On independent times and positions for Brownian motions

Abstract

Let $(B_t;t\ge 0)$, $(\text{resp. }((X_t,Y_t);t\ge 0))$ be a one (resp. two) dimensional Brownian motion started at 0. Let $T$ be a stopping time such that $(B_{t\wedge T};t\ge 0)$ \big(resp. $(X_{t\wedge T};t\ge 0);(Y_{t\wedge T};t\ge 0))$ is uniformly integrable. The main results obtained in the paper are:

1. if $T$ and $B_T$ are independent and $T$ has all exponential moments, then $T$ is constant.
2. If $X_T$ and $Y_T$ are independent and have all exponential moments, then $X_T$ and $Y_T$ are Gaussian.

We also give a number of examples of stopping times $T$, with only some exponential moments, such that $T$ and $B_T$ are independent, and similarly for $X_T$ and $Y_T$. We also exhibit bounded non-constant stopping times $T$ such that $X_T$ and $Y_T$ are independent and Gaussian.