arXiv subjects

Jean Jacod

Publications and source records attributed to Jean Jacod.

At least 19 recordsLinked to original sources

arXiv2017

A review of asymptotic theory of estimating functions

Asymptotic statistical theory for estimating functions is reviewed in a generality suitable for stochastic processes. Conditions concerning existence of a consistent estimator, uniqueness, rate of convergence, and the asymptotic distribution are treated separately. Our conditions are not minimal, but can be verified for many interesting stochastic process models. Several examples illustrate the wide applicability of the theory and why the generality is needed.

math.ST↗

arXiv2016

Backward stochastic differential equation driven by a marked point process: An elementary approach with an application to optimal control

We address a class of backward stochastic differential equations on a bounded interval, where the driving noise is a marked, or multivariate, point process. Assuming that the jump times are totally inaccessible and a technical condition holds (see Assumption (A) below), we prove existence and uniqueness results under Lipschitz conditions on the coefficients. Some counter-examples show that our assumptions are indeed needed. We use a novel approach that allows reduction to a (finite or infinite) system of deterministic differential equations, thus avoiding the use of martingale representation theorems and allowing potential use of standard numerical methods. Finally, we apply the main results to solve an optimal control problem for a marked point process, formulated in a classical way.

math.PR↗

arXiv2014

A remark on the rates of convergence for integrated volatility estimation in the presence of jumps

The optimal rate of convergence of estimators of the integrated volatility, for a discontinuous Itô semimartingale sampled at regularly spaced times and over a fixed time interval, has been a long-standing problem, at least when the jumps are not summable. In this paper, we study this optimal rate, in the minimax sense and for appropriate "bounded" nonparametric classes of semimartingales. We show that, if the $r$th powers of the jumps are summable for some $r\in[0,2)$, the minimax rate is equal to $\min(\sqrt{n},(n\log n)^{(2-r)/2})$, where $n$ is the number of observations.

math.ST↗

arXiv2014

Efficient estimation of integrated volatility in presence of infinite variation jumps

We propose new nonparametric estimators of the integrated volatility of an Itô semimartingale observed at discrete times on a fixed time interval with mesh of the observation grid shrinking to zero. The proposed estimators achieve the optimal rate and variance of estimating integrated volatility even in the presence of infinite variation jumps when the latter are stochastic integrals with respect to locally "stable" Lévy processes, that is, processes whose Lévy measure around zero behaves like that of a stable process. On a first step, we estimate locally volatility from the empirical characteristic function of the increments of the process over blocks of shrinking length and then we sum these estimates to form initial estimators of the integrated volatility. The estimators contain bias when jumps of infinite variation are present, and on a second step we estimate and remove this bias by using integrated volatility estimators formed from the empirical characteristic function of the high-frequency increments for different values of its argument. The second step debiased estimators achieve efficiency and we derive a feasible central limit theorem for them.

math.ST↗

arXiv2013

Quarticity and other functionals of volatility: Efficient estimation

We consider a multidimensional Ito semimartingale regularly sampled on [0,t] at high frequency $1/Δ_n$, with $Δ_n$ going to zero. The goal of this paper is to provide an estimator for the integral over [0,t] of a given function of the volatility matrix. To approximate the integral, we simply use a Riemann sum based on local estimators of the pointwise volatility. We show that although the accuracy of the pointwise estimation is at most $Δ_n^{1/4}$, this procedure reaches the parametric rate $Δ_n^{1/2}$, as it is usually the case in integrated functionals estimation. After a suitable bias correction, we obtain an unbiased central limit theorem for our estimator and show that it is asymptotically efficient within some classes of sub models.

math.ST↗

arXiv2013

Statistical Properties of Microstructure Noise

We study the estimation of moments and joint moments of microstructure noise. Estimators of arbitrary order of (joint) moments are provided, for which we establish consistency as well as central limit theorems. In particular, we provide estimators of auto-covariances and auto-correlations of the noise. Simulation studies demonstrate excellent performance of our estimators even in the presence of jumps and irregular observation times. Empirical studies reveal (moderate) positive auto-correlation of the noise for the stocks tested.

stat.ME↗

arXiv2012

A test for the rank of the volatility process: the random perturbation approach

In this paper we present a test for the maximal rank of the matrix-valued volatility process in the continuous Ito semimartingale framework. Our idea is based upon a random perturbation of the original high frequency observations of an Ito semimartingale, which opens the way for rank testing. We develop the complete limit theory for the test statistic and apply it to various null and alternative hypotheses. Finally, we demonstrate a homoscedasticity test for the rank process.

math.ST↗

arXiv2012

Estimation of volatility functionals: the case of a square root n window

We consider a multidimensional Ito semimartingale regularly sampled on [0,t] at high frequency 1/Δ_n, with Δ_n going to zero. The goal of this paper is to provide an estimator for the integral over [0,t] of a given function of the volatility matrix, with the optimal rate 1/\sqrt{Δ_n} and minimal asymptotic variance. To achieve this we use spot volatility estimators based on observations within time intervals of length k_nΔ_n. In [5] this was done with k_n tending to infinity and k_n\sqrt{Δ_n} tending to 0, and a central limit theorem was given after suitable de-biasing. Here we do the same with k_n of order 1/\sqrt{Δ_n}. This results in a smaller bias, although more difficult to eliminate.

