In this paper we propose a generalisation to the Markov Arrival Process (MAP) risk model, by allowing for a delayed receipt of required capital injections whenever the surplus of an insurance firm is negative. Delayed capital injections often appear in practice due to the time taken for administrative and processing purposes of the funds from a third party or the shareholders of a firm. We introduce a MAP risk model that allows for capital injections to be received instantaneously, or with a random delay, depending on the amount of deficit experienced by the firm. For this model, we derive a system of Fredholm integral equations of the second kind for the Gerber-Shiu function and obtain an explicit expression (in matrix form) in terms of the Gerber-Shiu function of the MAP risk model without capital injections. In addition, we show that the expected discounted accumulated capital injections and the expected discounted overall time in red, up to the time of ruin, satisfy a similar integral equation, which can also be solved explicitly. Finally, to illustrate the applicability of our results, numerical examples are given.

在本文中，我们提出了一种马尔可夫到达过程（MAP）风险模型的一般化方法，方法是在保险公司的盈余为负时允许延迟接收所需的注资。实际上，由于从第三方或公司股东那里进行资金的管理和处理需要时间，在实践中经常会出现延迟的注资。我们引入了MAP风险模型，该模型允许根据企业所遭受的赤字数量，立即或随机延迟地接受注资。对于此模型，我们针对Gerber-Shiu函数推导出第二类Fredholm积分方程组，并根据MAP风险模型的Gerber-Shiu函数获得无资本注入的显式表达式（以矩阵形式）。此外，我们表明，预期的折现累积资本注入和预期折现的总时间（用红色表示，直到破产前的时间）都满足类似的积分方程，也可以明确求解。最后，为了说明我们的结果的适用性，给出了数值示例。