In this paper, a classical risk model with liquid reserves and proportional investment is considered, and the expected total discounted dividend before ruin of insurance companies under the threshold dividend strategy is studied. First, the integral differential equations of the expected total discounted dividend before ruin satisfying certain boundary conditions is derived. Second, since the explicit solutions of the equations cannot be obtained, the numerical approximation solutions are obtained by the sinc approximation method. Finally, we discuss the effects of parameters such as risk capital ratio and liquid reserve on the expected total discounted dividend before ruin by some examples.

In this paper, an investment risk model with bilateral jumps was considered, assuming the insurer invested the surplus in two types of assets, namely, risk-free and risky ones, in a certain proportion. First, the integral-differential equations of the Gerber-Shiu function related to ruin and penalty were obtained, then, the sinc approximation method was used to obtain a numerical solution. Furthermore, we presented a special example for finding the explicit solutions (ES). By calculating the relative errors of the approximate solution (SA) and ES, we verified the superiority of the sinc method. Finally, several examples under different kinds of jumps were provided to show the impact of parameters such as investment ratio, discount factor or intensity of Poisson process on the ruin probability.

We mainly studied the dividend payout with a two-sided jumps risk model under random observation. The two-sided jumps in the model represent random claims and random returns. First, we obtained the integral differential equation of the expected dividend under the boundary conditions. Because the equations cannot be solved directly under normal circumstances, we chose the sinc numerical method here to approximate the solution of the equations. Then the error analysis of the approximate solution was carried out to illustrate the rationality of the numerical method. Finally, some concrete numerical examples were given.