A non-monotone Lambda-VaR can run through existing monotone-Lambda code when its confidence function has bounded variation and is right lower semicontinuous. Bellini and Liebrich make the computational implication explicit in their conclusion: the mixture representation of Section 7 "makes the numerical schemes developed in Peri and Wunderlich (2026) for monotone Λ available in the general case."
The object
A lambda quantile lets the VaR confidence level vary with the loss level. Instead of a fixed probability α, it uses a function Λ from R into [0,1]. Its left version is Q^-_Λ(F) = inf{x: F(x) ≥ Λ(x)}, where F is a cumulative distribution function (CDF); Q^+_Λ is the right version and uses the strict inequality. Setting Λ ≡ α recovers classical quantiles.
The cleanest example in the paper sets Λ to λ1 below a threshold z and λ2 above it. Outside the transition region, the result follows the ordinary λ1- or λ2-quantile. When Q^-{λ2}(F) ≤ z ≤ Q^-{λ1}(F), the Λ-quantile equals z.
Two properties keep this construction relevant to risk management. Infinite-mean distributions pose no definitional problem, unlike Expected Shortfall. Also, a strictly decreasing Λ makes the left and right versions coincide. They are then weakly continuous on all of M, a property classical quantiles lack.
The treatment is entirely measure theoretic. Its domain consists of M, the set of all univariate CDFs on R, the compactly supported CDFs within M, and L^0 over an atomless probability space. There is no dataset, sample period, estimation exercise or reported risk number. The authors instead characterise finiteness, constancy and the attainment property. They establish identifiability of Λ from a two-parameter family of shifted Bernoulli distributions, give if-and-only-if conditions for weak semicontinuity and convex level sets, derive a mixture representation for bounded-variation Λ, and identify the ordinal covariance group of a Λ-quantile.
Why allow Λ to rise and fall?
Closure supplies the first answer. Proposition 4.1 equates inf over an arbitrary index set of Q^+{Λ_i} with Q^+ formed from the pointwise infimum. It also gives the minimum rule for Q^- over finite index sets. Since a pointwise minimum of monotone functions can lose monotonicity, a closed class has to admit non-monotone Λ. Max-aggregation fails outright. For F uniform on [0,1], Example 4.2 constructs two parameters satisfying Q^-{Λ_1}(F) = Q^-{Λ_2}(F) = 1/4, yet Q^-{max{Λ_1,Λ_2}}(F) = 3/4. Classical quantiles have no corresponding asymmetry.
Mixing provides the other route. Proposition 7.4 replaces F with λF + (1-λ)G at a fixed weight. This operation preserves the bounded-variation, right-lower-semicontinuous class whenever (1-λ)G ≤ Λ_0 ≤ λ + (1-λ)G. Its variation obeys TV(Λ) ≤ (1/λ)(TV(Λ_0) + 1 - λ). Because the resulting parameter takes on the decreasing behaviour of -(1-λ)G, monotonicity can disappear even when Λ_0 is monotone.
The authors describe closure and mixing as a posteriori reasons for considering the broader class. They offer no economic or regulatory account of a threshold that changes direction. Their stated case is closure.
The reduction to keep
Theorem 7.1 carries the practical result. Let Λ have bounded variation and be right lower semicontinuous. There are then an increasing Λ_inc into [0,1], a weight λ in (0,1] satisfying λ ≥ Λ_inc(-∞), and a fixed G in M such that, for every F, Q^±Λ(F) = Q^±{Λ_inc}(λF + (1-λ)G). The Jordan decomposition Λ = a + G^+ - G^- produces the constants. Writing c = G^-(∞) gives λ = 1/(1+c) and Λ_inc = (a + G^+)/(1+c).
Corollary 7.3 makes the decreasing case especially concrete. The increasing parameter becomes a constant α, while λ lies in [1/2,1] and 0 ≤ α ≤ λ. When Λ(-∞) = 1 and Λ(∞) = 0, one may choose α = λ = 1/2. The resulting Λ-quantile is the median of an equal-weights mixture containing F and a fixed reference distribution. Wang (2024) is credited with communicating this observation and its connection to Moulin's (1980) characterisation of anonymous, efficient, strategy-proof voting schemes as generalised medians.
Implementation therefore means mixing the loss distribution with a fixed reference distribution at a fixed weight, then sending that mixture to a monotone-Λ solver. Choosing Λ, G and λ remains the user's problem. Remark 7.2 states that the representing triple is generally non-unique; uniqueness holds only for positive non-constant decreasing Λ. A risk committee should not be shown (Λ_inc, λ, G) as an identified decomposition.
The paper supplies no method for choosing Λ.
Right lower semicontinuity carries the proofs
The authors state their position directly: "it is right lower semicontinuity of Λ, not monotonicity, the property that drives most of our arguments." For Λ in standard representation, this condition yields the attainment property, turning the defining infimum into a minimum. It also supports identification within the class of right lower semicontinuous Λ in standard representation.
Inside that class, values of the Λ-quantile on the two-parameter shifted Bernoulli family recover Λ. The CDFs are B_{x,p} = (1-p)1_{[x,x+1)} + 1_{[x+1,∞)}, and the reconstruction is Λ(x) = inf{1-p: Q^-Λ(B{x,p}) = x} for x ≤ N_0. Classical quantiles and distortion riskmetrics require only the one-parameter family B_{0,p}; Λ-quantiles carry one more degree of freedom. This gives a practical audit. Feed a Λ-VaR engine two-point distributions and infer the Λ it actually applies. The procedure breaks outside the right-lsc class: Example 3.4 provides a family whose quantiles reveal nothing about the parameter ε in (1/2,1].
