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This work analyzes a discrete-time Markov Control Model (MCM) on Borel spaces when the performance index is the expected total discounted cost. This criterion admits unbounded costs. It is assumed that the discount rate in any period is obtained by using recursive functions and a known initial discount rate. The classic dynamic programming method for finite-horizon case is verified. Under slight conditions, the existence of deterministic non-stationary optimal policies for infinite-horizon case...

This paper studies a class of discrete-time discounted semi-Markov control model on Borel spaces. We assume possibly unbounded costs and a non-stationary exponential form in the discount factor which depends of on a rate, called the discount rate. Given an initial discount rate the evolution in next steps depends on both the previous discount rate and the sojourn time of the system at the current state. The new results provided here are the existence and the approximation of optimal policies for...

We study the problem in the title and show that it is equivalent to the fact that every set of reals is an increasing union of measurable sets. We also show the relationship of it with Sierpi'nski sets.

The paper deals with a class of discrete-time stochastic control processes under a discounted optimality criterion with random discount rate, and possibly unbounded costs. The state process $\left\{x_t\right\}$ and the discount process $\left\{{\alpha }_t\right\}$ evolve according to the coupled difference equations $x_{t+1}=F(x_t,{\alpha }_t,a_t,{\xi }_t),$ ${\alpha }_{t+1}=G({\alpha }_t,{\eta }_t)$ where the state and discount disturbance processes $\left\{{\xi }_t\right\}$ and $\left\{{\eta }_t\right\}$ are sequences of i.i.d. random variables with densities ${\rho }^{\xi }$ and ${\rho }^{\eta }$ respectively. The main objective is to introduce approximation algorithms of the optimal...