  Consider the motion of a Brownian particle in two or more dimensions, whose
coordinate processes are standard Brownian motions with zero drift initially,
and then at some random/unobservable time, one of the coordinate processes gets
a (known) non-zero drift permanently. Given that the position of the Brownian
particle is being observed in real time, the problem is to detect the time at
which a coordinate process gets the drift as accurately as possible. We solve
this problem in the most uncertain scenario when the random/unobservable time
is (i) exponentially distributed and (ii) independent from the initial motion
without drift. The solution is expressed in terms of a stopping time that
minimises the probability of a false early detection and the expected delay of
a missed late detection. To our knowledge this is the first time that such a
problem has been solved exactly in the literature.

Probability