TY - JOUR

T1 - Product formulas for multiple stochastic integrals associated with Lévy processes

AU - Di Tella, Paolo

AU - Geiss, Christel

AU - Steinicke, Alexander

N1 - Publisher Copyright: © The Author(s) 2024.

PY - 2024/11/7

Y1 - 2024/11/7

N2 - 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.

AB - 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.

KW - 60F05

KW - 60G44

KW - 60G51

KW - 60H05

KW - Central limit theorem

KW - Lévy processes

KW - Moment formulas

KW - Product formulas for multiple stochastic integrals

UR - http://www.scopus.com/inward/record.url?scp=85208257995&partnerID=8YFLogxK

U2 - 10.1007/s13348-024-00456-6

DO - 10.1007/s13348-024-00456-6

M3 - Article

VL - ??? Stand: 6. Dezember 2024

SP - ???

JO - Collectanea mathematica

JF - Collectanea mathematica

SN - 0010-0757

IS - ??? Stand: 6. Dezember 2024

ER -