Differential equations for time-dependent amplitudes of coupled waves at the Stokes frequency and the frequency of incident radiation in a spherical particle are obtained based on the method of expanding optical fields into a series over eigenfunctions of the stationary linear scattering problem. Solutions of the equations for the start of the SRS process and under steady state conditions are analyzed. The SRS threshold is determined, and the threshold for the steady state SRS at a given intensity is found for the case of double resonance between the fields. It is shown that to excite the SRS, one should compensate for the loss of the Stokes wave due to absorption and emission through the particle surface. To provide for steady state SRS generation, it is necessary to additionally compensate for the energy loss due to pump depletion.