This paper aims to give a design strategy on the prescribed performance H∞ tracking control problem for a class of strict-feedback stochastic nonlinear systems based on the backstepping technique. Generally, by using the backstepping design method, the stochastic nonlinear systems can only be made to be bounded in probability and it is difficult to achieve the H∞ performance criterion due to the positive constant term appeared in the stability analysis. Thus, a novel concept with regard to the bounded-H∞ performance is proposed in this paper to overcome the design difficulty. By using the new concept and the adaptive neural network technique as well as Gronwall inequality, an adaptive neural network prescribed performance bounded-H∞ tracking controller is designed. Therein, neural networks are used to approximate the unknown packaged nonlinear functions. The assumption that the approximation errors of neural networks are square-integrable in some literature is eliminated. The designed controller guarantees that all the signals in the closed-loop stochastic nonlinear systems are bounded in probability, the tracking error is constrained into an adjustable neighborhood of the origin with the prescribed performance bounds, and the controlled system has a given H∞ disturbance attenuation performance for external disturbances. Finally, the simulation results are provided to illustrate the effectiveness and feasibility of the proposed approach.