Joint Laws of Maximum Drawdown and Maximum Drawup for Spectrally Negative L\'evy Processes

Abstract

Let X be a spectrally negative L\'evy process observed up to an inde- pendent exponential time T with parameter {\gamma} > 0. We study the joint law of the maximum drawdown and the maximum drawup of X on [0, T ]. The path is decomposed according to the two possible orderings of its in- fimum and supremum. Conditional on the values of these extrema and on their ordering, the resulting pre-, intermediate, and post- components are independent. We identify their laws as Doob h-transforms of killed spec- trally negative L\'evy processes and express the corresponding distribution functions explicitly in terms of the {\gamma}-scale functions W ({\gamma}) and Z({\gamma}) and their derivatives. Combining the conditional laws with the joint densities of the extrema yields integral representations of the joint distribution and moments.