Earlier work imposed stronger conditions for finiteness. Lemma 2.3 asks only for lim inf Λ > 0 at minus infinity and, under standard representation, lim inf Λ < 1 at plus infinity. Burzoni et al. (2017) and Frittelli et al. (2014) required Λ to stay bounded away from both 0 and 1.
Theorem 5.1 matters immediately for step functions. Q^+Λ is weakly upper semicontinuous exactly when Λ(x) ≥ lim inf{y↓x} Λ(y), which rules out upward jumps from the right. As the authors write, the condition "requires neither monotonicity nor any right- or left-regularity of Λ."
Weak continuity asks much more. With right lsc Λ, Q^+_Λ can be weakly continuous only when Λ ≡ 1 below a threshold and is strictly decreasing above it. For Q^-_Λ, weak continuity requires Λ to decrease strictly up to N_0 and equal zero afterwards. The paper connects weak continuity at F to strong consistency of the empirical plug-in through Glivenko-Cantelli, as well as to qualitative robustness in Hampel's (1971) sense. A step Λ satisfies the semicontinuity condition and misses the continuity condition, placing it beyond that consistency argument.
We found no rates or finite-sample results in the paper. A validation team would want both. For backtesting, the introduction credits the earlier literature, including Corbetta and Peri (2018), with tests extending violation counting.
Convex level sets receive a parallel analysis. Since Osband (1985), they have been known as necessary for elicitability. Proposition 6.1 shows that right lower semicontinuity suffices for Q^-_Λ, though necessity does not follow. The if-and-only-if results in Theorem 6.4 and Proposition 6.5 require local right monotonicity; the left quantile also needs standard representation. The authors conclude that a fully general characterisation "does not seem possible in full generality."
An ordinal covariance group
Section 8 assigns a group structure to ordinal covariance. If Λ is in standard representation, meets the attainment condition and produces a finite left Λ-quantile, then G(Q^-_Λ) is the invariance group of Λ. For bounded-variation right-continuous Λ, the group is the pointwise stabiliser of the support of the corresponding signed measure. The proof uses the Poincaré recurrence theorem.
With n jump points, the group consists of increasing homeomorphisms fixing those n points. The comparisons are Aff^+ for the mean and translations for the entropic measure. G(Λ) tracks the distance from a classical quantile: constant Λ gives the full group S, while strictly monotone Λ gives the trivial group. I see no trade in that result.
Our equity overlay
We translated the idea into a daily long-only equity overlay and ran it on 50 liquid US stocks and ETFs. The daily-bar sample runs from 2015-01-01 to 2024-12-31. Across that decade, the construction returned 249.37% in total with a Sharpe of 0.68. Volatility reached 36.55%, and maximum drawdown was -84.84%. Since the theoretical paper reports no performance figure, there is no author result available for comparison.
The baseline sleeve combines trend and momentum. Eligibility requires the close to exceed the 200-day SMA and the 126-day return to be positive. We then select the top 20 eligible names by 126-day return, assign inverse 20-day volatility weights capped at 10% each, and leave any residual in cash.
The paper's object enters through the overlay. Each asset has a rolling 252-day empirical distribution of daily losses, subject to a 126-observation minimum. sigma_t is the 252-day standard deviation of those losses, and the traded estimate retained that 252-day window throughout. We evaluate the step Λ_t on z = loss/sigma_t. Its value is 0.800 when z is below 0.5, 0.900 for z in [0.5, 1.0), 0.950 for z in [1.0, 1.5), and 0.975 when z is at or above 1.5. Q^-_Λ is the first sorted loss where the empirical CDF reaches the level.
Exposure is halved above 2.5 sigma and cut to zero above 3.5 sigma. Re-entry waits until the estimate drops below 2.2 sigma. Commission was four tenths of a cent a share with a one dollar minimum. We modelled no slippage, which favours a book rebalanced daily.
The run has two direct limitations. Our implemented Λ increases with loss and is right-continuous. Theorem 7.1 therefore reduces to the identity map: c = 0, λ = 1, Λ_inc = Λ, any G. The overlay never uses the reduction that forms the paper's main result. Also, sigma_t rescales the step function each day, so our computed quantity falls outside the fixed functional parameter covered by the theorems.
The reported figures describe one automated pass at a volatility-scaled Λ gate on a long-or-cash equity book. Their scope ends there; this run supplies no verdict on the authors' work. The decade includes exactly one fast crash. A 252-day sigma reacts slowly, while the 2.2-sigma re-entry rule delays the return to market. The three choices produce the -84.84% drawdown and 36.55% volatility.
What could change the verdict?
For a desk already using a loss-dependent limit, the paper removes an implementation objection. Bounded variation and right lower semicontinuity permit reuse of monotone machinery, while the reconstruction method can check which Λ an engine applies. A desk without such a limit gets little reason here to adopt one. I would not expect a regulator to accept a threshold with a function's degrees of freedom in place of a scalar.
A finite-sample result would change my view: a standard error for a Λ-quantile estimated from 252 daily losses, shown beside the same number for VaR and Expected Shortfall. Until then, selecting Λ contributes more than the theorem used to compute it.