Nonparametric estimation of scalar diffusions based on low frequency data

Abstract

We study the problem of estimating the coefficients of a diffusion (Xt, t ≥ 0); the estimation is based on discrete data Xn∆, n = 0, 1, . . . , N . The sampling frequency ∆ −1 is constant, and asymptotics are taken as the number N of observations tends to infinity. We prove that the problem of estimating both the diffusion coefficient (the volatility) and the drift in a nonparametric setting is ill-posed: the minimax rates of convergence for Sobolev constraints and squarederror loss coincide with that of a, respectively, first-and second-order linear inverse problem. To ensure ergodicity and limit technical difficulties we restrict ourselves to scalar diffusions living on a compact interval with reflecting boundary conditions.Our approach is based on the spectral analysis of the associated Markov semigroup. A rate-optimal estimation of the coefficients is obtained via the nonparametric estimation of an eigenvalue-eigenfunction pair of the transition operator of the discrete time Markov chain (Xn∆, n = 0, 1, . . . , N ) in a suitable Sobolev norm, together with an estimation of its invariant density.where the last line follows from Proposition 4.2. 4.3. Bias estimates. In a first estimation step, we bound the deterministic error due to the finite-dimensional projection π µ J P ∆ of P ∆ . We start with a lemma stating that π µ J and π J have similar approximation properties.Lemma 4.4. Let m : [0, 1] → [m 0 , m 1 ] be a measurable function with m 1 ≥ m 0 > 0. Denote by π m J the L 2 (m)-orthogonal projection onto the multiresolution space V J . Then there is a constant C = C(m 0 , m 1 ) such thatOn the other hand, the Bernstein inequality in V J and the Jackson inequality for Id − π J in H 1 and L 2 yield, for f ∈ H 1 ,where the constants depend only on the approximation spaces.Proposition 4.5. Uniformly over Θ s we have π µ J P ∆ − P ∆ H 1 →H 1 2 −Js .Proof. The transition density p ∆ is the kernel of the operator P ∆ . Hence, from Lemma 6.7 it follows that P ∆ : H 1 → H s+1 is continuous with a uniform norm bound over Θ s . Lemma 4.4 yieldsThe Jackson inequality in H 1 gives the result.Corollary 4.6. Let κ J 1 be the largest eigenvalue smaller than 1 of π µ J with eigenfunction u J 1 . Then uniformly over Θ s the following estimate holds:

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