Product formulas for multiple stochastic integrals associated with Lévy processes

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Authors

External Organisational units

  • University of Jyvaskyla
  • TU Dresden

Abstract

In the present paper, we obtain an explicit product formula for products of multiple integrals w.r.t. a random measure associated with a Lévy process. As a building block, we use a representation formula for products of martingales from a compensated-covariation stable family. This enables us to consider Lévy processes with both jump and Gaussian part. It is well known that for multiple integrals w.r.t. the Brownian motion such product formulas exist without further integrability conditions on the kernels. However, if a jump part is present, this is, in general, false. Therefore, we provide here sufficient conditions on the kernels which allow us to establish product formulas. As an application, we obtain explicit expressions for the expectation of products of iterated integrals, as well as for the moments and the cumulants for stochastic integrals w.r.t. the random measure. Based on these expressions, we show a central limit theorem for the long time behaviour of a class of stochastic integrals. Finally, we provide methods to calculate the number of summands in the product formula.

Details

Original languageEnglish
Pages (from-to)???
Number of pages33
JournalCollectanea mathematica
Volume??? Stand: 6. Dezember 2024
Issue number??? Stand: 6. Dezember 2024
DOIs
Publication statusPublished - 7 Nov 2024