Generalised Brownian Bridges: Examples

#### - Xue-Mei Li

2018, v.24, Issue 1, 151-163

ABSTRACT

We observe that the probability distribution of the Brownian motion with drift $-c x/ (1-t)$ where $c\not =1$ is singular with respect to that of the classical Brownian bridge measure on $[0,1]$, while their Cameron\tire Martin spaces are equal set-wise if and only if $c> 1/2$, 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.

Keywords: Brownian bridges, time-dependent singular drifts, Gaussian measures, equivalence, Cameron\tire Martin spaces