  We consider the problem of detecting an abrupt change in the distribution of
a sequentially observed stochastic process. We establish the optimality of the
CUSUM test with respect to a modified version of Lorden's criterion for
arbitrary processes with continuous paths and apply this general result to the
special case of fractional diffusion-type processes. As a by-product, we show
that the CUSUM test optimizes Lorden's original criterion when a fractional
Brownian motion with Hurst index H adopts a polynomial drift term with exponent
H + 1/2 after the change.
