Joint Laws of Maximum Drawdown and Maximum Drawup for Spectrally Negative Lévy Processes

Source abstract

Let X be a spectrally negative Lévy process observed up to an inde- pendent exponential time T with parameter γ > 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 Lévy processes and express the corresponding distribution functions explicitly in terms of the γ-scale functions W (γ) and Z(γ) and their derivatives. Combining the conditional laws with the joint densities of the extrema yields integral representations of the joint distribution and moments.