Optimal feedback control is investigated for a second-order plant with white noise disturbances and with bounded control. The white noises are state-dependent, and the system is such that dimensional analysis can be applied with advantage. The optimal controller is found to be represented by a switching curve which is determined analytically to within a single parameter. To find the value of this parameter, the Bellman-Florentin partial differential equation of optimal stochastic control is converted to an ordinary differential equation by means of dimensional analysis; and a resulting two-point boundary-value problem is solved numerically. It is found that the optimal switching curve can deviate considerably from the corresponding curve for a deterministic system. It is also found that the stochastic system becomes unstable if the intensities of the white noises are increased sufficiently.