  In this paper, we investigate a sparse optimal control of continuous-time
stochastic systems. We adopt the dynamic programming approach and analyze the
optimal control via the value function. Due to the non-smoothness of the $L^0$
cost functional, in general, the value function is not differentiable in the
domain. Then, we characterize the value function as a viscosity solution to the
associated Hamilton-Jacobi-Bellman (HJB) equation. Based on the result, we
derive a necessary and sufficient condition for the $L^0$ optimality, which
immediately gives the optimal feedback map. Especially for control-affine
systems, we consider the relationship with $L^1$ optimal control problem and
show an equivalence theorem.
