# Registered Data

## [00672] Efficient inference for large and high-frequency data

**Session Date & Time**:- 00672 (1/3) : 5B (Aug.25, 10:40-12:20)
- 00672 (2/3) : 5C (Aug.25, 13:20-15:00)
- 00672 (3/3) : 5D (Aug.25, 15:30-17:10)

**Type**: Proposal of Minisymposium**Abstract**: In this minisymposium, the notion of asymptotically efficient estimation and asymptotically efficient statistical decision is discussed for various models appearing in finance and econometrics. For some applications, the data are acquired at high-frequency and in-fill observation scheme is considered. Here, asymptotical properties of the estimators for the parameters of the rough volatility models in quantitative finance and for the solutions of stochastic differential equations with jumps or with singular coefficients will be presented. In other applications, the large sample observation scheme is used. Asymptotical efficient statistical decisions and estimations are introduced for time series in econometrics (FARIMA, Threshold AR).**Organizer(s)**: Alexandre Brouste, Mathieu Rosenbaum**Classification**:__62Fxx__,__62Mxx__,__62Cxx__**Speakers Info**:- Elise Bayraktar (Université Gustave Eiffel)
- Laurent Denis (Le Mans Université)
- Ahmed Kebaier (Université d’Évry)
- Hiroki Masuda (The University of Tokyo)
- Grégoire Szymanski (École Polytechnique)
- Carsten Chong (Columbia University)
- Mikko Pakkanen (University of Waterloo)
- Tetsuya Takabatake ( Hiroshima, University)
- Yury Kutoyants (Le Mans Université)
- Youssef Esstafa (Le Mans University)
- Marius Soltane (UT Compiègne)
- Lionel Truquet (ENSAI)

**Talks in Minisymposium**:**[01388] Statistical inference for rough volatility: Central limit theorems****Author(s)**:**Carsten Chong**(Columbia University)- Marc Hoffmann (Université Paris Dauphine-PSL)
- Yanghui Liu (Baruch College CUNY)
- Mathieu Rosenbaum (Ecole Polytechnique)
- Grégoire Szymanski (Ecole Polytechnique)

**Abstract**: In recent years, there has been substantive empirical evidence that stochastic volatility is rough. In other words, the local behavior of stochastic volatility is much more irregular than semimartingales and resembles that of a fractional Brownian motion with Hurst parameter $H<0.5$. In this paper, we derive a consistent and asymptotically mixed normal estimator of $H$ based on high-frequency price observations. In contrast to previous works, we work in a semiparametric setting and do not assume any a priori relationship between volatility estimators and true volatility. Furthermore, our estimator attains a rate of convergence that is known to be optimal in a minimax sense in parametric rough volatility models.

**[01903] High-frequency estimation of pure jump alpha-CIR process****Author(s)**:**Elise Bayraktar**(Université Gustave Eiffel)

**Abstract**: We consider a pure-jump stable Cox-Ingersoll-Ross ($\alpha$-stable CIR) process defined by $$X_t=x_0+\int_0^t(a-bX_s)ds+\int_0^s\delta X_{s-}^{1/ \alpha}dL^\alpha_s$$ where $(L^\alpha_t)_{t \geq 0}$ is a compensated $\alpha$-stable Lévy process with non-negative jumps and $\alpha \in (1,2).$ We study the joint estimation of drift, scaling and jump activity parameters $(a,b,\delta,\alpha)$ from high-frequency observations of the process on a fixed time period. We prove the existence of a consistent and asymptotic mixed normal estimator based on an approximation of the likelihood function.

**[01940] Statistical inference for rough volatility: minimax theory****Author(s)**:**Grégoire Szymanski**(Ecole Polytechnique, CMAP)- Carsten Chong (Columbia University, 1Department of Statistics)
- Marc Hoffmann (Université Paris Dauphine-PSL, Ceremade)
- Yanghui Liu (Baruch College CUNY, 3Department of Mathematics)
- Mathieu Rosenbaum (Ecole Polytechnique, CMAP)

**Abstract**: Rough volatility models have gained considerable interest in the quantitative finance community in recent years. In this paradigm, the volatility of the asset price is driven by a fractional Brownian motion with a small value for the Hurst parameter $H$. In this work, we provide a rigorous statistical analysis of these models. To do so, we establish minimax lower bounds for parameter estimation and design procedures based on wavelets attaining them. We notably obtain an optimal speed of convergence of $n^{-1/(4H+2)}$ for estimating $H$ based on $n$ sampled data, extending results known only for the easier case $H>1/2$ so far. We therefore establish that the parameters of rough volatility models can be inferred with optimal accuracy in all regimes.

