Itô formula for generalized Lévy functionals (Q2735167)

The authors start with a brief introduction to the analysis of generalized Lévy functionals. They describe the construction of Lévy-Itô integrals, introduce the notions of test and generalized functionals and consider integral \(S\)-transform. The next step is the generalized Itô formula for Lévy processes with both Gaussian and compound Poisson parts. Let \(X(t)\) be a Lévy process, let be the function \(F\in S'\), where \(S'\) is the space of tempered distributions on \(\mathbb{R}\), then using the Segal-Bargmann transform for Lévy processes, the authors define the composite process \(F(X(t))\) as a generalized Lévy functional and decompose \(F(X(t))\) using first distribution derivative of \(F\) and two Bochner integrals with respect to positive measure \(d\lambda=dt\otimes (1+u^2)d\beta(u)\), \(\beta\) is the measure of jumps of \(X\). Finally, the Kubo-Takenaka formula is generalized to the Lévy process \(X\). The result obtained generalizes also the Itô formula introduced by Kunita-Watanabe for regular Lévy functionals.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00044].