We analyse the general relativistic oscillations of thin accretion discs around compact astrophysical objects interacting with the surrounding medium through non-gravitational forces. The interaction with the external medium (a thermal bath) is modelled via a friction force and a random force, respectively. The general equations describing the stochastically perturbed discs are derived by considering the perturbations of trajectories of the test particles in equatorial orbits, assumed to move along the geodesic lines. By taking into account the presence of a viscous dissipation and of a stochastic force, we show that the dynamics of the stochastically perturbed discs can be formulated in terms of a general relativistic Langevin equation. The stochastic energy transport equation is also obtained. The vertical oscillations of the discs in the Schwarzschild and Kerr geometries are considered in detail, and they are analysed by numerically integrating the corresponding Langevin equations. The vertical displacements, velocities and luminosities of the stochastically perturbed discs are explicitly obtained for both the Schwarzschild and the Kerr cases.