Stochastic control for insurers; what can we learn from finance, and what are the differences?.
Christian Hipp (Karlsruher Institute of Technology, Karlsruhe, Germany)
We give examples for stochastic control problems in insurance: optimal reinsurance (unlimited and limited excess of loss), optimal investment (without constraint: singularity, leverage, asymptotics), with constraints (no leverage, no shortselling and singularities caused by constraints), dividend optimisation and combinations. As methods for solution we discuss dynamic equations of Hamilton-Jacobi-Bellman type, the viscosity solution concept and a comparison argument for the insurance context. Emphasis is on numerical methods: we give an Euler type method which works in most cases and prove convergence.
Finally, we give a list of open problems together with heuristic solutions for a two objective problem: maximizing dividend payment under a ruin constraint.
Keywords: Stochastic Control, Viscosity Solutions, Euler type discretisations, Multi objective problem.