We consider stochastic optimal control problems in which the state is constrained to a proper cone. Specifically, we consider risk-neutral, risk-averse, and risk-seeking formulations within a single entropic-risk framework. We investigate classes of problems whose Bellman operator preserves affine functions and characterize their optimal value functions and feedback policies for both finite-horizon and time-average infinite-horizon problems. Additionally, we establish convergence and state-bound results under finite-step cost observability. Moreover, we demonstrate an exact connection between entropic-risk optimal control on proper cones and deterministic infsup control for systems whose dynamics and stage costs are affine in the disturbances. In addition, we provide examples of classes of systems for which explicit solutions can be obtained.
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