Malliavin calculus for the optimal estimation of the invariant density of discretely observed diffusions in intermediate regime
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Let (X_t)_t ≥ 0 be solution of a one-dimensional stochastic differential equation. Our aim is to study the convergence rate for the estimation of the invariant density in intermediate regime, assuming that a discrete observation of the process (X_t)_t ∈ [0, T] is available, when T tends to ∞. We find the convergence rates associated to the kernel density estimator we proposed and a condition on the discretization step Δ_n which plays the role of threshold between the intermediate regime and the continuous case. In intermediate regime the convergence rate is n^- 2 β/2 β + 1, where β is the smoothness of the invariant density. After that, we complement the upper bounds previously found with a lower bound over the set of all the possible estimator, which provides the same convergence rate: it means it is not possible to propose a different estimator which achieves better convergence rates. This is obtained by the two hypothesis method; the most challenging part consists in bounding the Hellinger distance between the laws of the two models. The key point is a Malliavin representation for a score function, which allows us to bound the Hellinger distance through a quantity depending on the Malliavin weight.
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