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Parameter estimation and singularity of laws on the path space for SDEs driven by Rosenblatt processes

Petr Čoupek, Pavel Kříž, Bohdan Maslowski

Abstract

In this paper, we study parameter identification for solutions to (possibly non-linear) SDEs driven by additive Rosenblatt process and singularity of the induced laws on the path space. We propose a joint estimator for the drift parameter, diffusion intensity, and Hurst index that can be computed from discrete-time observations with a bounded time horizon and we prove its strong consistency (as well as the speed of convergence) under in-fill asymptotics with a fixed time horizon. As a consequence of this strong consistency, singularity of measures generated by the solutions with different drifts is shown. This results in the invalidity of a Girsanov-type theorem for Rosenblatt processes.

Parameter estimation and singularity of laws on the path space for SDEs driven by Rosenblatt processes

Abstract

In this paper, we study parameter identification for solutions to (possibly non-linear) SDEs driven by additive Rosenblatt process and singularity of the induced laws on the path space. We propose a joint estimator for the drift parameter, diffusion intensity, and Hurst index that can be computed from discrete-time observations with a bounded time horizon and we prove its strong consistency (as well as the speed of convergence) under in-fill asymptotics with a fixed time horizon. As a consequence of this strong consistency, singularity of measures generated by the solutions with different drifts is shown. This results in the invalidity of a Girsanov-type theorem for Rosenblatt processes.
Paper Structure (7 sections, 7 theorems, 123 equations, 6 figures)

This paper contains 7 sections, 7 theorems, 123 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Scaled Rosenblatt process
  4. SDEs driven by additive Rosenblatt process
  5. Estimating drift
  6. Estimating all parameters
  7. Simulations

Key Result

Theorem 2.1

For any $H\in (1/2,1)$, there is the following convergence:

Figures (6)

  • Figure 1: Sample solutions to the equations (ROU), (RSDE) and (FOU) ordered from left to right.
  • Figure 2: Boxplots of estimates $\hat{H}_N$ of Hurst parameter $H$ (top) and log-log plots of corresponding RMSE (bottom) for equations (ROU), (RSDE) and (FOU) (ordered from left to right). Red lines represent true parameter values.
  • Figure 3: Boxplots of estimates $\hat{\sigma}_N$ of noise intensity $\sigma$ (top) and log-log plots of corresponding RMSE (bottom) for equations (ROU), (RSDE) and (FOU) (ordered from left to right). Red lines represent true parameter values.
  • Figure 4: Boxplots of drift estimates $\hat{\lambda}_{1,N}$ considering $H$ and $\sigma$ known (top), and the optimized plug-in estimator $\hat{\lambda}_{N}^{\delta^o}$ with $H$ and $\sigma$ unknown (bottom) for equations (ROU), (RSDE) and (FOU) ordered from left to right. Red line represents true values.
  • Figure 5: Log-log plots of sample RMSE for drift estimates $\hat{\lambda}_{1,N}$ (top) and $\hat{\lambda}_{N}^{\delta^o}$ (bottom) for equations (ROU), (RSDE) and (FOU) ordered from left to right.
  • ...and 1 more figures

Theorems & Definitions (21)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • Remark 4.2
  • Remark 4.3
  • ...and 11 more