Note on the Weak Convergence of Hyperplane $$\alpha $$-Quantile Functionals and Their Continuity in the Skorokhod J1 Topology

Abstract The $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> -quantile $$M_{t,\alpha }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:msub> </mml:math> of a stochastic process 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 $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> -quantile options. We straightforwardly extend the classical monodimensional setting to $${\mathbb {R}}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> by introducing the hyperplane $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> -quantile, and we find an explicit functional continuity set of the $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> -quantile as a functional mapping $${\mathbb {R}}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> -valued càdlàg functions to $${\mathbb {R}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> . This specification allows us to use continuous mapping and assert that if a $${\mathbb {R}}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> -valued càdlàg stochastic process X a.s. belongs to such continuity set, then $$X^n\Rightarrow X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>X</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>⇒</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> (i.e., weakly in the Skorokhod sense) implies $$M_{t,\alpha }(X^n)\rightarrow ^\text {w}M_{t,\alpha }(X)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>X</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:msup> <mml:mo>→</mml:mo> <mml:mtext>w</mml:mtext> </mml:msup> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> (i.e., weakly) in the usual sense. We further the discussion by considering the conditions for convergence of a ‘random time’ functional of $$M_{t,\alpha }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:msub> </mml:math> , the first time at which the $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> -quantile has been hit, applied to sequences of càdlàg 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 càdlàg 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 $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> -quantile—weakly in the usual sense.