Classical electrodynamics with the hypothesis of a universal, Lorentz invariant, background radiation (stochastic electrodynamics) has been proposed as a possible alternative to quantum electrodynamics. The stochastic equations of motion of a charged particle are derived according to this theory, and they are compared with those of Brownian motion. A development of the equations in powers of the fine-structure constant α is considered. The harmonic oscillator is studied with the result that the oscillator performs a simple harmonic motion very stable in phase. The amplitude changes slowly and at random. The mean values of the kinetic and potential energy are calculated and agree quite well with the results of quantum electrodynamics up to first order in α. The existence of excited states is shown which prove to be very similar to the coherent states of the quantum oscillator. The calculated rate of spontaneous emission of radiation agrees with the result of quantum electrodynamics but the line width does not agree. Arguments are given which show that the quantum line width calculated according to the Weisskopf-Wigner theory cannot be correct in the case of the oscillator. A general expression for the evolution of the expectation value of any observable of the oscillator in quantum electrodynamics is also derived.