= Solution
Under <quota share reinsurance>, $S_I=\alpha S$ and $S_R=(1-\alpha)S$. Write $v=\operatorname{Var}(S)=\lambda\mathbb E[X^2]$. Then
$$
\operatorname{Var}(S_I)+\operatorname{Var}(S_R)
=v\bigl(\alpha^2+(1-\alpha)^2\bigr)
=\frac v2+2v\left(\alpha-\frac12\right)^2.
$$
The <variance-minimizing quota share split> is therefore
$$
\boxed{\alpha^*=\frac12,\qquad \min\{\operatorname{Var}(S_I)+\operatorname{Var}(S_R)\}=\frac{\lambda\mathbb E[X^2]}2.}
$$
For positive claims and $\lambda>0$ this minimizer is unique. If the aggregate has zero <variance>, every retained fraction gives the same zero objective.
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