In this paper, adaptive tracking control problem is investigated for a class of switched stochastic nonlinear systems with an asymmetric output constraint. By introducing a nonlinear mapping (NM), the asymmetric output-constrained switched stochastic system is first transformed into a new system without any constraint, which achieves the equivalent control objective. The command filter technique is employed to handle the "explosion of complexity" in traditional backstepping design, and neural networks (NNs) are directly utilized to cope with the completely unknown nonlinear functions and stochastic disturbances existing in systems. At last, on the basis of stochastic Lyapunov function method, an adaptive neural controller is developed for the considered system. It is shown that the designed adaptive controller can guarantee that all the signals remain semi-globally uniformly ultimately bounded (SGUUB), while the output constraint is satisfied and the desired signal can be tracked with a small domain of the origin. Simulation results are offered to illustrate the feasibility of the newly designed control scheme.