This paper considers the channel state information (CSI) acquisition and exploitation problem in cloud radio access networks (Cloud-RAN). A novel CSI acquisition method, called compressive CSI acquisition, is adopted to effectively reduce the CSI signaling overhead by obtaining instantaneous coefficients of only a subset of all the channel links.To deal with the uncertainty in available CSI, we propose a stochastic power control (SPC) framework, which is a highly intractable joint chance constrained program (JCCP). In contrast to conventional works, which can only find sub-optimal solutions, we propose a novel stochastic power control algorithm with optimality guarantee. Simulation results will show that the proposed compressive CSI acquisition method can reduce the overhead significantly. Moreover, the proposed SPC algorithm provides performance close to the full CSI case while only partial CSI is required.