Upper bounds for superquantiles of martingales
Let \((M_n)_{n\in\mathbb{N}}\) be a real-valued martingale in \(L^p\) for either \(p\in(1,2]\) or \(p=3\); these two cases are considered separately in the present paper. For any \(u\in(0,1]\), the corresponding quantile (or value-at-risk) of \(M_n\) is defined by \[ Q_{M_n}(u)=\inf\{x\in\mathbb{R}:\mathbb{P}(M_n>x)\leq u\}\,, \] and the superquantile (or conditional value-at-risk) by \[ \widetilde{Q}_{M_n}(u)=u^{-1}\int_0^uQ_{M_n}(t)\,dt\,. \] The main results of this paper are upper bounds on the superquantiles of \(M_n\), and upper bounds on the quantiles and superquantiles of \(M_n^*=\max\{M_0,\dots,M_n\}\). Key ingredients in the proofs of these bounds are upper bounds on moments of the form \(\mathbb{E}[(M_n+t)_+^p]\). In the case \(p=3\), these need additional boundedness assumptions on the increments of the underlying martingale. Some numerical comparisons with other available bounds are also presented.
scientific article; zbMATH DE number 7398735
scientific article