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Nonparametric Estimation of the Transition Density Function for Diffusion Processes

Fabienne Comte, Nicolas Marie

TL;DR

We address the problem of nonparametrically estimating the transition density $p_t(x,y)$ of a diffusion process from $N$ independent copies observed on $[0,2T]$, under smoothness and nondegeneracy assumptions that guarantee the density's existence. The authors propose a projection-based least squares estimator on a product space $\\mathcal{S}_{\mathbf m}$ and analyze its risk under an $f$-weighted norm, with a data-driven model selection that yields adaptive rates. They derive nonadaptive risk bounds, establish rates for anisotropic Sobolev-Hermite regularity, and propose a penalized criterion to select model dimensions $(m_1,m_2)$, achieving near-optimal bias-variance tradeoffs up to a $\\log(N)$ factor. Numerical experiments on Ornstein-Uhlenbeck and Cox–Ingersoll–Ross type diffusions illustrate the estimator's accuracy and the effectiveness of the adaptive model selection in practice.

Abstract

We assume that we observe $N$ independent copies of a diffusion process on a time-interval $[0,2T]$. For a given time $t$, we estimate the transition density $p_t(x,y)$, namely the conditional density of $X_{t + s}$ given $X_s = x$, under conditions on the diffusion coefficients ensuring that this quantity exists. We use a least squares projection method on a product of finite dimensional spaces, prove risk bounds for the estimator and propose an anisotropic model selection method, relying on several reference norms. A simulation study illustrates the theoretical part for Ornstein-Uhlenbeck or square-root (Cox-Ingersoll-Ross) processes.

Nonparametric Estimation of the Transition Density Function for Diffusion Processes

TL;DR

We address the problem of nonparametrically estimating the transition density of a diffusion process from independent copies observed on , under smoothness and nondegeneracy assumptions that guarantee the density's existence. The authors propose a projection-based least squares estimator on a product space and analyze its risk under an -weighted norm, with a data-driven model selection that yields adaptive rates. They derive nonadaptive risk bounds, establish rates for anisotropic Sobolev-Hermite regularity, and propose a penalized criterion to select model dimensions , achieving near-optimal bias-variance tradeoffs up to a factor. Numerical experiments on Ornstein-Uhlenbeck and Cox–Ingersoll–Ross type diffusions illustrate the estimator's accuracy and the effectiveness of the adaptive model selection in practice.

Abstract

We assume that we observe independent copies of a diffusion process on a time-interval . For a given time , we estimate the transition density , namely the conditional density of given , under conditions on the diffusion coefficients ensuring that this quantity exists. We use a least squares projection method on a product of finite dimensional spaces, prove risk bounds for the estimator and propose an anisotropic model selection method, relying on several reference norms. A simulation study illustrates the theoretical part for Ornstein-Uhlenbeck or square-root (Cox-Ingersoll-Ross) processes.
Paper Structure (24 sections, 10 theorems, 187 equations, 5 figures, 1 table)

This paper contains 24 sections, 10 theorems, 187 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Two motivating examples
  3. A projection least squares estimator of the transition density function
  4. Assumptions on the coefficients of Equation (\ref{['main_equation']})
  5. The projection least squares estimator and some related definitions
  6. Examples of bases
  7. Risk bounds on the projection least squares estimator
  8. Risk bound with respect to the empirical norm
  9. Risk bound with respect to the $f$-weighted norm, on a truncated version of the projection least squares estimator
  10. Rates in the anisotropic Sobolev-Hermite spaces
  11. Model selection
  12. General case
  13. Two special cases: $t > t_0 > 0$ and compactly supported bases
  14. Numerical experiments
  15. Concluding remarks
  16. ...and 9 more sections

Key Result

Proposition 3.1

There exist two positive constants $\mathfrak c_T$ and $\mathfrak m_T$, depending on $T$ but not on $t$, such that for every $x,y\in\mathbb R$, Then, the density function is well-defined and satisfies the following properties:

Figures (5)

  • Figure 1: Example 1. Transition density (left) and the estimation (right). Selected dimensions (4,5), 100*MISE = 0.22. $N = 200$, $T = 10$, $\Delta = 0.01$, $t = 1$.
  • Figure 2: Example 1. Full red line, the true and the estimation in dotted blue. Left: $x\mapsto p_t(x,y)$ for a fixed value of $y$ ($y = -0.27$ top and $y = -1$ bottom). Right: $y\mapsto p_t(x,y)$ for a fixed value of $x$ ($x = 0.03$ top and $x = -0.55$ bottom). $N = 200$, $T = 10$, $\Delta = 0.01$, $t = 1$.
  • Figure 3: Example 2. Transition density (left) and the estimation (right). Selected dimensions (2,41), 100*MISE = 0.16. $N = 200$, $T = 10$, $\Delta = 0.01$, $t = 1$.
  • Figure 4: Example 3. Transition density (left) and the estimation (right). Selected dimensions (6,9), 100*MISE = 0.29. $N = 200$, $T = 10$, $\Delta = 0.01$, $t = 1$.
  • Figure 5: Example 3. Full red line, the true and the estimation in dotted blue. Left: $x\mapsto p_t(x,y)$ for a fixed value of $y$ ($y = 0.71$). Right: $y\mapsto p_t(x,y)$ for a fixed value of $x$ ($x = 0.76$). $N = 200$, $T = 10$, $\Delta = 0.01$, $t = 1$.

Theorems & Definitions (12)

  • Proposition 3.1
  • proof
  • Theorem 4.2
  • Theorem 4.3
  • Definition 4.4
  • Proposition 4.5
  • Theorem 5.2
  • Proposition 5.4
  • Lemma A.1
  • Lemma A.2
  • ...and 2 more