Some Results on Α-Quantiles

The aim of the paper is to present a simple method to rederive some recent results concerning distribution identities for the so called α-quantile for a real-valued stochastic process (see [1] and [2]). Also, the first discrete time version of this result, which seems due to [4], shall appear in continuous time. We first propose a (new) theorem presenting an upper and lower bound for the α-quantile for an arbitrary piecewise continuous real-valued function. The bounds are identical if the function is linear, so if the function represents sample paths for a stochastic process having stationary and independent increments, we can use the similar linearity in distribution to rederive the fundamental distribution equality as studied in [1]