  We present an analytical method to estimate the continuous-time collision
probability of motion plans for autonomous agents with linear controlled Ito
dynamics. Motion plans generated by planning algorithms cannot be perfectly
executed by autonomous agents in reality due to the inherent uncertainties in
the real world. Estimating end-to-end risk is crucial to characterize the
safety of trajectories and plan risk optimal trajectories. In this paper, we
derive upper bounds for the continuous-time risk in stochastic robot navigation
using the properties of Brownian motion as well as Boole and Hunter's
inequalities from probability theory. Using a ground robot navigation example,
we numerically demonstrate that our method is considerably faster than the
naive Monte Carlo sampling method and the proposed bounds perform better than
the discrete-time risk bounds.
