Minimax Rates for Nonparametric Drift Estimation in Affine Stochastic Delay Differential Equations

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Abstract

Let X be a stationary process satisfying the stochastic differential equation with time delay

The weight function g∈L2([−r,0]) is estimated nonparametrically from the continuous observation of a trajectory of X up to time T > 0. The estimation problem is transformed into an illposed inverse problem with stochastically perturbed operator and data so that an estimator of g may be constructed by the Ritz–Galerkin projection method. The L2-risk of the estimator is asymptotically for T → ∞ of order T−s/2s+3, where g is assumed to lie in some Sobolev ball of order s. This rate is shown to be optimal in a minimax sense.

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