A description intermediate between the usual stochastic description of a Brownian particle and the deterministic description of a classical particle is proposed. It is based on a model which utilizes the notions of a current velocity and of an osmotic velocity, and which generates a random process which allows us to associate with any given initial and final conditions a unique differentiable trajectory. This intermediate description of the Brownian motion, in terms of quasiparticles with quasideterministic behavior, gives back the same mean and the same variance as does the usual stochastic description.