  Stochastic optimal control problems have a long tradition in applied
probability, with the questions addressed being of high relevance in a
multitude of fields. Even though theoretical solutions are well understood in
many scenarios, their practicability suffers from the assumption of known
dynamics of the underlying stochastic process, raising the statistical
challenge of developing purely data-driven strategies. For the mathematically
separated classes of continuous diffusion processes and L\'evy processes, we
show that developing efficient strategies for related singular stochastic
control problems can essentially be reduced to finding rate-optimal estimators
with respect to the sup-norm risk of objects associated to the invariant
distribution of ergodic processes which determine the theoretical solution of
the control problem. From a statistical perspective, we exploit the exponential
$\beta$-mixing property as the common factor of both scenarios to drive the
convergence analysis, indicating that relying on general stability properties
of Markov processes is a sufficiently powerful and flexible approach to treat
complex applications requiring statistical methods. We show moreover that in
the L\'evy case $-$ even though per se jump processes are more difficult to
handle both in statistics and control theory $-$ a fully data-driven strategy
with regret of significantly better order than in the diffusion case can be
constructed.
