Generalised Brownian bridges: examples

Abstract

We observe that the probability distribution of the Brownian motion with drift $-c\frac{x}{1-t}$ where $c\ne 1$ is singular with respect to that of the classical Brownian bridge measure on $[0,1]$, while their Cameron-Martin spaces are equal set-wise if and only if $c>\frac{1}{2}$, providing also examples of exponential martingales on $[0,1)$ not extendable to a continuous martingale on $[0,1]$. Other examples of generalised Brownian bridges are also studied.

To appear in Markov Processes and Related Fields

References in corpus (2)

• Strict local martingales: examples
• On the Semi-Classical Brownian Bridge Measure

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