Minimizing infinite time horizon discounted cost with mean, variance, and bounded variation controls

We consider an infinite time horizon discounted cost minimization problem for a class of Ito processes. The available controls are drift and diffusion coefficients and the added bounded variation process. We show that the value function is twice continuously differentiable and derive an optimal policy which has feedback-type drift and diffusion coefficients. When the absolute value of the optimal drift grows faster than the running cost function, the optimal bounded variation process is identically zero. When it grows weaker than the running cost function, optimal bounded variation process is a local time-type process. In this case, we relate the control problem with an optimal stopping problem. We also establish the Abelian limit relations between the value functions of the discounted cost problem and the stationary problem.