arXiv subjects

Yoann Potiron

Publications and source records attributed to Yoann Potiron.

13 recordsLinked to original sources

arXiv2026

Parametric estimation of Hawkes processes based on ordinary least squares

We develop a parametric estimation framework for self-exciting Hawkes processes whose intensity functions admit a parametric form. The estimation procedure is based on ordinary least squares. To apply the least squares estimation, we restrict to a kernel class that can be expressed as a sum of the product of a parameter and a function. We first establish the central limit theorem for the proposed estimator. We then show the consistency of the asymptotic variance estimator. Finally, we introduce a Wald test statistic and derive its asymptotic distribution. We apply the proposed methodology to trade times from high-frequency financial asset data. Our empirical results provide evidence for three heterogeneous trader types. In particular, we identify two trader types as high-frequency traders and one trader type as fundamental trader.

stat.ME↗

arXiv2026

The realized copula of volatility

We study a new measure of codependency in the second moment of a continuous-time multivariate asset price process, which we name the realized copula of volatility. The statistic is based on local volatility estimates constructed from high-frequency asset returns and affords a nonparametric estimator of the empirical copula of the latent stochastic volatility. We show consistency of our estimator with in-fill asymptotic theory, either with a fixed or increasing time span. In the latter setting, we derive a functional central limit theorem for the empirical process associated with the measurement error of the time-invariant marginal copula of volatility. We also develop a goodness-of-fit test to evaluate hypotheses about the shape of the latter. In a simulation study, we demonstrate that our estimator is a good proxy of both the empirical and marginal copula of volatility, even with a moderate amount of high-frequency data recorded over a relatively short sample. The goodness-of-fit test is found to exhibit size control and excellent power. We implement our framework on high-frequency transaction data from futures contracts that track the U.S. equity and treasury bond market. A Gumbel copula is found to offer a near-perfect bind between the realized variance processes in these data.

econ.EM↗

arXiv2023

Explicit formula of boundary crossing probabilities for continuous local martingales to constant boundary

An explicit formula for the probability that a continuous local martingale crosses a one or two-sided random constant boundary in a finite time interval is derived. We obtain that the boundary crossing probability of a continuous local martingale to a constant boundary is equal to the boundary crossing probability of a standard Wiener process to a constant boundary up to a time change of quadratic variation value. This relies on the constancy of the boundary and the Dambis, Dubins-Schwarz theorem for continuous local martingale. The main idea of the proof is the scale invariant property of the time-changed Wiener process and thus the scale invariant property of the first-passage time. As an application, we also consider an inverse first-passage time problem of quadratic variation.

math.PR↗

arXiv2023

Non-explicit formula of boundary crossing probabilities by the Girsanov theorem

This paper derives several formulae for the probability that a Wiener process, which has a stochastic drift and random variance, crosses a one-sided stochastic boundary within a finite time interval. A non-explicit formula is first obtained by the Girsanov theorem when considering an equivalent probability measure in which the boundary is constant and equal to its starting value. A more explicit formula is then achieved by decomposing the Radon-Nikodym derivative inverse. This decomposition expresses it as the product of a random variable, which is measurable with respect to the Wiener process's final value, and an independent random variable. We also provide an explicit formula based on a strong theoretical assumption. To apply the Girsanov theorem, we assume that the difference between the drift increment and the boundary increment, divided by the standard deviation, is absolutely continuous. Additionally, we assume that its derivative satisfies Novikov's condition.

math.PR↗

arXiv2023

Approximation convergence in the inverse first-passage time problem

The inverse first-passage time problem determines a boundary such that the first-passage time of a Wiener process to this boundary has a given distribution. An approximation which is based on the starting value of the boundary to a smooth boundary by a piecewise linear boundary is given by equating the probability of the first-passage time to a linear boundary and the increment of the distribution on each interval. We propose a modification of that approximation which also approximates the starting value of the boundary. First, we show that the approximation is well-defined when assuming that the boundary is absolutely continuous. Second, we show that a subsequence of this new approximation uniformly converges to the boundary when the length of each interval of linear approximation goes to 0 asymptotically. The results are obtained using Arzela-Ascoli theorem on any compact space on which we further assume that the boundary admits uniformly dominated derivative. As the starting value of the boundary is unknown, this makes the new approximation more suitable for applications. The results are also proved in the first-passage time problem of a reflected Wiener process.

math.PR↗

arXiv2021

Cointegration in high frequency data

In this paper, we consider a framework adapting the notion of cointegration when two asset prices are generated by a driftless Itô-semimartingale featuring jumps with infinite activity, observed regularly and synchronously at high frequency. We develop a regression based estimation of the cointegrated relations method and show the related consistency and central limit theory when there is cointegration within that framework. We also provide a Dickey-Fuller type residual based test for the null of no cointegration against the alternative of cointegration, along with its limit theory. Under no cointegration, the asymptotic limit is the same as that of the original Dickey-Fuller residual based test, so that critical values can be easily tabulated in the same way. Finite sample indicates adequate size and good power properties in a variety of realistic configurations, outperforming original Dickey-Fuller and Phillips-Perron type residual based tests, whose sizes are distorted by non ergodic time-varying variance and power is altered by price jumps. Two empirical examples consolidate the Monte-Carlo evidence that the adapted tests can be rejected while the original tests are not, and vice versa.

