Stochastic electrodynamics is classical electrodynamics with the hypothesis of a random background radiation in the whole space. The probability distributions of the Fourier components of this electromagnetic radiation are assumed Gaussian. Lorentz invariance fixes the spectrum of the radiation except for a constant, measuring its intensity, which is identified with Planck’s constant. A formalism is developed to deal with general stochastic problems, associating a Hilbert space with every set of random variables. Then, a mapping is defined of the algebra of continuous functions of the random variables onto the algebra of operators on the Hilbert space. A correspondence is found between probability distributions of the random variables and vectors in the Hilbert space. The case of Gaussian random variables is considered in detail. The formalism is applied to the study of the radiation field when some known amount of radiation is present besides the random background. This closely resembles the usual quantization of the free radiation field, although there are significant differences. In particular, not all vectors of the Hilbert space represent physical states in the present theory.