استراتژی سرمایه گذاری بهینه جهت به حداقل رساندن احتمال تباهی یک شرکت بیمه تحت محدودیت استقراض

We consider that the surplus of an insurance company follows a Cramér–Lundberg process. The management has the possibility of investing part of the surplus in a risky asset. We consider that the risky asset is a stock whose price process is a geometric Brownian motion. Our aim is to find a dynamic choice of the investment policy which minimizes the ruin probability of the company. We impose that the ratio between the amount invested in the risky asset and the surplus should be smaller than a given positive bound a. For instance the case a=1 means that the management cannot borrow money to buy stocks. [Hipp, C., Plum, M., 2000. Optimal investment for insurers. Insurance: Mathematics and Economics 27, 215–228] and [Schmidli, H., 2002. On minimizing the ruin probability by investment and reinsurance. Ann. Appl. Probab. 12, 890–907] solved this problem without borrowing constraints. They found that the ratio between the amount invested in the risky asset and the surplus goes to infinity as the surplus approaches zero, so the optimal strategies of the constrained and unconstrained problems never coincide. We characterize the optimal value function as the classical solution of the associated Hamilton–Jacobi–Bellman equation. This equation is a second-order non-linear integro-differential equation. We obtain numerical solutions for some claim-size distributions and compare our results with those of the unconstrained case.

An important issue in actuarial theory is to study the ruin probability of an insurance company when the management has the possibility of investing in the financial market. Paulsen and Gjessing (1997) and Paulsen (1998) treated this problem assuming that all the surplus is invested in the risky asset. Frovola et al. (2002) studied the asymptotic behavior of the ruin probability for light tails in the case where a constant proportion of the surplus is invested in the risky asset (stocks). The problem of finding the optimal dynamic investment strategy which minimizes the ruin probability was first studied by Browne (1995) under the assumptions that the uncontrolled surplus follows a Brownian motion (diffusion approximation) and that the financial market follows a classical Black–Scholes model consisting on a risk-free asset (bond) and a risky asset (stock). Hipp and Plum (2000) found the investment strategy which minimizes the ruin probability modeling the aggregate claim amount as in the classical risk model. They found that, in the optimal investment strategy, the management should borrow money to invest in the risky asset; in this optimal strategy, the ratio between the amount invested in the risky asset and the surplus goes to infinity as the surplus approaches zero. In fact, there are examples where, under the optimal investment strategy, the company should be in debt no matter the surplus (see Example 6.3). In this paper we treat the same problem but imposing borrowing constraints; namely we impose that the ratio between the amount invested in the risky asset and the surplus should be smaller than a given positive bound a. For instance the case a=1 means that the management has the possibility to invest in the risky asset any proportion of the surplus but cannot borrow money to buy stocks. Schmidli (2002) solved a related problem, he found the optimal investment strategy and the optimal proportional reinsurance policy which minimizes the ruin probability, with no borrowing constraints. In the setting of the diffusion approximation, Promislow and Young (2005) and Luo (2008) found the optimal investment strategy and the optimal proportional reinsurance policy which minimizes the ruin probability under different borrowing constraints. Vila and Zariphopoulou (1997) and Bayraktar and Young (2007) studied other optimal control problems with borrowing constraints. In this paper, we first obtain the Hamilton–Jacobi–Bellman equation associated to the optimization problem, which is a non-linear degenerate second-order integro-differential equation. Then we construct a weak solution of the HJB equation as a fixed point of a non-linear integral operator. We study the regularity of the weak solution and prove that it is twice continuously differentiable and so, a classical solution of the HJB equation. Finally, we use Itô’s formula and martingale techniques to prove that the optimal survival probability is a multiple of the solution obtained previously. Moreover, we obtain the optimal constrained strategy and show that it depends only on the current surplus. We also prove that for small surpluses, the optimal policy is to invest the maximum allowed in the risky asset. In order to obtain the optimal survival probability, we construct via a fixed-point operator, the survival probability corresponding to invest in the risky asset a fixed proportion of the surplus and proved that it is a classical solution of the corresponding integro-differential equation. We include numerical examples comparing the optimal survival probabilities and the optimal investment strategies in the unconstrained and constrained cases. This paper is organized as follows: In Section 2, we state the optimization problem and the HJB equation. In Sections 3 and 4, we construct via a fixed-point operator a weak solution of the HJB equation and show that it is a classical solution. In Section 5, we use the solution obtained in Section 3 to obtain the optimal ruin probability and describe the optimal investment strategy. In Section 6, we show some numerical examples.