Controlled risk processes in discrete time: Lower and upper approximations to the optimal probability of ruin

We consider the controlled random walk as a flexible model of risk process which takes into account the possibility of intervention of the insurer. We derive a method of approximating the optimal probability of ruin from below and from above. The approximations are based on iterations of the Bellman operator. To initialize the sequence of upper approximations we can use the Lundberg bound or another upper bound on the probability of ruin.