I will introduce a stochastic version of the optimal transport problem. This problem is stated in the usual Benamou-Brenier dynamic formulation. It consists in studying the case of a stochastic terminal condition. I will give some results of controllability for several versions of the problem. I will then show an analysis by means of the study of the associated Hamilton-Jacobi-Bellman equation, which is set on the set of probability measures. I will give a new notion of viscosity solutions of this equation, which yields general comparison principles, in particular for cases involving terms modeling stochasticity in the optimal control problem. Using this notion, I will be able to establish results of existence and uniqueness of viscosity solutions of the Hamilton-Jacobi-Bellman equation.