4 Envelope and extremizer

This chapter shows that the stop-loss envelope from Chapter 3 is sharp. The constants \(a\) and \(p_0\) are tuned so that the piecewise formula for \(J_G\) is tangent to the Gaussian comparator at the correct contact point, and a canonical probability measure \(\mu ^\star \) is constructed to saturate the tail cap and attain the envelope exactly.