arXiv subjects

Thomas Delerue

Publications and source records attributed to Thomas Delerue.

4 recordsLinked to original sources

arXiv2022

Rate-optimal estimation of mixed semimartingales

Consider the sum $Y=B+B(H)$ of a Brownian motion $B$ and an independent fractional Brownian motion $B(H)$ with Hurst parameter $H\in(0,1)$. Even though $B(H)$ is not a semimartingale, it was shown in [\textit{Bernoulli} \textbf{7} (2001) 913--934] that $Y$ is a semimartingale if $H>3/4$. Moreover, $Y$ is locally equivalent to $B$ in this case, so $H$ cannot be consistently estimated from local observations of $Y$. This paper pivots on another unexpected feature in this model: if $B$ and $B(H)$ become correlated, then $Y$ will never be a semimartingale, and $H$ can be identified, regardless of its value. This and other results will follow from a detailed statistical analysis of a more general class of processes called \emph{mixed semimartingales}, which are semiparametric extensions of $Y$ with stochastic volatility in both the martingale and the fractional component. In particular, we derive consistent estimators and feasible central limit theorems for all parameters and processes that can be identified from high-frequency observations. We further show that our estimators achieve optimal rates in a minimax sense.

math.ST↗

arXiv2021

When Frictions are Fractional: Rough Noise in High-Frequency Data

The analysis of high-frequency financial data is often impeded by the presence of noise. This article is motivated by intraday return data in which market microstructure noise appears to be rough, that is, best captured by a continuous-time stochastic process that locally behaves as fractional Brownian motion. Assuming that the underlying efficient price process follows a continuous It\^o semimartingale, we derive consistent estimators and asymptotic confidence intervals for the roughness parameter of the noise and the integrated price and noise volatilities, in all cases where these quantities are identifiable. In addition to desirable features such as serial dependence of increments, compatibility between different sampling frequencies and diurnal effects, the rough noise model can further explain divergence rates in volatility signature plots that vary considerably over time and between assets.

math.ST↗

arXiv2021

Normal approximation of the solution to the stochastic wave equation with Lévy noise

For a sequence $\dot{L}^{\varepsilon}$ of Lévy noises with variance $σ^2(\varepsilon)$, we prove the Gaussian approximation of the solution $u^{\varepsilon}$ to the stochastic wave equation driven by $σ^{-1}(\varepsilon) \dot{L}^{\varepsilon}$ and thus extend the result of C. Chong and T. Delerue [Stoch. Partial Differ. Equ. Anal. Comput. (2019)] to the class of hyperbolic stochastic PDEs. That is, we find a necessary and sufficient condition in terms of $σ^2(\varepsilon)$ for $u^{\varepsilon}$ to converge in law to the solution to the same equation with Gaussian noise. Furthermore, $u^{\varepsilon}$ is shown to have a space-time version with a càdlàg property determined by the wave kernel, and its derivative $\partial_t u^{\varepsilon}$ a càdlàg version when viewed as a distribution-valued process. These two path properties are essential to our proof of the normal approximation as the limit is characterized by martingale problems that necessitate both random elements. Our results apply to additive as well as to multiplicative noises.

math.PR↗

arXiv2019

Normal approximation of the solution to the stochastic heat equation with Lévy noise

Given a sequence $\dot{L}^{\varepsilon}$ of Lévy noises, we derive necessary and sufficient conditions in terms of their variances $σ^2(\varepsilon)$ such that the solution to the stochastic heat equation with noise $σ(\varepsilon)^{-1} \dot{L}^\varepsilon$ converges in law to the solution to the same equation with Gaussian noise. Our results apply to both equations with additive and multiplicative noise and hence lift the findings of S. Asmussen and J. Rosiński [J. Appl. Probab. 38 (2001) 482-493] and S. Cohen and J. Rosiński [Bernoulli 13 (2007) 195-210] for finite-dimensional Lévy processes to the infinite-dimensional setting without making distributional assumptions on the solutions such as infinite divisibility. One important ingredient of our proof is to characterize the solution to the limit equation by a sequence of martingale problems. To this end, it is crucial to view the solution processes both as random fields and as càdlàg processes with values in a Sobolev space of negative real order.

math.PR↗