This paper proposes a safe speed control strategy for an autonomous vehicle while approaching unsignalized crosswalks with pedestrians, which relies on a receding horizon stochastic model predictive control (SMPC). It is assumed that cruising at a certain speed is a main vehicle preference to provide comfortable driving. A stochastic formulation of MPC is employed to account for uncertainties related to pedestrian crossing decisions. The prediction horizon is divided in two phases, i.e., prior and after pedestrian reaching the crosswalk edge. The deterministic cost related to the first phase is calculated online for different vehicle control trajectory parameters, while the probabilistic cost for the second phase is derived from two offline-calculated maps corresponding to two characteristic scenarios, i.e., pedestrian opting for cross and yield when reaching the crosswalk edge. The probabilities of cross and yield decisions depend on vehicle states, and they are obtained from a binary logistic regression prediction model. SMPC is verified against a baseline control strategy by means of large-scale Monte Carlo simulations over a wide range of randomly generated initial conditions.