Chaos expansions: applications to a generalized eigenvalue problem for the Malliavin derivative

Abstract

We study the chaos expansion transform, in short, chaos expansions, in a class of white noise spaces with series expansions by means of Hermite polynomials and functions, with certain weight sequences. Since Hermite polynomials are eigenfunctions for the Ornstein–Uhlenbeck operator, we apply the chaos expansion transform in solving of a class of equations. Moreover, we solve a generalized eigenvalue problem for the Malliavin derivative by means of chaos expansions.

Keywords:

• Hermit polynomials and functions
• series expansions of distributions
• generalized random process
• chaos expansion
• Malliavin derivative
• Ornstein–Uhlenbeck operator

AMS Subject Classification :

• 33E20
• 44A05
• 46F12
• 60H40
• 60H15

Acknowledgements

This paper was supported by the project No. 144016, financed by the Ministry of Science, Republic of Serbia.