math.PR↗

arXiv2012

Testing whether jumps have finite or infinite activity

We propose statistical tests to discriminate between the finite and infinite activity of jumps in a semimartingale discretely observed at high frequency. The two statistics allow for a symmetric treatment of the problem: we can either take the null hypothesis to be finite activity, or infinite activity. When implemented on high-frequency stock returns, both tests point toward the presence of infinite-activity jumps in the data.

math.ST↗

arXiv2012

Identifying the successive Blumenthal-Getoor indices of a discretely observed process

This paper studies the identification of the Lévy jump measure of a discretely-sampled semimartingale. We define successive Blumenthal-Getoor indices of jump activity, and show that the leading index can always be identified, but that higher order indices are only identifiable if they are sufficiently close to the previous one, even if the path is fully observed. This result establishes a clear boundary on which aspects of the jump measure can be identified on the basis of discrete observations, and which cannot. We then propose an estimation procedure for the identifiable indices and compare the rates of convergence of these estimators with the optimal rates in a special parametric case, which we can compute explicitly.

math.ST↗

arXiv2010

Is Brownian motion necessary to model high-frequency data?

This paper considers the problem of testing for the presence of a continuous part in a semimartingale sampled at high frequency. We provide two tests, one where the null hypothesis is that a continuous component is present, the other where the continuous component is absent, and the model is then driven by a pure jump process. When applied to high-frequency individual stock data, both tests point toward the need to include a continuous component in the model.

math.ST↗

arXiv2010

Do price and volatility jump together?

We consider a process $X_t$, which is observed on a finite time interval $[0,T]$, at discrete times $0,Δ_n,2Δ_n,\ldots.$ This process is an Itô semimartingale with stochastic volatility $σ_t^2$. Assuming that $X$ has jumps on $[0,T]$, we derive tests to decide whether the volatility process has jumps occurring simultaneously with the jumps of $X_t$. There are two different families of tests for the two possible null hypotheses (common jumps or disjoint jumps). They have a prescribed asymptotic level as the mesh $Δ_n$ goes to $0$. We show on some simulations that these tests perform reasonably well even in the finite sample case, and we also put them in use on S&P 500 index data.

q-fin.ST↗

arXiv2010

Limit theorems for moving averages of discretized processes plus noise

This paper presents some limit theorems for certain functionals of moving averages of semimartingales plus noise which are observed at high frequency. Our method generalizes the pre-averaging approach (see [Bernoulli 15 (2009) 634--658, Stochastic Process. Appl. 119 (2009) 2249--2276]) and provides consistent estimates for various characteristics of general semimartingales. Furthermore, we prove the associated multidimensional (stable) central limit theorems. As expected, we find central limit theorems with a convergence rate $n^{-1/4}$, if $n$ is the number of observations.

math.ST↗

arXiv2009

Mod-Gaussian convergence: new limit theorems in probability and number theory

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-Kác Theorem.

math.NT↗

arXiv2009

Estimating the degree of activity of jumps in high frequency data

We define a generalized index of jump activity, propose estimators of that index for a discretely sampled process and derive the estimators' properties. These estimators are applicable despite the presence of Brownian volatility in the process, which makes it more challenging to infer the characteristics of the small, infinite activity jumps. When the method is applied to high frequency stock returns, we find evidence of infinitely active jumps in the data and estimate their index of activity.

math.ST↗

arXiv2009

Testing for common arrivals of jumps for discretely observed multidimensional processes

We consider a bivariate process $X_t=(X^1_t,X^2_t)$, which is observed on a finite time interval $[0,T]$ at discrete times $0,Δ_n,2Δ_n,....$ Assuming that its two components $X^1$ and $X^2$ have jumps on $[0,T]$, we derive tests to decide whether they have at least one jump occurring at the same time ("common jumps") or not ("disjoint jumps"). There are two different tests for the two possible null hypotheses (common jumps or disjoint jumps). Those tests have a prescribed asymptotic level, as the mesh $Δ_n$ goes to 0. We show on some simulations that these tests perform reasonably well even in the finite sample case, and we also put them in use for some exchange rates data.

math.ST↗

arXiv2009

Testing for jumps in a discretely observed process

We propose a new test to determine whether jumps are present in asset returns or other discretely sampled processes. As the sampling interval tends to 0, our test statistic converges to 1 if there are jumps, and to another deterministic and known value (such as 2) if there are no jumps. The test is valid for all Itô semimartingales, depends neither on the law of the process nor on the coefficients of the equation which it solves, does not require a preliminary estimation of these coefficients, and when there are jumps the test is applicable whether jumps have finite or infinite-activity and for an arbitrary Blumenthal--Getoor index. We finally implement the test on simulations and asset returns data.

math.ST↗

arXiv2008

Estimation of the Brownian dimension of a continuous Itô process

In this paper, we consider a $d$-dimensional continuous Itô process which is observed at $n$ regularly spaced times on a given time interval $[0,T]$. This process is driven by a multidimensional Wiener process and our aim is to provide asymptotic statistical procedures which give the minimal dimension of the driving Wiener process, which is between 0 (a pure drift) and $d$. We exhibit several different procedures, all similar to asymptotic testing hypotheses.

math.ST↗