Drawdowns of Diffusions

In this paper, using the excursion theory, we provide new proofs of, firstly, Lehoczky’s formula (in an extended form allowing a lower bound for the underlying diffusion) for the joint distribution of the first drawdown time and the maximum before this time, and, secondly, of Malyutin’s formula for the joint distribution of the first hitting time and the maximum drawdown before this time. It is remarkable – but there is a clean explanation – that the excursion theoretical approach which we developed first for Lehoczky’s formula also provides a proof for Malyutin’s formula. Moreover, we discuss some generalizations and analyze the pure jump process describing the maximum before the first drawdown time when the size of the drawdown is varying.