This paper develops the mathematical framework to use semi-analytic higher-order state transition tensor (STT) approximations of a nonlinear system to solve an optimal control problem with impulsive controls and stochastic constraints. It is shown that STTs can be used to construct several different constraint formulations to account for the nonlinear evolution of state uncertainties. An analytical optimization algorithm is developed which takes advantage of the easily differentiable properties of STTs. The proposed methodology is promising for situations with highly nonlinear dynamics but limited computational resources, such as on-board a spacecraft.