Product formulas for multiple stochastic integrals associated with Lévy processes
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In: Collectanea mathematica, Vol. ??? Stand: 13. November 2024, 07.11.2024, p. ???.
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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: 13. November 2024
SP - ???
JO - Collectanea mathematica
JF - Collectanea mathematica
SN - 0010-0757
ER -