  An explicit formula for the probability that a continuous local martingale
crosses a one or two-sided random constant boundary in a finite time interval
is derived. We obtain that the boundary crossing probability of a continuous
local martingale to a constant boundary is equal to the boundary crossing
probability of a standard Wiener process to a constant boundary up to a time
change of quadratic variation value. This relies on the constancy of the
boundary and the Dambis, Dubins-Schwarz theorem for continuous local
martingale. The main idea of the proof is the scale invariant property of the
time-changed Wiener process and thus the scale invariant property of the
first-passage time. As an application, we also consider an inverse
first-passage time problem of quadratic variation.
