5.1 Closed form and tangent point
If \(G \sim \mathcal N(0,1)\) and \(c{\gt}0\), then the stop-loss transform of \(cG\) is
\[ g_c(u) = c\, \varphi (u/c) - u\, \overline{\Phi }(u/c). \]
Proof
Evaluate the integral \(\int _u^\infty (x-u)\, d\gamma _c(x)\) by integrating by parts to obtain the standard Gaussian call-function formula.
At the point \(u_\star := c_0 z\), the Gaussian stop-loss at scale \(c_0\) has slope \(-p_0\) and value \(B - p_0 u_\star \). By convexity of the Gaussian stop-loss,
\[ g_{c_0}(u) \ge B - p_0 u \qquad \text{for all } u \in \mathbb {R}. \]
Proof
The slope identity is \(g_{c_0}'(u)=-\overline{\Phi }(u/c_0)\) evaluated at \(u=c_0 z\), where \(\overline{\Phi }(z)=p_0\) by definition. The value identity is \(c_0 \varphi (z)=B\). Convexity then gives the global supporting-line inequality.