q-fin.ST↗

arXiv2020

Estimation for high-frequency data under parametric market microstructure noise

We develop a general class of noise-robust estimators based on the existing estimators in the non-noisy high-frequency data literature. The microstructure noise is a parametric function of the limit order book. The noise-robust estimators are constructed as plug-in versions of their counterparts, where we replace the efficient price, which is non-observable, by an estimator based on the raw price and limit order book data. We show that the technology can be applied to five leading examples where, depending on the problem, price possibly includes infinite jump activity and sampling times encompass asynchronicity and endogeneity.

math.ST↗

arXiv2019

Testing if the market microstructure noise is fully explained by the informational content of some variables from the limit order book

In this paper, we build tests for the presence of residual noise in a model where the market microstructure noise is a known parametric function of some variables from the limit order book. The tests compare two distinct quasi-maximum likelihood estimators of volatility, where the related model includes a residual noise in the market microstructure noise or not. The limit theory is investigated in a general nonparametric framework. In the presence of residual noise, we examine the central limit theory of the related quasi-maximum likelihood estimation approach.

q-fin.ST↗

arXiv2018

Local Parametric Estimation in High Frequency Data

In this paper, we give a general time-varying parameter model, where the multidimensional parameter possibly includes jumps. The quantity of interest is defined as the integrated value over time of the parameter process $Θ= T^{-1} \int_0^T θ_t^* dt$. We provide a local parametric estimator (LPE) of $Θ$ and conditions under which we can show the central limit theorem. Roughly speaking those conditions correspond to some uniform limit theory in the parametric version of the problem. The framework is restricted to the specific convergence rate $n^{1/2}$. Several examples of LPE are studied: estimation of volatility, powers of volatility, volatility when incorporating trading information and time-varying MA(1).

q-fin.ST↗

arXiv2018

Efficient asymptotic variance reduction when estimating volatility in high frequency data

This paper shows how to carry out efficient asymptotic variance reduction when estimating volatility in the presence of stochastic volatility and microstructure noise with the realized kernels (RK) from [Barndorff-Nielsen et al., 2008] and the quasi-maximum likelihood estimator (QMLE) studied in [Xiu, 2010]. To obtain such a reduction, we chop the data into B blocks, compute the RK (or QMLE) on each block, and aggregate the block estimates. The ratio of asymptotic variance over the bound of asymptotic efficiency converges as B increases to the ratio in the parametric version of the problem, i.e. 1.0025 in the case of the fastest RK Tukey-Hanning 16 and 1 for the QMLE. The impact of stochastic sampling times and jump in the price process is examined carefully. The finite sample performance of both estimators is investigated in simulations, while empirical work illustrates the gain in practice.

q-fin.ST↗

arXiv2017

Statistical inference for the doubly stochastic self-exciting process

We introduce and show the existence of a Hawkes self-exciting point process with exponentially-decreasing kernel and where parameters are time-varying. The quantity of interest is defined as the integrated parameter $T^{-1}\int_0^Tθ_t^*dt$, where $θ_t^*$ is the time-varying parameter, and we consider the high-frequency asymptotics. To estimate it naïvely, we chop the data into several blocks, compute the maximum likelihood estimator (MLE) on each block, and take the average of the local estimates. The asymptotic bias explodes asymptotically, thus we provide a non-naïve estimator which is constructed as the naïve one when applying a first-order bias reduction to the local MLE. We show the associated central limit theorem. Monte Carlo simulations show the importance of the bias correction and that the method performs well in finite sample, whereas the empirical study discusses the implementation in practice and documents the stochastic behavior of the parameters.

q-fin.ST↗

arXiv2016

Classifying Patents Based on their Semantic Content

In this paper, we extend some usual techniques of classification resulting from a large-scale data-mining and network approach. This new technology, which in particular is designed to be suitable to big data, is used to construct an open consolidated database from raw data on 4 million patents taken from the US patent office from 1976 onward. To build the pattern network, not only do we look at each patent title, but we also examine their full abstract and extract the relevant keywords accordingly. We refer to this classification as semantic approach in contrast with the more common technological approach which consists in taking the topology when considering US Patent office technological classes. Moreover, we document that both approaches have highly different topological measures and strong statistical evidence that they feature a different model. This suggests that our method is a useful tool to extract endogenous information.

physics.soc-ph↗

arXiv2016

Estimation of integrated quadratic covariation with endogenous sampling times

When estimating high-frequency covariance (quadratic covariation) of two arbitrary assets observed asynchronously, simple assumptions, such as independence, are usually imposed on the relationship between the prices process and the observation times. In this paper, we introduce a general endogenous two-dimensional nonparametric model. Because an observation is generated whenever an auxiliary process called observation time process hits one of the two boundary processes, it is called the hitting boundary process with time process (HBT) model. We establish a central limit theorem for the Hayashi-Yoshida (HY) estimator under HBT in the case where the price process and the observation price process follow a continuous Ito process. We obtain an asymptotic bias. We provide an estimator of the latter as well as a bias-corrected HY estimator of the high-frequency covariance. In addition, we give a consistent estimator of the associated standard error.

q-fin.ST↗