Karhunen–Loeve expansion for the additive detrended Brownian motion

ABSTRACT

In this article, we obtain the Karhunen–Loeve expansion and Pythagorean type distribution identities for the additive detrended Brownian motion. As applications, the corresponding Laplace transform and small-deviation asymptotic behavior for the L₂ norm are given.

KEYWORDS:

• Additive detrended Brownian motion
• Fredholm integral equations
• Karhunen–Loeve expansion
• Laplace transform
• Small deviation

MATHEMATICS SUBJECT CLASSIFICATION:

• 60G15
• 60G12
• 60F10

Funding

The author is supported by the NSFC grant (No.11401085) and the Fundamental Research Funds for the Central Universities (No. 2572015BB14).