A1 Journal article (refereed)
Product formulas for multiple stochastic integrals associated with Lévy processes (2024)

Di Tella, P., Geiss, C., & Steinicke, A. (2024). Product formulas for multiple stochastic integrals associated with Lévy processes. Collectanea mathematica, Early online. https://doi.org/10.1007/s13348-024-00456-6

JYU authors or editors

Publication details

All authors or editors: Di Tella, Paolo; Geiss, Christel; Steinicke, Alexander

Journal or series: Collectanea mathematica

ISSN: 0010-0757

eISSN: 2038-4815

Publication year: 2024

Publication date: 07/11/2024

Volume: Early online

Publisher: Springer

Publication country: Spain

Publication language: English

DOI: https://doi.org/10.1007/s13348-024-00456-6

Publication open access: Openly available

Publication channel open access: Partially open access channel

Publication is parallel published (JYX): https://jyx.jyu.fi/handle/123456789/98348

Web address of parallel published publication (pre-print):  https://doi.org/10.48550/arXiv.2309.11150

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.

Keywords: stochastic processes; probability calculation

Free keywords: product formulas for multiple stochastic integrals; moment formulas; Lévy processes; central limit theorem

Contributing organizations

Ministry reporting: Yes

VIRTA submission year: 2024

Preliminary JUFO rating: 1