Inequalities on the ruin probability for light-tailed distributions with some restrictions

When there are some restrictions on the random variables of insurance risk model, it is impossible to calculate the exact value of ruin probabilities. For these cases, even finding a suitable approximation, is very important from a practical point of view. In the present paper, we consider the renewal insurance surplus model with light-tailed claim amount distributions and try to find some inequalities on the infinite time ruin probability depending on the amount of initial reserve using statistical and mathematical approaches if the assumption of net profit does not hold but there exist some other restrictions on the mathematical functions of random variables of model. The assertions depend on the amount of initial reserve, distribution of nonnegative claim occurrences times and successive claim amounts are obtained. Finally, to show the application and effectiveness of given theorems two examples are presented. Through these examples, the infinite time ruin probabilities are estimated using Monte Carlo simulation and give an intuitive way to understand the nature of ruin.