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Jean Jacod

Publications and source records attributed to Jean Jacod.

24 records · Page 2Linked to original sources
arXiv2006

Asymptotic properties of realized power variations and related functionals of semimartingales

This paper is concerned with the asymptotic behavior of sums of terms which are a test function f evaluated at successive increments of a discretely sampled semimartingale. Typically the test function is a power function (when the power is 2 we get the realized quadratic variation) . We prove a variety of ``laws of large numbers'', that is convergence in probability of these sums, sometimes after normalization. We also exhibit in many cases the rate of convergence, as well as associated central limit theorems.

math.PR↗
arXiv2005

Asymptotic properties of power variations of Lévy processes

We determine the asymptotic behavior of the realized power variations, or more generally of sums of a given test function evaluated at the successive increments of a Lévy process. One can completely elucidate the first order behavior (convergence in probability, possibly after normalization). As for the associated CLT, one can show some versions of it, but only in a limited number of cases. In some other cases, a CLT just does not exist.

math.PR↗
arXiv2005

Limit theorems for bipower variation in financial econometrics

In this paper we provide an asymptotic analysis of generalised bipower measures of the variation of price processes in financial economics. These measures encompass the usual quadratic variation, power variation and bipower variations which have been highlighted in recent years in financial econometrics. The analysis is carried out under some rather general Brownian semimartingale assumptions, which allow for standard leverage effects.

math.PR↗
arXiv2004

Fisher's Information for Discretely Sampled Levy Processes

This paper studies the asymptotic behavior of the Fisher information for a Levy process discretely sampled at an increasing frequency. We show that it is possible to distinguish not only the continuous part of the process from its jumps part, but also different types of jumps, and derive the rates of convergence of efficient estimators.

math.PR↗
arXiv2004

The Euler scheme for Levy driven stochastic differential equations: limit theorems

We study the Euler scheme for a stochastic differential equation driven by a Levy process Y. More precisely, we look at the asymptotic behavior of the normalized error process u_n(X^n-X), where X is the true solution and X^n is its Euler approximation with stepsize 1/n, and u_n is an appropriate rate going to infinity: if the normalized error processes converge, or are at least tight, we say that the sequence (u_n) is a rate, which, in addition, is sharp when the limiting process (or processes) is not trivial. We suppose that Y has no Gaussian part (otherwise a rate is known to be u_n=\sqrt n). Then rates are given in terms of the concentration of the Levy measure of Y around 0 and, further, we prove the convergence of the sequence u_n(X^n-X) to a nontrivial limit under some further assumptions, which cover all stable processes and a lot of other Levy processes whose Levy measure behave like a stable Levy measure near the origin. For example, when Y is a symmetric stable process with index α\in(0,2), a sharp rate is u_n=(n/\log n)^{1/α}; when Y is stable but not symmetric, the rate is again u_n=(n/\log n)^{1/α} when α>1, but it becomes u_n=n/(\log n)^2 if α=1 and u_n=n if α<1.

math.PR↗