Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions

By Richard Nickl, Kolyan Ray. Annals of Statistics, 2018

59 citations recorded by the source corpus.

The problem of determining a periodic Lipschitz vector field $b=(b_1, \dots, b_d)$ from an observed trajectory of the solution $(X_t: 0 \le t \le T)$ of the multi-dimensional stochastic differential equation \begin{equation*} dX_t = b(X_t)dt + dW_t, \quad t \geq 0, \end{equation*} where $W_t$ is a standard $d$-dimensional Brownian motion, is considered. Convergence rates of a penalised least squares estimator, which equals the maximum a posteriori (MAP) estimate corresponding to a high-dimensional Gaussian product prior, are derived. These results are deduced from corresponding contraction rates for the associated posterior distributions. The rates obtained are optimal up to log-factors in $L^2$-loss in any dimension, and also for supremum norm loss when $d \le 4$. Further, when $d \le 3$, nonparametric Bernstein-von Mises theorems are proved for the posterior distributions of $b$. From this we deduce functional central limit theorems for the implied estimators of the invariant measure $\mu_b$. The limiting Gaussian process distributions have a covariance structure that is asymptotically optimal from an information-theoretic point of view.

Subfields: Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/5624e23be041e215b3d6ef1f12f3a4f97f2cca23

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In Finance

  1. Richard Nickl
  2. Evarist Gin'e

In Statistics and Probability

  1. Gabriel P. Paternain
  2. F. Monard
  3. Richard Nickl
  4. Kolyan Ray
  5. I. Castillo
  6. Evarist Gin'e
  7. D. Chetverikov
  8. Sara A. van de Geer

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