Translated Brownian Motions and Associated Wick Products

Abstract

The concept of Wick product is strongly related to the underlying Brownian motion we have fixed on the probability space. Via the Girsanov's theorem we construct a family of new Brownian motions, obtained as translations of the original one, and to each of them we associate a Wick product. This produces a family of Wick products, named γ-Wick products, parameterized by the performed translations. We aim to describe this family of products. We also define a new family of stochastic integrals, which are related in a natural way to the γ-Wick products.

Keywords:

• Girsanov's theorem
• Stochastic differential equations
• Wick product

AMS 2000 Classification:

• 60H05
• 60H40
• 60H20

ACKNOWLEDGMENTS

The author thanks to Prof. Isabel Simão for her interest in this work and for stimulating discussions. Financial support through the project POCTI/ISFL/209 (Portugal-FEDER EU) is also acknowledged.