arXiv subjects

Pierre Gruet

Publications and source records attributed to Pierre Gruet.

10 recordsLinked to original sources

arXiv2026

Learning to Bid in FCR Markets: A Best-of-Both-Worlds Approach

Bidding in the European Frequency Containment Reserve (FCR) market is challenging for flexibility providers because competing offers are hidden and bidders observe only partial feedback form the market, such as, clearing price and awarded quantity. For a participant active in a single country, we show that the multi-country FCR clearing problem can be recast as a repeated multi-unit uniform-price auction against an endogenous vector of opposing bids. This reformulation yields an online learning problem and allows us to adapt a Best-of-Both-Worlds combinatorial semi-bandit algorithm implementable from this standard market feedback. The resulting bidder achieves logarithmic pseudo-regret in stochastic environments and $\mathcal{O}(\sqrt{T})$ regret in adversarial ones. Synthetic experiments confirm the expected scaling, and backtests on historical European FCR data show competitive performance in practice: the method performs especially well on stable products, while EXP3-type baselines can be safer under stronger non-stationarity. Overall, the results show that learning-based bidding in FCR markets is theoretically grounded and practically useful when the learning rule matches product-level market stability.

cs.LG↗

arXiv2025

Fast Gaussian process inference by exact Mat\'ern kernel decomposition

To speed up Gaussian process inference, a number of fast kernel matrix-vector multiplication (MVM) approximation algorithms have been proposed over the years. In this paper, we establish an exact fast kernel MVM algorithm based on exact kernel decomposition into weighted empirical cumulative distribution functions, compatible with a class of kernels which includes multivariate Mat\'ern kernels with half-integer smoothness parameter. This algorithm uses a divide-and-conquer approach, during which sorting outputs are stored in a data structure. We also propose a new algorithm to take into account some linear fixed effects predictor function. Our numerical experiments confirm that our algorithm is very effective for low-dimensional Gaussian process inference problems with hundreds of thousands of data points. An implementation of our algorithm is available at https://gitlab.com/warin/fastgaussiankernelregression.git.

stat.ML↗

arXiv2025

A non-local estimator for locally stationary Hawkes processes

We consider the problem of estimating the parameters of a non-stationary Hawkes process with time-dependent reproduction rate and baseline intensity. Our approach relies on the standard maximum likelihood estimator (MLE), coinciding with the conventional approach for stationary point processes characterised by [Ogata, 1978]. In the fully parametric setting, we find that the MLE over a single observation of the process over $[0, T]$ remains consistent and asymptotically normal as $T \to \infty$. Our results extend partially to the semi-nonparametric setting where no specific shape is assumed for the reproduction rate $g \colon [0, 1] \mapsto \mathbb{R}_+$. We construct a time invariance test with null hypothesis that g is constant against the alternative that it is not, and find that it remains consistent over the whole space of continuous functions of [0, 1]. As an application, we employ our procedure in the context of the German intraday power market, where we provide evidence of fluctuations in the endogeneity rate of the order flow.

math.ST↗

arXiv2025

Game Theory and Multi-Agent Reinforcement Learning for Zonal Ancillary Markets

We characterize zonal ancillary market coupling relying on noncooperative game theory. To that purpose, we formulate the ancillary market as a multi-leader single follower bilevel problem, that we subsequently cast as a generalized Nash game with side constraints and nonconvex feasibility sets. We determine conditions for equilibrium existence and show that the game has a generalized potential game structure. To compute market equilibrium, we rely on two exact approaches: an integrated optimization approach and Gauss-Seidel best-response, that we compare against multi-agent deep reinforcement learning. On real data from Germany and Austria, simulations indicate that multi-agent deep reinforcement learning achieves the smallest convergence rate but requires pretraining, while best-response is the slowest. On the economics side, multi-agent deep reinforcement learning results in smaller market costs compared to the exact methods, but at the cost of higher variability in the profit allocation among stakeholders. Further, stronger coupling between zones tends to reduce costs for larger zones.

cs.MA↗

arXiv2025

Some limit theorems for locally stationary Hawkes processes

We prove a law of large numbers and functional central limit theorem for a class of multivariate Hawkes processes with time-dependent reproduction rate. We address the difficulties induced by the use of non-convolutive Volterra processes by recombining classical martingale methods introduced in Bacry et al. [3] with novel ideas proposed by Kwan et al. [19]. The asymptotic theory we obtain yields useful applications in financial statistics. As an illustration, we derive closed-form expressions for price distortions under liquidity constraints.

