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Controlled discrete time Markov processes are studied first with long rungeneral discounting functional. It is shown that optimal strategies for averagereward per unit time problem are also optimal for average generally discountingfunctional. Then long run risk sensitive reward functional with generaldiscounting is considered. When risk factor is positive then optimal value ofsuch reward functional is dominated by the reward functional corresponding tothe long run risk sensitive control. In the case of negative risk factor we getan asymptotical result, which says that optimal average reward per unit timecontrol is nearly optimal for long run risk sensitive reward functional withgeneral discounting, assuming that risk factor is close to $0$. For thispurpose we show in Appendix upper estimates for large deviations of weightedempirical measures, which are of independent interest.

Long run stochastic control problems with general discounting