Marc Hoffmann

Modelling microstructure noise with mutually exciting point processes

We introduce a new stochastic model for the variations of asset prices at the tick-by-tick level in dimension 1 (for a single asset) and 2 (for a pair of assets). The construction is based on marked point processes and relies on mutually exciting stochastic intensities as introduced by Hawkes. We associate a counting process with the positive and negative jumps of an asset price. By suitably coupling the stochastic intensities of upward and downward changes of prices for several assets simultaneously, we can reproduce microstructure noise (i.e. strong microscopic mean reversion at the level of seconds to a few minutes) and the Epps effect (i.e. the decorrelation of the increments in microscopic scales) while preserving standard Brownian diffusion behaviour on large scales. More effectively, we obtain analytical closed-form formulae for the mean signature plot and the correlation of two price increments that enable us to track across scales the effect of the mean-reversion up to the diffusive limit of the model. We show that the theoretical results are consistent with empirical fits on futures Euro–Bund and Euro–Bobl in several situations.

Adaptive estimation in diffusion processes

We study the nonparametric estimation of the coefficients of a 1-dimensional diffusion process from discrete observations. Different asymptotic frameworks are considered. Minimax rates of convergence are studied over a wide range of Besov smoothness classes. We construct estimators based on wavelet thresholding which are adaptive (with respect to an unknown degree of smoothness). The results are comparable with simpler models such as density estimation or nonparametric regression.

Nonparametric Estimation of the Division Rate of a Size-Structured Population

We consider the problem of estimating thedivision rate of a size-structured population in anonparametric setting. The size of the system evolvesaccording to a transport-fragmentation equation: eachindividual grows with a given transport rate, and splitsinto two offsprings of the same size, followinga binary fragmentation process with unknown division ratethat depends on its size. In contrast to a deterministicinverse problem approach, we take in this talk theperspective of statistical inference: our data consists ina large sample of the size of individuals, when theevolution of the system is close to its time-asymptoticbehavior, so that it can be related to the eigenproblem ofthe considered transport-fragmentation equation. Byestimating statistically each term of the eigenvalueproblem and by suitably inverting a certain linearoperator, we are able to construct a more realisticestimator of the division rate that achieves the sameoptimal error bound as in related deterministic inverseproblems. Our procedure relies on kernel methods withautomatic bandwidth selection.

ON ADAPTIVE POSTERIOR CONCENTRATION RATES

tion of Holder balls and that moreover achieve our lower bound. We analyse the consequences in terms of asymptotic behaviour of poste- rior credible balls as well as frequentist minimax adaptive estimation. Our results are appended with an upper bound for the contraction rate under an arbitrary loss in a generic regular experiment. The up- per bound is attained for certain sieve priors and enables to extend our results to density estimation.

HAWKES PROCESSES ON LARGE NETWORKS

We generalise the construction of multivariate Hawkes processes to a possibly infinite network of counting processes on a directed graph G. The process is constructed as the solution to a system of Poisson driven stochastic differential equations, for which we prove pathwise existence and uniqueness under some reasonable conditions. We next investigate how to approximate a standard N -dimensional Hawkes process by a simple inhomogeneous Poisson process in the mean-field framework where each pair of individuals interact in the same way, in the limit N → ∞. In the so-called linear case for the interaction, we further investigate the large time behaviour of the process. We study in particular the stability of the central limit theorem when exchanging the limits N, T → ∞ and exhibit different possible behaviours. We finally consider the case G = Z d with nearest neighbour interactions. In the linear case, we prove some (large time) laws of large numbers and exhibit different behaviours, reminiscent of the infinite setting. Finally we study the propagation of a single impulsion started at a given point of Z d at time 0. We compute the probability of extinction of such an impulsion and, in some particular cases, we can accurately describe how it propagates to the whole space. Mathematics Subject Classification (2010): 60F05, 60G55, 60G57.

ESTIMATION OF THE HURST PARAMETER FROM DISCRETE NOISY DATA

We estimate the Hurst parameter

Adaptive wavelet estimation of the diffusion coefficient under additive error measurements.

H

Blockwise SVD with error in the operator and application to blind deconvolution

of a fractional Brownian motion from discrete noisy data observed along a high frequency sampling scheme. The presence of systematic experimental noise makes recovery of

ON ESTIMATING THE DIFFUSION COEFFICIENT: PARAMETRIC VERSUS NONPARAMETRIC

H

Minimax estimation of the diffusion coefficient through irregular samplings

more difficult since relevant information is mostly contained in the high frequencies of the signal. We quantify the difficulty of the statistical problem in a min-max sense: we prove that the rate