  We consider optimal control problems for diffusion processes, where the
objective functional is defined by a time-consistent dynamic risk measure. We
focus on coherent risk measures defined by $g$-evaluations. For such problems,
we construct a family of time and space perturbed systems with
piecewise-constant control functions. We obtain a regularized optimal value
function by a special mollification procedure. This allows us to establish a
bound on the difference between the optimal value functions of the original
problem and of the problem with piecewise-constant controls.
