統計学輪講 第11回

Deep neural networks have been shown to achieve nearly minimax-optimal rates in nonparametric regression when the target function has a hierarchical compositional structure, thereby mitigating the classical curse of dimensionality [1]. Motivated by this theory, Oga and Koike [2] studied drift estimation for discretely observed multidimensional diffusion processes using least-squares estimators over sparse ReLU neural-network classes. In this talk, we consider a Lévy-driven extension of their framework. The statistical problem is to estimate the drift coefficient of a multidimensional stochastic differential equation driven by both Brownian noise and jump noise from high-frequency discrete observations. After normalizing increments by the sampling interval, the problem can be viewed as a nonparametric regression problem with dependent observations, where the main new difficulty is the control of the compensated jump martingale term. We show that, under suitable light-tail assumptions on the Lévy measure, the DNN-based drift estimator attains the same convergence rate as in the diffusion case. We also discuss a diffusion plus Gaussian compound-Poisson jump submodel, on which the estimator is minimax optimal up to logarithmic factors.

[1] J. Schmidt-Hieber. Nonparametric regression using deep neural networks with ReLU activation function. The Annals of Statistics, 48(4):1875–1897, 2020.
[2] A. Oga and Y. Koike. Drift estimation for a multi-dimensional diffusion process using deep neural networks. Stochastic Processes and their Applications, 170:104240, 2024.