Limit behavior of the Esscher premium (Q289615)

The authors define the Esscher premium as NEWLINE\[NEWLINE\pi_{\mathrm{Esscher}(\alpha)}(X)=\frac{E[Xe^{\alpha X}]}{E[e^{\alpha X}]},NEWLINE\]NEWLINE for an arbitrary risk \(X\) with \(\alpha\geq 0\), and show the convergence from below of the Esscher premium (for an arbitrary insurance risk \(X\)) to the essential supremum of the priced risk when the premium parameter tends to \(+\infty\).NEWLINENEWLINEAs a consequence of this result, they also present convergence of the expected/averaged Esscher transform of an arbitrary random variable \(X\) to its essential infimum from above when the transformation parameter tends to \(-\infty\).