3.1 Layer-cake and stop-loss identities
For every probability measure \(\mu \) on \(\mathbb {R}\) and every \(u \ge 0\),
\[ \int (x-u)_+\, d\mu (x) = \int _u^\infty \mathbb {P}_\mu (x {\gt} t)\, dt. \]
Proof
Expand \((x-u)_+\) as \(\int _u^\infty \mathbf{1}_{\{ x{\gt}t\} }\, dt\) and apply Tonelli’s theorem to swap the order of integration.
If \(\mu \) is a probability measure with finite first moment, then for every \(u \ge 0\) its stop-loss transform equals the tail integral from Lemma 4.
Proof
Immediate from Lemma 4 and the definition of the stop-loss transform.