Markov control models with unknown random state–action-dependent discount factors

The paper deals with a class of discounted discrete-time Markov control models with non-constant discount factors of the form $$\tilde{\alpha } (x_{n},a_{n},\xi _{n+1})$$ α ~ ( x n , a n , ξ n + 1 ) , where $$x_{n},a_{n},$$ x n , a n , and $$\xi _{n+1}$$ ξ n + 1 are the state, the action, and a random disturbance at time $$n,$$ n , respectively, taking values in Borel spaces. Assuming that the one-stage cost is possibly unbounded and that the distributions of $$\xi _{n}$$ ξ n are unknown, we study the corresponding optimal control problem under two settings. Firstly we assume that the random disturbance process $$\left\{ \xi _{n}\right\} $$ ξ n is formed by observable independent and identically distributed random variables, and then we introduce an estimation and control procedure to construct strategies. Instead, in the second one, $$\left\{ \xi _{n}\right\} $$ ξ n is assumed to be non-observable whose distributions may change from stage to stage, and in this case the problem is studied as a minimax control problem in which the controller has an opponent selecting the distribution of the corresponding random disturbance at each stage. Copyright Sociedad de Estadística e Investigación Operativa 2015