Infinite dimensional analysis of pure jump Lévy processes on the Poisson space

Abstract

Abstract. We develop a white noise calculus for pure jump Lévy processes on the Poisson space. This theory covers the treatment of Lévy processes of unbounded variation. The starting point of the theory is the construction of a distribution space. This space has many of the same nice properties as the classical Schwartz space, but is modified in a certain way in order to be more suitable for pure jump Lévy processes. We apply Minlos's theorem to this space and obtain a white noise measure which satisfies the first condition of analyticity, and which is non-degenerate. Furthermore, we obtain generalized Charlier polynomials for all Lévy measures. We introduce Kondratiev test function and distribution spaces, the S-transform and the Wick product. We proceed by using a transfer principle on Poisson spaces to establish a differential calculus.

Stochastic differential equations—some new ideas

“…Thus Y b t is the unique strong solution of (26). Note that Y b t as an approximation of adapted solutions in L 2 ðm; R d Þ is also adapted, since our underlying filtration is m-augmented (see (27)). This completes the proof.…”

Differential equations driven by Lévy white noise in spaces of Hilbert space-valued stochastic distributions

“…In this Section, we quickly recall the definition and some properties of Hilbert spacevalued Lévy processes as well as some aspects of the white noise theory for Poisson random measures developed in [20]. These concepts are going to be at the base of the next Section.…”

“…Recently, a white noise space for general Lévy processes has been constructed in Ref. [20]. Another direction was taken by the authors in Ref.…”

“…[11] and [20] and to develop a white noise framework for Hilbert space-valued Lévy processes. As an application we then look in Section 4 at stochastic evolution equations driven by additive Lévy noise with values in spaces of stochastic H-valued distributions of the following type where O ¼ {x [ R n ; 0 , x k , a k ; k ¼ 1; .…”

Stochastic partial differential equations driven by multi-parameter white noise of Lévy processes

“…physics, biology and economics. A white noise theory for Lévy processes was developed in the papers [5], [3], [14], [18] and [16].…”