We suppose that a Lévy process is observed at discrete time points. A rather general construction of minimum-distance estimators is shown to give consistent estimators of the Lévy-Khinchine characteristics as the number of observations tends to infinity, keeping the observation distance fixed. For a specific C 2 -criterion this estimator is rate-optimal. The connection with deconvolution and inverse problems is explained. A key step in the proof is a uniform control on the deviations of the empirical characteristic function on the whole real line.

Abstract. We investigate the problem of calibrating an exponential Lévy model based on market prices of vanilla options. We show that this inverse problem is in general severely ill-posed and we derive exact minimax rates of convergence. The estimation procedure we propose is based on the explicit inversion of the option price formula in the spectral domain and a cut-off scheme for high frequencies as regularisation.

In this paper, we clarify the relations between the existing sets of regularity conditions for convergence rates of nonparametric indirect regression (NPIR) and nonparametric instrumental variables (NPIV) regression models. We establish minimax risk lower bounds in mean integrated squared error loss for the NPIR and the NPIV models under two basic regularity conditions that allow for both mildly ill-posed and severely ill-posed cases. We show that both a simple projection estimator for the NPIR model, and a sieve minimum distance estimator for the NPIV model, can achieve the minimax risk lower bounds, and are rate-optimal uniformly over a large class of structure functions, allowing for mildly ill-posed and severely ill-posed cases.KEY WORDS: Nonparametric instrumental regression; Nonparametric indirect regression; Statistical ill-posed inverse problems; Minimax risk lower bound; Optimal rate.

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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: