  In this work we study drawdowns and drawups of general diffusion processes.
The drawdown process is defined as the current drop of the process from its
running maximum, while the drawup process is defined as the current increase
over its running minimum. The drawdown and the drawup are the first hitting
times of the drawdown and the drawup processes respectively. In particular, we
derive a closed-form formula for the Laplace transform of the probability
density of the drawdown of a units when it precedes the drawup of b units. We
then separately consider the special case of drifted Brownian motion, for which
we derive a closed form formula for the above-mentioned density by inverting
the Laplace transform. Finally, we apply the results to a problem of interest
in financial risk-management and to the problem of transient signal detection
and identification of two-sided changes in the drift of general diffusion
processes.
