Regularity of the law

Regularities and logarithmic derivatives of densities for SDEs with jumps.

Atsushi Takeuchi (Osaka City University)

Consider stochastic differential equations with jumps, which include the drift, the diffusion, and the jump terms. In this talk, we shall study the existence of the smooth density and its logarithmic derivatives with respect to the initial point of the equation, under some nice conditions on the coefficients and the L ́evy measure. Our approach is based upon the Malliavin calculus on the Wiener-Poisson space introduced by Picard and Ishikawa-Kunita. The key tools are the fundamental inequalities for semi- martingales, and the martingale representation theorem via the Kolmogorov backward equation for the integro-differential operator associated with the equation. The results obtained here reflect the effect not only from the diffusion term, but also from the jump term.

Dirichlet forms applied to Poisson measures : simplified construction and the double Fock space.

Nicolas Bouleau (Université Paris-Est - École des Ponts ParisTech)

We develop the Malliavin calculus for Poisson measures in the way of acting on the jumps size with general local Dirichlet forms. The construction that we present replace the Friedrichs method by a Monte Carlo argument. The choice of the gradient as a marked point process yields a very simple structure in terms of chaos and Fock spaces. We give a collection of remarkable formulas including the lent particle formulas for the gradient, the generator and the ”divergence”. We begin to study the celebrated functional inequalities in this framework.

Some applications of the Lent Particle Method.

Laurent Denis (Université d'Évry)

The lent particle method introduced in the previous talk by N. Bouleau, gives rise to a new explicit calculus and permits to develop a Malliavin calculus on the Poisson space in a simple way. In this talk, we shall first construct Sobolev spaces, study functional inequalities (Khintchine and Meyer inequalities) and establish a criterion which ensures existence of density and regularity of laws of Poisson functionals. Then, we’ll apply it to solutions of SDE’s driven by Poisson measure. We also shall consider non classical examples such as a new kind of (non linear) subordination for which the space of marks (bottom space) is infinite dimensional.

Integration by parts formula for SDE’s with jumps.

Vlad Bally (Université Paris-Est - Marne-la-Vallée)

We establish an integration by parts formula in an abstract framework in order to study the regularity of the law for processes solution of stochastic differential equations with jumps, including equations with discontinuous coefficients for which the Malliavin calculus developed by Bismut and Bichteler, Gravereaux and Jacod fails.

Regularization for the 2D Boltzmann equation.

Nicolas Fournier (Université Paris-Est Créteil)

We present a probabilistic interpretation of the Boltzmann equation in terms of a jumping S.D.E. Based on this and on the Malliavin calculus for jumps processes intro- duced by Bally-Clément, we prove that the solution enjoys some regularization properties.

Local Malliavin calculus for Lévy processes and applications.

Josep L. Solé (Universitat Autonoma de Barcelona)

We will develop a Malliavin calculus for L ́evy processes based on a family of true derivative operators. The starting point is an extension to Lvy processes of the pioneering paper by Carlen and Pardoux for the standard Poisson process. Our extension includes the classical Malliavin derivative for Gaussian processes. We obtain a sufficient condition for the absolute continuity of functionals of the Lvy process. As an application, we will analyze the absolute continuity of the laws of the solutions of some stochastic differential equations driven by Lvy processes. The talk is based in a joint work with Jorge Leon (Cinvestav), Josep Vives (UB) and Frederic Utzet (UAB).

On an absolute continuity criterion for Ornstein-Uhlenbeck processes

Thomas Simon (Université Lille 1)

We consider multidimensional Ornstein-Uhlenbeck processes with Lévy noise. Under a non-singularity assumption on the drift term, we give a NASC for the absolute continuity of the laws at fixed time. This condition is not time-dependent, contrary to infinitely divisible laws in general. It is expressed as a geometric condition between the drift, the Lévy measure, and the Brownian component. The proof relies on a suitable stratification method and basic control theory.

Convergence in variation for the laws of Poisson functionals under weak regularity assumptions.

Alexey Kulik (University of Kiev)

The talk is devoted to the criteria, proved in [1], for a sequence of laws of $\Re^m$-valued  random vectors $\xi_n, n\geq 1$, restricted to a non-trivial part of the probability space, to converge in variation. These criteria are formulated in the terms of a group ${\mathcal G}$ of admissible transformations of the basic probability space, and demand in particular that every component of $\xi_n$ is either $L_p$-differentiable ($p>m$) or a.s.  differentiable w.r.t. ${\mathcal G}$. The difference between $L_p$- and a.s.- based criteria is essential; the latter one is more technically complicated and contains a specific assumption  on the family $\{\xi_n\}$ to have uniformly dominated increments w.r.t. ${\mathcal G}$.  On the other hand, the a.s.-criterion, unlike the $L_p$- one, typically leads to  weak regularity assumptions on a model.

To demonstrate that difference and the intrinsic further applications, we consider Markov process $X$  solution to an Itô-Lévy type SDE driven by a Poisson point measure,  and show that convergence in variation criteria, combined with natural Lyapunov-type assumptions, perform an efficient tool for proving following $\phi$-ergodic rates for the transition probability $P_t$ of the  process $X$:$\int_{\Re^m}\phi(y)|P_t(x,\cdot)-\pi|(dy)\leq r(t)\psi(x), \quad x\in \Re^m, \quad t\geq 0 $

([2]; here $\pi$ is the invariant distribution). In that context, the $L_p$-criterion is applicable when the coefficients of initial SDE are smooth enough and either the Lévy measure of $\nu$ is absolutely continuous or the jump perturbations are additive. Otherwise, the $L_p$-criterion fails and one should imply the a.s.-criterion. Another field of applications is provided by  the approach developed in [3], where a spectral gap property for the $L_2$-generator of the process $X$ was established in the terms of explicit $\phi$-ergodic rates both for the process $X$ and its dual one $X^*$. In that context, the  a.s.-criterion is of essential importance because, even in the simplest case where $X$ is a Lévy-driven Ornstein-Uhlenbeck process, it's dual process follows an Itô-Lévy type SDE with discontinuous coefficients and hence can not be treated in  terms of the $L_p$-criterion.

References:

[1] A.M. Kulik, Absolute continuity and convergence in variation for distributions of a functionals of Poisson point measure, arXiv:0803.2389 (2008).
[2] A.M. Kulik, Exponential ergodicity of the solutions to SDE?s with a jump noise. Stoch. Proc. Appl. 119, 602 – 632 (2009).
[3] A.M. Kulik, Asymptotic and spectral properties of exponentially φ-ergodic Markov pro- cesses, arXiv:0911.5473 (2009).