math.PR↗

arXiv2024

A spectral mixture representation of isotropic kernels with application to random Fourier features

Rahimi and Recht (2007) introduced the idea of decomposing positive definite shift-invariant kernels by randomly sampling from their spectral distribution for machine learning applications. This famous technique, known as Random Fourier Features (RFF), is in principle applicable to any such kernel whose spectral distribution can be identified and simulated. In practice, however, it is usually applied to the Gaussian kernel because of its simplicity, since its spectral distribution is also Gaussian. Clearly, simple spectral sampling formulas would be desirable for broader classes of kernels. In this paper, we show that the spectral distribution of positive definite isotropic kernels in $\mathbb{R}^{d}$ for all $d\geq1$ can be decomposed as a scale mixture of $\alpha$-stable random vectors, and we identify the mixing distribution as a function of the kernel. This constructive decomposition provides a simple and ready-to-use spectral sampling formula for many multivariate positive definite shift-invariant kernels, including exponential power kernels, and generalized Cauchy kernels, as well as newly introduced kernels such as the generalized Mat\'ern, Tricomi, and Fox $H$ kernels. In particular, we retrieve the fact that the spectral distributions of these kernels, which can only be explicited in terms of the Fox $H$ special function, are scale mixtures of the multivariate Gaussian distribution, along with an explicit mixing distribution formula. This result has broad applications for support vector machines, kernel ridge regression, Gaussian processes, and other kernel-based machine learning techniques for which the random Fourier features technique is applicable.

cs.LG↗

arXiv2021

A survey of electricity spot and futures price models for risk management applications

This review presents the set of electricity price models proposed in the literature since the opening of power markets. We focus on price models applied to financial pricing and risk management. We classify these models according to their ability to represent the random behavior of prices and some of their characteristics. In particular, this classification helps users to choose among the most suitable models for their risk management problems.

q-fin.MF↗

arXiv2021

Electricity intraday price modeling with marked Hawkes processes

We consider a 2-dimensional marked Hawkes process with increasing baseline intensity in order to model prices on electricity intraday markets. This model allows to represent different empirical facts such as increasing market activity, random jump sizes but above all microstructure noise through the signature plot. This last feature is of particular importance for practitioners and has not yet been modeled on those particular markets. We provide analytic formulas for first and second moments and for the signature plot, extending the classic results of Bacry et al. (2013) in the context of Hawkes processes with random jump sizes and time dependent baseline intensity. The tractable model we propose is estimated on German data and seems to fit the data well. We also provide a result about the convergence of the price process to a Brownian motion with increasing volatility at macroscopic scales, highlighting the Samuelson effect.

q-fin.TR↗

arXiv2019

Efficient volatility estimation in a two-factor model

We statistically analyse a multivariate HJM diffusion model with stochastic volatility. The volatility process of the first factor is left totally unspecified while the volatility of the second factor is the product of an unknown process and an exponential function of time to maturity. This exponential term includes some real parameter measuring the rate of increase of the second factor as time goes to maturity. From historical data, we efficiently estimate the time to maturity parameter in the sense of constructing an estimator that achieves an optimal information bound in a semiparametric setting. We also identify nonparametrically the paths of the volatility processes and achieve minimax bounds. We address the problem of degeneracy that occurs when the dimension of the process is greater than two, and give in particular optimal limit theorems under suitable regularity assumptions on the drift process. We consistently analyse the numerical behaviour of our estimators on simulated and real datasets of prices of forward contracts on electricity markets.

math.ST↗

arXiv2015

An optimal trading problem in intraday electricity markets

We consider the problem of optimal trading for a power producer in the context of intraday electricity markets. The aim is to minimize the imbalance cost induced by the random residual demand in electricity, i.e. the consumption from the clients minus the production from renewable energy. For a simple linear price impact model and a quadratic criterion, we explicitly obtain approximate optimal strategies in the intraday market and thermal power generation, and exhibit some remarkable properties of the trading rate. Furthermore, we study the case when there are jumps on the demand forecast and on the intraday price, typically due to error in the prediction of wind power generation. Finally, we solve the problem when taking into account delay constraints in thermal power production.

q-fin.TR↗