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Nonparametric Drift Estimation from Diffusions with Correlated Brownian Motions

Fabienne Comte, Nicolas Marie

TL;DR

This work addresses nonparametric drift estimation for a collection of diffusion processes driven by correlated Brownian motions over a fixed horizon. It introduces a projection-based least squares estimator that does not require knowledge of the correlation matrix $R$, and it derives nonasymptotic risk bounds that explicitly depend on the dependence structure through terms like $1+ frac{1}{N} ext{sum}_{i eq k}|R_{i,k}|$. An adaptive estimator is proposed via model selection with a data-driven penalty, achieving near-optimal bias-variance trade-offs across a model collection. Numerical experiments validate robustness to varying correlation levels and demonstrate effective dimension selection with both cosine and Hermite bases. The results extend drift estimation theory to dependent diffusion systems and provide practical guidance for handling correlation without fully specifying $R$ in applications.

Abstract

In the present paper, we consider that $N$ diffusion processes $X^1,\dots,X^N$ are observed on $[0,T]$, where $T$ is fixed and $N$ grows to infinity. Contrary to most of the recent works, we no longer assume that the processes are independent. The dependency is modeled through correlations between the Brownian motions driving the diffusion processes. A nonparametric estimator of the drift function, which does not use the knowledge of the correlation matrix, is proposed and studied. Its integrated mean squared risk is bounded and an adaptive procedure is proposed. Few theoretical tools to handle this kind of dependency are available, and this makes our results new. Numerical experiments show that the procedure works in practice.

Nonparametric Drift Estimation from Diffusions with Correlated Brownian Motions

TL;DR

This work addresses nonparametric drift estimation for a collection of diffusion processes driven by correlated Brownian motions over a fixed horizon. It introduces a projection-based least squares estimator that does not require knowledge of the correlation matrix , and it derives nonasymptotic risk bounds that explicitly depend on the dependence structure through terms like . An adaptive estimator is proposed via model selection with a data-driven penalty, achieving near-optimal bias-variance trade-offs across a model collection. Numerical experiments validate robustness to varying correlation levels and demonstrate effective dimension selection with both cosine and Hermite bases. The results extend drift estimation theory to dependent diffusion systems and provide practical guidance for handling correlation without fully specifying in applications.

Abstract

In the present paper, we consider that diffusion processes are observed on , where is fixed and grows to infinity. Contrary to most of the recent works, we no longer assume that the processes are independent. The dependency is modeled through correlations between the Brownian motions driving the diffusion processes. A nonparametric estimator of the drift function, which does not use the knowledge of the correlation matrix, is proposed and studied. Its integrated mean squared risk is bounded and an adaptive procedure is proposed. Few theoretical tools to handle this kind of dependency are available, and this makes our results new. Numerical experiments show that the procedure works in practice.
Paper Structure (18 sections, 7 theorems, 145 equations, 3 figures, 2 tables)

This paper contains 18 sections, 7 theorems, 145 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminary motivation and example in the parametric framework
  3. A projection least squares estimator of the drift function
  4. The objective function
  5. The projection least squares estimator and some related matrices
  6. Risk bound on the projection least squares estimator
  7. Model selection
  8. Numerical experiments
  9. Proofs
  10. Proof of Lemma \ref{['preliminary_lemma_variance']}
  11. Proof of Theorem \ref{['risk_bound']}
  12. Steps of the proof
  13. Proof of Lemma \ref{['bound_variance_remainder']}
  14. Proof of Lemma \ref{['deviation_probabilities']}
  15. Proof of Theorem \ref{['bound_adaptive_estimator']}
  16. ...and 3 more sections

Key Result

Lemma 1

Under Assumption assumption_basis, for $\sigma$ belonging to $\mathbb L^2(\mathbb R,f_T(x)dx)$ but possibly unbounded, with If in addition $\sigma$ is bounded, then

Figures (3)

  • Figure 1: Example 3. True functions in bold red and beam of 25 estimated drift $b_3$ with Hermite (left) and cosine (right) bases, $\rho = 0.5$. The MISE$\times 100$ are $0.12$, $0.33$ and the mean of selected dimensions are $8.4$, $4.3$.
  • Figure 2: Example 4. True functions in bold red and beam of 25 estimated drift $b_4$ with Hermite (left) and cosine (right) bases, $\rho = 0.5$. The MISE$\times 100$ are $0.36$, $0.26$ and the mean of selected dimensions are $4.4$, $6.0$.
  • Figure 3: Example 5. True functions in bold red and beam of 25 estimated drift $b_5$ with Hermite basis for $\rho = 0$ (left) and $\rho = 0.9$ (right). The MISE$\times 100$ are $0.87$, $1.67$ and the mean of selected dimensions are $12.5$ in both cases.

Theorems & Definitions (7)

  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5