Within the quantum theory of measurements continuous in time, a photon detection theory was formulated by using quantum stochastic calculus; this is a purely quantum formulation, where the usual notions of quantum mechanics appear: the dynamics is given by unitary operators and the observables are represented by commuting self-adjoint operators. In this paper we show how this theory can be equivalently developed by means of classical stochastic differential equations for vectors in a Hilbert space and for trace-class operators. This second formulation is linked to Belavkin's equation for a posteriori states and to quantum trajectory theory. A great part of this paper is dedicated to proving this equivalence between the purely quantum formulation and the stochastic one. The theory of direct and heterodyne detection is developed, with emphasis on the fact that the theory of heterodyne detection can be obtained as a limiting case (strong local oscillator) from the theory of direct detection. We discuss also the connections between the quantum Monte Carlo wavefunction method and some of the stochastic equations which appear in the theory of direct detection.