Risky investments and survival probability in the insurance model with two-sided jumps: Problems for integrodifferential equations and ordinary differential equation and their equivalence

We consider a model of an insurance portfolio that includes both non-life and life annuity insurance while assuming that the surplus (or some of its fraction) is invested in risky assets with the price dynamics given by a geometric Brownian motion. The portfolio surplus (in the absence of investments) is described by a stochastic process involving two-sided jumps and a continuous drift. Downward jumps correspond to the claim sizes and upward jumps are interpreted as random gains that arise at the final moments of the life annuity contracts realizations (i.e. at the moments of the death of policyholders). The drift is determined by the difference between premiums in the non-life insurance contracts and the annuity payments. We study the ruin problem for the model with investment using an approach based on integrodifferential equations (IDE) for the survival probabilities as a function of initial surplus. The main problem in calculating the survival probability as a solution of the IDE is that the initial value of the probability itself or its derivative at a zero initial surplus is priori unknown. For the case of the exponential distributions of the jumps, we propose a solution to this problem based on the assertion that the problem for an IDE is equivalent to a problem for an ordinary differential equation (ODE) with some nonlocal condition added. As a result, a solution to the original problem can be obtained as a solution to the ODE problem with an unknown parameter, which is finally determined using the specified nonlocal condition and a normalization condition.