This paper proposes an improved particle swarm optimization named PSO with Controllable Random Exploration Velocity (PSO-CREV) behaving an additional exploration behavior. Different from other improvements on PSO, the updating principle of PSO-CREV is constructed in terms of stochastic approximation diagram. Hence a stochastic velocity independent on cognitive and social components of PSO can be added to the updating principle, so that particles have strong exploration ability than those of conventional PSO. The conditions and main behaviors of PSO-CREV are described. Two properties in terms of "divergence before convergence" and "controllable exploration behavior" are presented, which promote the performance of PSO-CREV. An experimental method based on a complex test function is proposed by which the proper parameters of PSO-CREV used in practice are figured out, which guarantees the high exploration ability, as well as the convergence rate is concerned. The benchmarks and applications on FCRNN training verify the improvements brought by PSO-CREV.