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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...