Solvency of an insurance company in a dual risk model with investment: analysis and numerical study of singular boundary value problems

The survival probability of an insurance company in a collective pension insurance model (so-called dual risk model) is investigated in the case when the whole surplus (or its fixed fraction) is invested in risky assets, which are modeled by a geometric Brownian motion. A typical insurance contract for an insurer in this model is a life annuity in exchange for the transfer of the inheritance right to policyholder's property to the insurance company. The model is treated as dual with respect to the Cramér–Lundberg classical model. In the structure of an insurance risk process, this is expressed by positive random jumps (compound Poisson process) and a linearly decreasing deterministic component corresponding to pension payments. In the case of exponentially distributed jump sizes, it is shown that the survival probability regarded as a function of initial surplus defined on the nonnegative real half-line is a solution of a singular boundary value problem for an integro-differential equation with a non-Volterra integral operator. The existence and uniqueness of a solution to this problem is proved. Asymptotic representations of the survival probability for small and large values of the initial surplus are obtained. An efficient algorithm for the numerical evaluation of the solution is proposed. Numerical results are presented, and their economic interpretation is given. Namely, it is shown that, in pension insurance, investment in risky assets plays an important role in an increase of the company's solvency for small values of initial surplus.