Finding control policy for one discrete-time Markov chain on $[0, 1]$ with a given invariant measure

A discrete-time Markov chain on the interval $[0,1]$ with two possible transitions (left or right) at each step has been considerred. The probability of transition towards $0$ (and towards $1$) is a function of the current value of the chain. Having chosen the direction, the chain moves to the randomly chosen point from the appropriate interval. The authors assume that the transition probabilities depend on the current value of the chain only through a finite number of real-valued numbers. Under this assumption, they seek the transition probabilities, which guarantee the $L_2$ distance between the stationary density of the Markov chain and the given invariant measure on $[0,1]$ is minimal. Since there is no reward function in this problem, it does not fit in the MDP (Markov decision process) framework. The authors follow the sensitivity-based approach and propose the gradient- and simulation-based method for estimating the parameters of the transition probabilities. Numerical results are presented which show the performance of the method for various transition probabilities and invariant measures on $[0,1]$.