On Optimal Dividend and Investment Strategy under Renewal Risk Models

Abstract: In this paper we continue investigating the optimal dividend and investment problems under the Sparre Andersen model. More precisely, we assume that the claim frequency is a renewal process instead of a standard compound Poisson process, whence semi-Markovian. Building on our previous work \cite{BaiMa17}, where we established the dynamic programming principle via a {\it backward Markovization} procedure and proved that the value function is the unique {\it constrained} viscosity solution of the HJB equation, in this paper we focus on the construction of the optimal strategy. The main difficulties in this effort is two fold: the regularity of the viscosity solution to a non-local, nonlinear, and degenerate parabolic PDE on an unbounded domain, which seems to be new in its own right; and the well-posedness of the closed-loop stochastic system. By introducing an auxiliary PDE, we construct an $\e$-optimal strategy, and prove the well-posedness of the corresponding closed-loop system, via a "bootstrap" technique with the help of a Krylov estimate.