A linear equation of motion for the state vector is presented, in which an anti-Hermitian Hamiltonian that fluctuates randomly is added to the usual Hamiltonian of the Schrödinger equation. It is shown how the resulting theory describes the continuous evolution of a state vector to an ensemble of reduced state vectors while retaining important physical features of the Ghirardi, Rimini, and Weber [Phys. Rev. D 34, 470 (1986)] theory of spontaneous localization, in which the state vector reduction occurs discontinuously. A novel aspect, compared with ordinary quantum theory, is that the state-vector norm changes with time. The squared norm of each state vector is interpreted as being proportional to the probability possessed by that state vector in the ensemble of state vectors. This interpretation is shown to be consistent with the independent Markovian evolution of each state vector.