Jean Jacod

Volatility estimators for discretely sampled Lévy processes

This paper studies the estimation of the volatility parameter in a model where the driving process is a Brownian motion or a more general symmetric stable process that is perturbed by another Levy process. We distinguish between a parametric case, where the law of the perturbing process is known, and a semiparametric case, where it is not. In the parametric case, we construct estimators which are asymptotically efficient. In the semiparametric case, we can obtain asymptotically efficient estimators by sampling at a sufficiently high frequency, and these estimators are efficient uniformly in the law of the perturbing process.

Non‐parametric Kernel Estimation of the Coefficient of a Diffusion

In this work we exhibit a non‐parametric estimator of kernel type, for the diffusion coefficient when one observes a one‐dimensional diffusion process at times i/n for i = , ..., n and study its asymptotics as n←∞. When the diffusion coefficient has regularity r≥ 1, we obtain a rate 1/nr/(1+2r), both for pointwise estimation and for estimation on a compact subset of R: this is the same rate as for non‐parametric estimation of a density with i.i.d. observations.

MOD-GAUSSIAN CONVERGENCE: NEW LIMIT THEOREMS IN PROBABILITY AND NUMBER THEORY

Abstract We introduce a new type of convergence in probability theory, which we call “mod-Gaussian convergence”. It is directly inspired by theorems and conjectures, in random matrix theory and number theory, concerning moments of values of characteristic polynomials or zeta functions. We study this type of convergence in detail in the framework of infinitely divisible distributions, and exhibit some unconditional occurrences in number theory, in particular for families of L-functions over function fields in the Katz–Sarnak framework. A similar phenomenon of “mod-Poisson convergence” turns out to also appear in the classical Erdős–Kac Theorem.

Rates of convergence to the local time of a diffusion

Abstract In this paper we consider the approximation of the local time Lt of a 1-dimensional diffusion process X at some level x, say x = 0 by normalized sums, say Utn, of functions of the values Xi/n for i ≤ nt as n → ∞. Our main aim is to prove an associated functional central limit theorem fiving a mixed normal limiting value to the sequence of processes n∞(Utn − Lt), for a suitable value of α.