FIELDS OF RESEARCH

Integral calculus on trees

To each real-valued continuous path one can associate a real tree. Our aim was to read some properties of the path on the tree. More precisely, it appears that

In particular, we prove that the Young integral (defined under some conditions with respect to paths with unbounded variation) can actually be viewed as a Lebesgue integral on the tree. This interpretation can also be extended to some integrals of the rough paths theory. Moreover, our technique can be generalised to paths with jumps.

We can apply this study to paths of fractional Brownian motion or of Lévy processes. In the case of the standard Brownian motion, the stochastic analysis of Brownian excursions also provides an interpretation of Itô integrals, and we obtain a new integral representation formula for Wiener functionals (formula of Clark-Ocone type).

For the fractional Brownian motion, we have also reviewed some classical integral representations. These formulae can be applied to the study of this process viewed as a Gaussian process, and in particular to results about its Cameron-Martin space and to its comparison with other fractional processes.

Publications

  1. J. Picard, Representation formulae for the fractional Brownian motion, in: C. Donati-Martin, A. Lejay, A. Rouault (Eds.), Séminaire de Probabilités XLIII, Lect. N. in Math. 2006, Springer, 2011.
  2. J. Picard, A tree approach to p-variation and to integration, The Annals of Probability 36 (2008), 6, 2235-2279.
  3. J. Picard, Brownian excursions, stochastic integrals, and representation of Wiener functionals, Electronic J. Probab. 11 (2006), 199-248.

Stochastic calculus on manifolds

If N is a manifold, the notions of N-valued martingales and of N-valued harmonic maps are closely related (like real martingales and real harmonic functions); more precisely, the maps h from M into N which are harmonic for a diffusion on M (for instance the Brownian motion if M is Riemannian) map this diffusion to a N-valued martingale. In particular, the construction of harmonic maps with fixed value on the boundary of M is related to the construction of martingales on N with fixed terminal value (this is the non linear Dirichlet problem). We can also consider Markov processes with jumps on M and martingales with jumps on N.

In order to construct these martingales, we have considered two methods:

The stochastic techniques can also be used to study the smoothness of harmonic maps; these maps are indefinitely differentiable if the diffusion satisfies Hörmander’s conditions. This can be proved by studying the martingales with values in the tangent bundle of N, and by using estimates on the smoothness of real harmonic functions (one can apply Malliavin’s calculus to get these estimates). In the elliptic case, a simple method also yields estimates on the gradient of harmonic maps.

We are also interested in the case where the manifold N becomes a singular metric space, in particular a tree. Some of the previous results can be extended to this case, but need an adaptation; for instance, in the case of processes with jumps, we have to introduce a new notion of martingales with jumps.

Publications

  1. J. Picard, Stochastic calculus and martingales on trees, Annales Institut Henri Poincaré, Prob. Stat. 41 (2005), 4, 631-683.
  2. J. Picard, Gradient estimates for some diffusion semigroups, Probability Theory and Related Fields 122 (2002), 593-612.
  3. J. Picard, The manifold-valued Dirichlet problem for symmetric diffusions, Potential Analysis 14 (2001), 1, 53-72.
  4. J. Picard, Smoothness of harmonic maps for hypoelliptic diffusions, The Annals of Probability 28 (2000), 2, 643-666.
  5. J. Picard, Barycentres et martingales sur une variété, Annales Institut Henri Poincaré, Prob. Stat. 30 (1994), 4, 647-702.
  6. J. Picard, Martingales on Riemannian manifolds with prescribed limit, Journal of Functional Analysis 99 (1991), 2, 223-261.
  7. J. Picard, Calcul stochastique avec sauts sur une variété, in: J. Azéma, P.A. Meyer et M. Yor (eds.), Séminaire de Probabilités XXV, Lect. N. in Math. 1485, Springer, 1991.
  8. J. Picard, Martingales sur le cercle, in: J. Azéma, P.A. Meyer et M. Yor (eds.), Séminaire de Probabilités XXIII, Lect. N. in Math. 1372, Springer, 1989.

Stochastic analysis on the Poisson space

We first study a class of transformations of the Poisson space which can be viewed as the analogue of the Cameron-Martin transformations on the Wiener space; these transformations add or remove masses to the Poisson point measure, and verify an integration by parts formula which is similar to the basic formula of Malliavin’s calculus.

Then we can extend to diffusions with jumps one of the main results of Malliavin’s calculus, namely the smoothness of the law. More precisely, we look for the existence of a smooth density for solutions of stochastic differential equations driven by a Lévy process; previous results enabled to deal with “regular” Lévy processes (with an absolutely continuous Lévy measure); with our method, we can also consider singular Lévy measures; the smoothness of the law is actually a consequence of the existence of a large number of small jumps. Then we derive some applications of this result, like the behaviour of the density in small time, and the smoothness of harmonic functions. A localisation technique also enables to study manifold valued processes, for instance Lévy processes on Lie groups.

