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On a Projection Least Squares Estimator for Jump Diffusion Processes

Hélène Halconruy, Nicolas Marie

Abstract

This paper deals with a projection least squares estimator of the drift function of a jump diffusion process $X$ computed from multiple independent copies of $X$ observed on $[0,T]$. Risk bounds are established on this estimator and on an associated adaptive estimator. Finally, some numerical experiments are provided.

On a Projection Least Squares Estimator for Jump Diffusion Processes

Abstract

This paper deals with a projection least squares estimator of the drift function of a jump diffusion process computed from multiple independent copies of observed on . Risk bounds are established on this estimator and on an associated adaptive estimator. Finally, some numerical experiments are provided.
Paper Structure (16 sections, 8 theorems, 76 equations, 3 figures, 1 table)

This paper contains 16 sections, 8 theorems, 76 equations, 3 figures, 1 table.

Table of Contents

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

Key Result

Lemma 1

${\rm trace}(\mathbf\Psi_{m}^{-1/2}\mathbf\Psi_{m,\sigma}\mathbf\Psi_{m}^{-1/2})\leqslant (\|\sigma\|_{\infty}^{2} +\lambda\mathfrak c_{\zeta^2}\|\gamma\|_{\infty}^{2})m$.

Figures (3)

  • Figure 1: Plots of $b$ and of $10$ adaptive estimations for Model 1 ($\overline{\widehat{m}} = 5.3$).
  • Figure 2: Plots of $b$ and of $10$ adaptive estimations for Model 2 ($\overline{\widehat{m}} = 4.2$).
  • Figure 3: Plots of $b$ and of $10$ adaptive estimations for Model 3 ($\overline{\widehat{m}} = 4.1$).

Theorems & Definitions (9)

  • Remark 1
  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6