DescriptionThe classical optimal control problems for discrete-time, transient Markov processes are infinite horizon, undiscounted expected total cost or reward models. Some examples of these models are optimal stopping problems and stochastic shortest or longest path problems, which may have applications in health-care, finance, and maintenance. However, such expected value models implicitly assume the decision maker is risk-neutral, so they may not be appropriate for several real-life problems. In this study, we use Markov risk measures to formulate a risk-averse version of the optimal control problem for transient Markov processes with general state and compact control spaces. We derive risk-averse dynamic programming equations and show that they have a unique solution which is also the optimal value of the Markov control problem. Furthermore, it is shown that a randomized policy may be strictly better than deterministic policies, when risk measures are employed. We suggest two algorithms, value iteration and policy iteration methods, for solving the dynamic programming equations and show their convergence. In general, each policy evaluation step of the policy iteration algorithm requires solving a system of nonsmooth equations. We use a version of nonsmooth Newton method to solve these equations and show its global convergence. We further consider a risk-averse finite horizon Markov control problem under randomized policies and derive a value iteration method for its solution. Finally, we work on asset selling, organ transplant, and credit card examples to illustrate the theory for infinite horizon problem, and present numerical results.