The tracking control problem for a class of stochastic and uncertain non-linear systems is addressed. The proposed controller uses suitable radial basis function neural network designs for the approximation of the unknown non-linearities while it is arbitrarily regulated in order to effectively penalize the tracking error. This regulation is implemented through a risk-sensitivity parameter. A stability analysis based on Lyapunov functions obtained by the backstepping technique, proves that all the error variables are bounded in probability; simultaneously, for any given risk-sensitivity parameter the system performance is regulated with respect to both a desired small average tracking error and low long-term average cost in accordance to a risk-sensitive cost criterion. Moreover, the larger this parameter is, the mean square tracking error becomes semiglobally uniformly ultimately bounded in a smaller area while a lower level of a long-term average cost is achieved. The effectiveness of the design approach is illustrated by simulation results wherein it becomes clear how one can achieve a tradeoff between good response and control effort.