Asymptotic expansion of generalized JES for ξ>1 (equivalently JMES with β>α)

Establish asymptotic expansions for the generalized Joint Expected Shortfall JES_{α,ξ}^{G}[Y|X] = E[Y | X > VaR_{α}(X), Y > ξ VaR_{α}(Y)] in the regime ξ ∈ (1, ∞), which corresponds to the Joint Marginal Expected Shortfall JMES_{α,β}[Y|X] with β > α. This extends existing extreme-value asymptotic results that cover ξ ≤ 1 to the case ξ > 1, in order to characterize the joint tail behavior of (X, Y) under asymmetric stress levels.

Background

The paper introduces the Joint Marginal Expected Shortfall (JMES) as a generalization of the Joint Expected Shortfall (JES) to allow different stress levels for X and Y. Specifically, JMES_{α,β}[Y|X] = E[Y | X > VaR_{α}(X), Y > VaR_{β}(Y)], while the generalized JES of Ji et al. (2021) uses a single α and a multiplicative factor ξ for Y’s threshold.

Ji et al. (2021) derived asymptotic expansions for JES in an extreme-value setting (e.g., Fréchet max-domain of attraction) when ξ ∈ (0, 1], effectively corresponding to β ≤ α. The present paper raises the question of establishing analogous asymptotics for the case ξ > 1 (i.e., β > α under JMES), which would extend the asymptotic theory to asymmetric tail conditioning where Y’s threshold exceeds that of X.