Company value with ruin constraint in a discrete model

Optimal dividend payment under a ruin constraint is a two objective control problem 1 which – in simple models – can be solved numerically by three essentially different methods. One 2 is based on a modified Bellman equation and the policy improvement method (see (2003)). In this 3 paper we use explicit formulas for running allowed ruin probabilities which avoid a complete search 4 and speed up and simplify the computation. The second is also a policy improvement method, but 5 without the use of a dynamic equation (see (2003)). It is based on closed formulas for first entry 6 probabilities and discount factors for the time until first entry (see (2016)). Third a new, faster and 7 more intuitive method which uses appropriately chosen barrier levels and a closed formula for 8 the corresponding dividend value. Using the running allowed ruin probabilities, a simple test for 9 admissibility – concerning the ruin constraint – is given. All these methods work for the discrete 10 De Finetti model and are applied in a numerical example. The non stationary Lagrange multiplier 11 method suggested in (2016), section 2.2.2 does also yield optimal dividend strategies which differ 12 from those in all other methods, and Lagrange gaps are present here. These gaps always exist in De 13 Finetti models, see (2017). 14