  This paper explores an optimal investment and reinsurance problem involving
both ordinary and catastrophe insurance businesses. The catastrophic events are
modeled as following a compound Poisson process, impacting the ordinary
insurance business. The claim intensity for the ordinary insurance business is
described using a Cox process with a shot-noise intensity, the jump of which is
proportional to the size of the catastrophe event. This intensity increases
when a catastrophe occurs and then decays over time. The insurer's objective is
to maximize their terminal wealth under the Monotone Mean-Variance (MMV)
criterion. In contrast to the classical Mean-Variance (MV) criterion, the MMV
criterion is monotonic across its entire domain, aligning better with
fundamental economic principles. We first formulate the original MMV
optimization problem as an auxiliary zero-sum game. Through solving the
Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation, explicit forms of the value
function and optimal strategies are obtained. Additionally, we provides the
efficient frontier within the MMV criterion. Several numerical examples are
presented to demonstrate the practical implications of the results.
