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Let $Y=(\log X-\mu )/\sigma$ and write $\varphi$ for the standard normal density, with $\sigma >0$. The change of variables converts the density integral intoComplete the square using $e^{k\sigma y}\varphi (y)=e^{k^2{\sigma }^2/2}\varphi (y-k\sigma )$. Integration gives the truncated lognormal momentHere $\Phi$ is the standard normal distribution function; take $\log 0=-\infty$ and $\log \infty =\infty$. The identity holds for every real $k$, including the full-interval limits, because the shifted Gaussian integral remains finite.

The limited excess of loss reinsurance payment is $L(X)=\min \{(X-M{)}_{+},A\}$. It is zero below $M$, equals $X-M$ for $M<X<M+A$, and equals $A$ above $M+A$. Since the lognormal distribution has a continuous density,Put $d_j(t)=(\log t-\mu -j{\sigma }^2)/\sigma$, $j=0,1$, and $\overline{\Phi }=1-\Phi$. Applying the moment identity with $k=1$ and $k=0$ gives the conditional lognormal layer recoveryThe denominator is $\mathbb{P}(X>M)$, and payments outside that event are zero. The three numerator terms are respectively the truncated first moment, the attachment subtracted from partial payments, and the capped payments on claims above the layer. This proves the conditional expectation including the limit on each reinsurer payment.

For claims inflation by $h=1.1$, set $X_h=hX$. Its log-mean is ${\mu }_h=\mu +\log h$ and its log-standard-deviation stays $\sigma$. The attachment and payment limit stay fixed. Therefore the requested new conditional mean isExplicitly, in the preceding fraction replace $d_j(t)$ by $d_j^{(h)}(t)=(\log t-\mu -\log 1.1-j{\sigma }^2)/\sigma$ and replace $e^{\mu +{\sigma }^2/2}$ by $1.1e^{\mu +{\sigma }^2/2}$. Equivalently, claims inflation of a fixed reinsurance layer givesbecause $L_{M,A}(hX)=h{L}_{M/h,A/h}(X)$ and the new conditioning event is $X>M/h$. Merely multiplying the original conditional mean by $1.1$ would omit both of these threshold changes.