Nt1310 Unit 1

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\section{Preliminaries}
Lemma $2.1$ in Cai and Wang (2016) has provided a formula to determine the first derivative for the function $g(c)=E[ D((X-c)_{+})]$, where $D$ is a convex and nondecreasing function defined on $\R^{+}$.
\begin{lemma}(Lemma 2.1 Cai and Wang (2016))\label{3-lm1}
Let $g(c)=E[\xi D((X-c)_{+})]$, where $\xi$ is a non-negative random variable. $D$ is a convex and increasing function defined on $\R^{+}$. Assume $g(c)c\}}],$$
$$g'_{-}(c)=-E[\xi D'_{+}((X-c)_{+})I_{\{X\geq c\}}],$$ and both of them are finite. If $D$ is differentiable with $D'(0)=0$, then
$$g'(c)=-E[\xi D'((X-c)_{+})].$$
\end{lemma}

\noindent Lemma $2.2$ in Cai and Wang (2016) introduced a result about the well-known conclusion about the minimizers of convex program with no constraints according to Rockafellar (2011).
\begin{lemma}(Lemma 2.2 Cai and Wang …show more content…
In addition, the objective function defined in \eqref{3-eq01} satisfies $g\geq 0$, $h\geq 0$, $0 c\}}], \\
E[g(X)\varphi_{1+}^{'}((I(X)-c)_{+})I_{\{I(X)\geq c\}}]&\geq E[h(X)\varphi_{2-}^{'}(X-I(X)+c)],
\end{aligned}
\right.
\end{equation}
where $\varphi_{i-}^{'}$ and $\varphi_{i+}^{'}$ represent the corresponding left and right derivatives of $\varphi_{i}$, $i=1,2$. If $\varphi_{1}$ and $\varphi_{2}$ are strictly convex, the solution is unique. If $\varphi_{1}$ and $\varphi_{2}$ are differentiable and $\varphi'_{1}(0)=\varphi'_{2}(0)=0$, the optimal reinsurance premium principles $c^{*}_{X,I,g,h}$ are solutions to
\begin{equation}\label{3-eq2}
E[h(X)\varphi_{2}^{'}(X-I(X)+c)]= E[g(X)\varphi_{1}^{'}((I(X)-c)_{+})].
\end{equation}
\end{theorem}

\begin{proof}
Similar as Proposition 3.1 (a) in Cai and Wang (2016), we can get that $f_{1}(c)$ is finite, non-negative, convex, and satisfies

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