Note on the Weak Convergence of Hyperplane $α$-Quantile Functionals and Their Continuity in the Skorokhod J1 Topology
The $α$-quantile of a stochastic process $M_{t,α}$ has been introduced in Miura (Hitotsubashi J Commerce Manag 27(1):15-28, 1992), and important distributional results have been derived in Akahori (Ann Appl Probab 5(2):383-388, 1995), Dassios (Ann Appl Probab 5(2):389-398, 1995) and Yor (J Appl Probab 32(2):405-416, 1995), with special attention given to the problem of pricing $α$-quantile options. We straightforwardly extend the classical monodimensional setting to $\mathbb{R}^d$ by introducing the hyperplane $α$-quantile, and we find an explicit functional continuity set of the $α$-quantile as a functional mapping $\mathbb{R}^d$-valued cadlag functions to $\mathbb{R}$. This specification allows us to use continuous mapping and assert that if a $\mathbb{R}^d$-valued cadlag stochastic process $X$ a.s. belongs to such continuity set, then $X^n \Rightarrow X$ (i.e., weakly in the Skorokhod sense) implies $M_{t,α}(X^n) \to^\textrm{w} M_{t,α}(X)$ (i.e., weakly) in the usual sense. We further the discussion by considering the conditions for convergence of a 'random time' functional of $M_{t,α}$, the first time at which the $α$-quantile has been hit, applied to sequences of cadlag functions converging in the Skorokhod topology. The Brownian distribution of this functional is studied, e.g., in Chaumont (J Lond Math Soc 59(2):729-741, 1999) and Dassios (Bernoulli 11(1):29-36, 2005). We finally prove the fact that if the limit process of a sequence of cadlag stochastic processes is a multidimensional Brownian motion with nontrivial covariance structure, such random time functional applied to the sequence of processes converges, jointly with the $α$-quantile, weakly in the usual sense.