Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions
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The problem of determining a periodic Lipschitz vector field b=(b_1, ..., b_d) from an observed trajectory of the solution (X_t: 0 < t < T) of the multi-dimensional stochastic differential equation dX_t = b(X_t)dt + dW_t, t ≥ 0, 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 < 4. Further, when d < 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 μ_b. The limiting Gaussian process distributions have a covariance structure that is asymptotically optimal from an information-theoretic point of view.
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