Nicolas Bouleau

Space-Time Setting and Examples

We are now going to focus on examples involving a Levy process, where the time does play a role. This means that the bottom space is of the form \(X={\mathbb R} _+ \times \Xi \). We will be able to define a filtration and the notion of predictability, and to apply our tools to the setting of stochastic processes with jumps. Let us give more precisely the hypotheses.

Reminders on Poisson Random Measures

This chapter is devoted to the notions which constitute the foundation of our inquiry. They are presented briefly together with references allowing a deeper investigation. Poisson random measures are random distributions of points in abstract spaces widely used in applications in order to represent spatial independence (see for instance references in Neveu Processus Ponctuels, 1997, [270]). Levy processes i.e. processes with independent increments, may be defined through Poisson random measures so that equipping Poisson random measures with Dirichlet forms make this tool available for studying Levy functionals . Next we present the framework adopted in the book and the famous chaos decomposition of the \(L^2\) space of Poisson random measures.

The Lent Particle Formula

The lent particle formula allows to compute either the gradient or the carre du champ of a Dirichlet structure associated with a Poissonian distribution of points. Let us explain the case of the carre du champ on two examples.

Applications to Stochastic Differential Equations Driven by a Random Measure

The aim of this chapter is to apply the lent particle formula to Stochastic Differential Equations (SDE’s) driven by a Poisson random measure to obtain criteria ensuring that the law of the solution admits a density. By iteration, we also establish criteria of smoothness for the density of the law of the solution.

Affine Processes, Rates Models

In this chapter we apply the Malliavin calculus with jumps, under the form of the lent particle method, to Levy processes with values in the paths spaces of Markov processes in order to deduce the existence of density for some of their functionals. Our approach of the Markov valued Levy processes—i.e. affine processes cf. [Duffie et al. Ann Appl Probab, 13:984–1053, 2003, 135]—uses no general result on these processes (cf. [Duffie et al. Ann Appl Probab, 13:984–1053, 2003, 135, Pitman, Yor, Z f Wahr 59:425–457, 1982, 290, Pitman, Yor, Lect N in Math 923, p.276 et seq 1982, 291, Shiga, Watanabe, Z. f. Wahr. 27:37–46, 1982, 337] etc.) but a direct elementary construction based on finite order marginals and subordination in Bochner sense (cf. Bouleau and Chateau [ Bouleau, Chateau, C R Acad Sc Paris t 309, s1:625–628, 1989, 74] and [Chateau, Quelques remarques sur les processes a accroissements independants et la sub ordination au sens de Bochner These Univ. Paris VI, 1990, 97]). As soon as the construction of the functional Levy process is achieved, the computation of the carre du champ operator is quite easy by the lent particle method.

Sobolev Spaces and Distributions on Poisson Space

If E and F are Hilbert spaces, \(E\hat{\otimes }F\) denotes the Hilbertian tensor product of E and F, i.e. the space of Hilbert-Schmidt operators from E into F. For \(e\in E\) and \(f\in F\), \(e\otimes f\) is the linear map \(x\mapsto (x,e)_Ef\) from E into F and \(\Vert e\otimes f\Vert _{E_F}=\Vert e\Vert _E\Vert f\Vert _F\). The set of simple tensors \(e\otimes f\) is total in \(E\hat{\otimes }F\), in other words \(E\hat{\otimes }F\) can be viewed as the completion of the algebraic tensor product \(E\otimes F\) with respect of the Hilbert-Schmidt norm.

Construction of the Dirichlet Structure on the Upper Space

As soon as a Dirichlet structure has been defined on the bottom space , in a natural way, following a probabilistic construction, one obtains a symmetric strongly continuous contraction semigroup on \(L^2\) of the upper space, hence a closed form. This form is actually Dirichlet. We will give three proofs of this result which are more or less simple depending on the adopted point of view.

Introduction to the Theory of Dirichlet Forms

A Dirichlet form is a generalization of the energy form \(f\mapsto \int _\Omega |\nabla f|^2 d\lambda \) introduced in the 1840s especially by William Thomson (Lord Kelvin) (cf. Temple, 100 Years of Mathematics, 1981, [351], Chap. 15) in order to solve by minimization the problem without second member \(\Delta f=0\) in the open set \(\Omega \) (Dirichlet principle). Riemann adopted the expression Dirichlet form (Riemann Grundlagen fur eine allgemeine Theorie der Funktionen einer veranderlischen komplexen Grosse, 1851, [314]). The generalization now known as a Dirichlet form keeps the notion in the same relationship with the semigroup as the energy form holds with the heat semigroup.