**[01999] Asymptotically Efficient Estimation for Fractional Brownian Motion with Additive Noise****Author(s)**:**Tetsuya Takabatake**(Hiroshima University)- Grégoire Szymanski (Ecole Polytechnique, CMAP)

**Abstract**: We will talk about an asymptotically efficient estimation of the Hurst index and the volatility parameter for a fractional Brownian motion with additive noise based on discrete observations. We will propose an asymptotically efficient estimator combining the ideas of the one-step method and a quadratic variation-type estimator using pre-averaged data.

**[02004] A GMM approach to estimate the roughness of stochastic volatility****Author(s)**:- Anine Bolko (Aarhus University)
- Kim Christensen (Aarhus University)
**Mikko Pakkanen**(University of Waterloo)- Bezirgen Veliyev (Aarhus University)

**Abstract**: I will present an approach to estimate log normal stochastic volatility models, including rough volatility models, using the generalised method of moments, GMM. In this GMM approach, estimation is done directly using realised measures, e.g., realised variance, avoiding the biases that arise from using a proxy of spot volatility. I will also present asymptotic theory for the GMM estimator, lending itself to inference, and apply the methodology to Oxford-Man realised volatility data.

**[02309] Local asymptotic properties for the growth rate of a jump-type CIR process****Author(s)**:- Mohamed Ben Alaya (University of Rouen)
**Ahmed Kebaier**(University of Evry)- Gyula Pap (University of Szeged)
- Ngoc Khue Tran (Hanoi University of Science and Technology)

**Abstract**: In this paper, we consider a one-dimensional jump-type Cox-Ingersoll-Ross pro- cess driven by a Brownian motion and a subordinator, whose growth rate is an unknown parameter. The Lévy measure of the subordinator is finite or infinite. Considering the process observed continuously or discretely at high frequency, we derive the local asymptotic properties for the growth rate in both ergodic and non-ergodic cases. Three cases are distinguished: subcritical, critical and supercritical. Local asymptotic normality (LAN) is proved in the subcritical case, local asymptotic quadraticity (LAQ) is derived in the critical case, and local asymptotic mixed normality (LAMN) is shown in the supercritical case. To do so, techniques of Malliavin calculus and a subtle analysis on the jump structure of the subordinator involving the amplitude of jumps and number of jumps are essentially used.

**[03713] Asymptotics for Student-Lévy regression****Author(s)**:**Hiroki Masuda**(University of Tokyo)

**Abstract**: We consider the quasi-likelihood analysis for a linear regression model driven by a Student Lévy process with constant scale and arbitrary degrees of freedom. We consider joint estimation of trend, scale, and degrees of freedom when the model is observed at a high frequency over an extending period. The bottleneck in this problem is that the Student distribution is not closed under convolution, making it difficult to estimate all the parameters fully based on the high-frequency time scale. To efficiently deal with that intricate nature, we propose a two-step quasi-likelihood analysis: first, we make use of the Cauchy quasi-likelihood for estimating the regression-coefficient vector and the scale parameter; then, we construct the sequence of the unit-period cumulative residuals to estimate the remaining degrees of freedom. We will present the hopefully asymptotically efficient theoretical behavior of the proposed estimator, which quantitatively clarifies the need for data thinning.

**[03747] Local asymptotic property for the Euler approximation of SDE driven by a stable Lévy process****Author(s)**:- Emmanuelle Clément (University Gustave Eiffel)
- Alexandre Brouste (Le Mans University)
- Thi Bao Trâm Ngô (Le Mans University)
**Laurent DENIS**(Le Mans University )

**Abstract**: We study the stochastic differential equations driven by a symmetric stable Lévy process, in which the joint parametric estimation of the drift coefficient, the scale coefficient and the jump activity of the process based on high frequency observations on a fixed time interval is considered. For these experiments, due to the non-explicit form of the likelihood function, our methodology will be to identify a simpler experiment, where the likelihood function has a traceable form, which is asymptotically equivalent in the Le Cam distance at the process observed at high frequency. To cover all values of jumping activity, the most appropriate experiment is to consider a numerical scheme that combines Euler's approximation of the scale coefficient with the solution of the ordinary equation given by the coefficient of derivative. We therefore prove the LAMN property for this corresponding Euler scheme with the ordinary differential equation. Thanks to the obtained LAMN property, we show that the one-step estimator is efficient. With an easy-to-compute initial estimator with good asymptotic behavior, it can exhibit a performance quite similar to that of the maximum likelihood estimator and reduce a lot of computation time. We illustrate our results by numerical simulations with the one-step procedure.

**[05359] Fast calibration of weak FARIMA models****Author(s)**:**Youssef Esstafa**(Le Mans Université )

**Abstract**: In this work, we investigate the asymptotic properties of Le Cam's one-step estimator for weak FARIMA models. For these models, noises are uncorrelated but neither necessarily independent nor martingale differences errors. We show under some regularity assumptions that the one-step estimator is strongly consistent and asymptotically normal with the same asymptotic variance as the least squares estimator. We show through simulations that the proposed estimator reduces computational time compared with the least squares estimator.