Publications

  1. J. Picard and C. Savona, Smoothness of the law of manifold-valued Markov processes with jumps, Bernoulli 19 (2013), 1880-1919.
  2. J. Picard and C. Savona, Smoothness of harmonic functions for processes with jumps, Stochastic Processes and their Applications 87 (2000), 69-91.
  3. J. Picard, Density in small time at accessible points for jump processes, Stochastic Processes and their Applications 67 (1997), 251-279.
  4. J. Picard, Density in small time for Lévy processes, ESAIM: Probab. Stat. 1 (1997), 357-389.
  5. J. Picard, Formules de dualité sur l’espace de Poisson, Annales Institut Henri Poincaré, Prob. Stat. 32 (1996), 4, 509-548.
  6. J. Picard, Transformations et équations anticipantes pour les processus de Poisson, Annales Mathématiques Blaise Pascal 3 (1996), 1, 111-123 (special issue dedicated to the memory of A. Badrikian).
  7. J. Picard, On the existence of smooth densities for jump processes, Probability Theory and Related Fields 105 (1996), 481-511. Erratum: Probability Theory and Related Fields 147 (2010), 711-713.

Non linear filtering

Our work is mainly devoted to the filtering of diffusions with small observation noise. More precisely, we look for filters which should be asymptotically efficient when the observation noise tends to 0. We develop techniques which enable the asymptotic study of the filtering error. The problem is simpler when the observation function is one-to-one, but we also consider more general cases.

We also studied other approximation problems, such as the “robustness” of the filter, the time discretization, or the case of small non-linearities.

Publications

  1. P. Milheiro de Oliveira et J. Picard, Approximate nonlinear filtering for a two-dimensional diffusion with one-dimensional observations in a low noise channel, SIAM J. Control Optim. 41 (2003), 6, 1801-1819.
  2. J. Picard, Estimation of the quadratic variation of nearly observed semimartingales with application to filtering, SIAM J. Control Optim. 31 (1993), 2, 494-517.
  3. J. Picard, A nonlinear filter with two time scales, in: I. Karatzas et D. Ocone (eds.), Applied Stochastic Analysis (Rutgers University 1991), Lect. N. in Control and Inform. Sc. 177, Springer, 1992.
  4. J. Picard, Efficiency of the extended Kalman filter for nonlinear systems with small noise, SIAM Journal on Applied Mathematics 51 (1991), 3, 843-885.
  5. J. Picard, Nonlinear filtering and smoothing with high signal-to-noise ratio, in: S. Albeverio et al. (eds.), Stochastic Processes in Physics and Engineering (4th BiBoS Symposium, Bielefeld 1986), Reidel, 1988.
  6. J. Picard, Asymptotic study of estimation problems with small observation noise, in: A. Germani (ed.), Stochastic Modelling and Filtering (IFIP Conference, Rome 1984), Lect. N. in Control and Inform. Sc. 91, Springer, 1987.
  7. J. Picard, Filtrage de diffusions vectorielles faiblement bruitées, in: A. Bensoussan et J.L. Lions (eds.), Analysis and Optimization of Systems (Conférence INRIA, Antibes 1986), Lect. N. in Control and Inform. Sc. 83, Springer, 1986.
  8. J. Picard, An estimate of the error in time discretization of nonlinear filtering problems, in: C. Byrnes et A. Lindquist (eds.), Theory and Applications of Nonlinear Control Systems (MTNS Symposium, Stockholm 1985), North-Holland, 1986.
  9. J. Picard, Nonlinear filtering of one-dimensional diffusions in the case of a high signal-to-noise ratio, SIAM Journal on Applied Mathematics 46 (1986), 1098-1125.
  10. J. Picard, A filtering problem with a small nonlinear term, Stochastics 18 (1986), 313-341.
  11. J. Picard, Approximation of nonlinear filtering problems and order of convergence, in: H. Korezlioglu, G. Mazziotto et J. Szpirglas (eds.), Filtering and Control of Random Processes (ENST-CNET Colloquium, Paris 1983), Lect. N. in Control and Inform. Sc. 61, Springer, 1984.
  12. J. Picard, Robustesse de la solution des problèmes de filtrage avec bruit blanc indépendant, Stochastics 13 (1984), 229-245.

Publications about other topics

  1. J. Picard, Convergence in probability for perturbed stochastic integral equations, Probability Theory and Related Fields 81 (1989), 383-452.
  2. J. Picard, Une classe de processus stable par retournement du temps, in: J. Azéma et M. Yor (eds.), Séminaire de Probabilités XX, Lect. N. in Math. 1204, Springer, 1986.