Numerical methods for optimal annuity purchasing and dividend optimization strategies under regime-switching models: Review of recent results

This chapter is concerned with the handling of insurance risk. We focus on numerical methods for regime-switching models that are widely used in analyzing optimization problems arising from dividend payment strategy and ruin probability, which are commonly formulated as stochastic control and optimization problems. The regime-switching models better describe the market conditions (such as bull and bear) and other economic conditions. The model is more versatile and realistic, but more complex to deal with. It is virtually impossible to obtain closed-form solutions for the associated optimal control problems that require the solutions of the Hamilton–Jacobi–Bellman (HJB) equations. For the regime-switching models, these HJB equations become HJB systems of equations. Closed-form solutions are virtually impossible to obtain, It is thus necessary to solve the problem numerically. We will use Markov chain approximation techniques to develop numerical solutions, which have the advantages that the regularity of the value function and/or analytic properties of the associated systems of HJB equations and/or quasi-variational inequalities need not be known. A feasible numerical approximation schemes is constructed to find a good approximation to the underlying problems, and the numerical examples are provided to illustrate the performance of the algorithm.