OPTIMAL INVESTMENT AND DIVIDEND STRATEGY UNDER RENEWAL RISK MODEL

In this paper we continue investigating the optimal dividend and investment problems under the Sparre Andersen model. More precisely, we try to give a more complete description of the optimal strategy when the claim frequency is a renewal process and therefore semi-Markovian, for which it is well-known that the barrier strategy is no longer optimal (cf. [H. Albrecher and the dynamic programming principle via a backward Markovization procedure and proved that the value function is the unique constrained viscosity solution of the Hamilton-Jacobi-Bellman (HJB) equation, which is a nonlocal, nonlinear, and degenerate parabolic partial integro-differential equation on an unbounded domain, in this paper we show that the optimal strategy is still of a band type but in a more complicated dynamic fashion. The main technical obstacles in constructing and validating the optimal strategy include the regularity of the value function, due to the fundamental degeneracy of the HJB equation caused by the Markovization procedure, and the well-posedness of the closedloop stochastic system, given the ``band"" nature of the optimal strategy. Some of the technical results in this paper are purely analytical and therefore interesting in their own right.