α-Wiener Bridges: Singularity of Induced Measures and Sample Path Properties

Abstract

Let us consider the process given by the SDE , t ∊ [0, T), where α ∊ ℝ, T ∊ (0, ∞), and (B _t ) _t≥0 is a standard Wiener process. In case of α > 0, the process X ^(α) is known as an α-Wiener bridge, in case of α = 1 as the usual Wiener bridge. We prove that for all α, β ∊ ℝ, α ≠ β, the probability measures induced by the processes X ^(α) and X ^(β) are singular on (C[0, T), ℬ(C[0, T))). Further, we investigate regularity properties of as t ↑ T.

Keywords:

• Induced measures
• Sample path properties
• Singularity
• α-Wiener bridges

Mathematics Subject Classification:

• 60G30
• 60G17
• 60J60

The first author has been supported by the Hungarian Scientific Research Fund under Grants No. OTKA–F046061/2004 and OTKA T-048544/2005. The second author has been supported by the Hungarian Scientific Research Fund under Grant No. OTKA T-048544/2005.