Low-thrust propulsion is a key technology for space exploration, and much work in astrodynamics has focused on the mathematical modeling and the optimization of low-thrust trajectories. Typically, a nominal trajectory is designed in a deterministic system. To account for model and execution errors, mission designers heuristically add margins, for example, by reducing the thrust and specific impulse or by computing penalties for specific failures. These conventional methods are time-consuming, done by hand by experts, and lead to conservative margins. This paper introduces a new method to compute nominal trajectories, taking into account disturbances. The method is based on stochastic differential dynamic programming, which has been used in the field of reinforcement learning but not yet in astrodynamics. A modified version of stochastic differential dynamic programming is proposed, where the stochastic dynamical system is modeled as the deterministic dynamical system with random state perturbations, the perturbed trajectories are corrected by linear feedback control policies, and the expected value is computed with the unscented transform method, which enables solving trajectory design problems. Finally, numerical examples are presented, where the solutions of the proposed method are more robust to errors and require fewer penalties than those computed with traditional approaches, when uncertainties are introduced.