The model under consideration is a natural extension of the classical Cramér-Lundberg risk model: it is assumed that an insurance company which has the wealth \(Y(t)\) at time \(t\) may invest an amount \(K(t)\) in a stock or market index described by a geometric Brownian motion and invest the remaining the reserve \(Y(t)-K(t)\) in a bond with a positive interest \(r>0\). The authors studied the behaviour of the ruin probability \(\psi(x,K)\), which depends on the initial capital \(x\) and a strategy of unvestment \(K=K(t)\). In particular, they focused on the optimal strategy \(K^*\) which minimizes \(\psi(x,K)\) over all predictable strategies. The paper consists of three parts.
In the first part, the authors derived the Hamilton-Jacobi-Bellman equation for the survival probability for the optimal strategy and obtained the asymptotic behaviour of the optimal ruin probability in the presence of positive interest force and claims with tails of regular variation. These results extend previous authors’ results [see Insur. Math. Econ. 30, No. 2, 211–217 (2002; Zbl 1055.91049)] with zero interest and results of C. Klüppelberg and U. Stadtmüller [Scand. Actuarial J. 1998, No. 1, 49–58 (1998; Zbl 1022.60083)] without investment possibility.
The second part presents an existence theorem for integro-differential equations for the survival probability of an insurer who invests a constant fraction \(k\) of his wealth \(Y(t)\) in a risky stock and the remaining wealth \(Y(t)-kY(t)\) in a bond. This theorem is rather general and extends a result by G. Wang and R. Wu [Stochastic Processes Appl. 95, No. 2, 329–341 (2001; Zbl 1064.91051)]. Finally, in the third part, the authors derived the asymptotic behaviour of the ruin probability of an insurer, introduced in the second part, in the presence of claims with regularly varying tails.