3.2 The sharp stop-loss envelope
Assume that \(\mu \) is centered and satisfies the tail cap \(\mathbb {P}_\mu (|x|{\gt}t)\le s_G(t)\) for every \(t \ge 0\). Then for every \(u \ge 0\),
where \(J_G\) is the explicit envelope built from the constants \(B\), \(a\), and \(p_0\).
Split into the two regions \(u \le a\) and \(u \ge a\). For \(u \le a\), the mean-zero condition and the tail cap bound the positive part of the first moment by \(B\), giving the affine upper bound \(B - p_0 u\). For \(u \ge a\), the layer-cake identity rewrites the stop-loss as a tail integral and the cap is integrated directly.
The stop-loss bound extends from \(u \ge 0\) to every real threshold \(u\) by applying the same argument to \(-X\) and using the mean-zero identity relating the stop-loss transforms of \(X\) and \(-X\).
The negative-threshold case is rewritten in terms of the positive-threshold stop-loss of \(-X\). Because the tail assumption is symmetric and the mean is zero, the transformed problem satisfies the same hypotheses.