On the Integro-Differential Equation Arising in the Ruin Problem for Non-Life Insurance Models with Investment

In the classical non-life insurance models, the capital reserve of an insurance company increases at a constant rate and decreases by downward jumps. We consider a generalization of this model by supposing that a fixed portion of the capital reserve is continuously invested in a risky asset whose price follows a geometric Brownian motion, while the complementary part is placed in a bank account with a constant rate of return. The quantity of interest is the ruin probability on the infinite time horizon as a function of the initial capital. In the present note, we assume only the continuity of the distribution of claims together with a standard moment restriction called “light tails.” Our main contribution is that we reveal, under such “minimalistic” hypotheses, that the ruin probability is smooth and satisfies a second-order integro-differential equation in the classical sense. We obtain the exact asymptotics for large values of the initial capital with “computable” constants and present results of numerical experiments. In contrast with other methods used in the theory, we rely upon only standard mathematics, allowing implementation in lecture courses for master’s students.