Publications

2013

• Log-normal continuous cascade model of asset returns: aggregation properties and estimation
  
  • Bacry Emmanuel
  • Kozhemyak Alexey
  • Muzy Jean-François
  Quantitative Finance, Taylor & Francis (Routledge), 2013, 13 (5), pp.795-818. no abstract (10.1080/14697688.2011.647411)
  DOI : 10.1080/14697688.2011.647411
• Some limit theorems for Hawkes processes and application to financial statistics
  
  • Bacry Emmanuel
  • Delattre Sylvain
  • Hoffmann Marc
  • Muzy Jean-François
  Stochastic Processes and their Applications, Elsevier, 2013, 123 (7), pp.2475 - 2499. Abstract In the context of statistics for random processes, we prove a law of large numbers and a functional central limit theorem for multivariate Hawkes processes observed over a time interval [ 0 , T ] when T ? ? . We further exhibit the asymptotic behaviour of the covariation of the increments of the components of a multivariate Hawkes process, when the observations are imposed by a discrete scheme with mesh ? over [ 0 , T ] up to some further time shift ? . The behaviour of this functional depends on the relative size of ? and ? with respect to T and enables to give a full account of the second-order structure. As an application, we develop our results in the context of financial statistics. We introduced in Bacry et al. (2013) [7] a microscopic stochastic model for the variations of a multivariate financial asset, based on Hawkes processes and that is confined to live on a tick grid. We derive and characterise the exact macroscopic diffusion limit of this model and show in particular its ability to reproduce the important empirical stylised fact such as the Epps effect and the lead?lag effect. Moreover, our approach enables to track these effects across scales in rigorous mathematical terms. (10.1016/j.spa.2013.04.007)
  DOI : 10.1016/j.spa.2013.04.007
• The 2d-Directed Spanning Forest is almost surely a tree
  
  • Coupier David
  • Tran Viet Chi
  Random Structures and Algorithms, Wiley, 2013, 42 (1), pp.59-72. We consider the Directed Spanning Forest (DSF) constructed as follows: given a Poisson point process N on the plane, the ancestor of each point is the nearest vertex of N having a strictly larger abscissa. We prove that the DSF is actually a tree. Contrary to other directed forests of the literature, no Markovian process can be introduced to study the paths in our DSF. Our proof is based on a comparison argument between surface and perimeter from percolation theory. We then show that this result still holds when the points of N belonging to an auxiliary Boolean model are removed. Using these results, we prove that there is no bi-infinite paths in the DSF. (10.1002/rsa.20400)
  DOI : 10.1002/rsa.20400
• Strong solutions to the equations of flow and heat transfer in magnetic fluids with internal rotations
  
  • Amirat Youcef
  • Hamdache Kamel
  Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2013, 33 (8), pp.3289-3320. In this paper we study the equations of flow and heat transfer in a magnetic fluid with internal rotations, when the fluid is subjected to the action of an external magnetic field. The system of equations is a combination of the Navier-Stokes equations, the magnetization relaxation equation of Bloch type, the magnetostatic equations and the temperature equation. We prove the local-in-time existence of the unique strong solution to the system equipped with initial and boundary conditions and establish a blow-up criterium for strong solutions. We then prove the global-in-time existence of strong solutions, under smallness assumptions on the initial data and the external magnetic field. (10.3934/dcds.2013.33.3289)
  DOI : 10.3934/dcds.2013.33.3289