Leo Gumpert: Optimal dividends in with-profit insurance using stochastic control

Abstract:

We study optimal dividend payments and investments of the surplus of with-profit life insurance policies using continuous-time stochastic control. Under some simplifying  assumptions, the control problem studied can be treated as a generalisation of the  investment-consumption problem first set up and studied by Merton.  We use the dynamic programming method, by which the control problem boils down  to solving a second order partial differential equation (PDE) called a Hamilton-Jacobi- Bellman equation. We consider cases where the policy holders display constant relative  risk aversion, which implies first that the PDE has a semi-explicit solution and second  that the optimal investment process is constant. The optimal dividend process is linear  in the surplus.  We illustrate the results with simulations for a simple life annuity, where the PDE has an explicit solution.