dense functional data in high dimensions: Revisiting phase transitions from a non-asymptotic perspective

Nonparametric estimation of the mean and covariance functions is ubiquit...

Please sign up or login with your details

Forgot password? Click here to reset