It is well known that in classical control theory, the controller has a certain objective to achieve, and the plant to be controlled does not have its own objective. However, this is not the case in many practical situations in, for example, social, economic, and rapidly developing "intelligent" engineering systems. For these kinds of systems, the classical control theory cannot be applied directly. This motivates us to introduce a new control framework called game-based control systems (GBCSs), which has a hierarchical decision-making structure, i.e., a higher level regulator and lower level multiple agents. The regulator is regarded as the macrocontroller that makes decisions first, and then the agents try to optimize their respective objective functions, where a possible Nash equilibrium may be reached as a result of a noncooperative differential game. A fundamental issue in GBCSs is whether it is possible for the regulator to change the macrostates of the system by regulating the Nash equilibrium formed by the agents at the lower level. The investigation of this problem was initiated recently by the authors for deterministic systems. In this paper, we formulate this problem in the general stochastic nonlinear framework, and then focus on linear stochastic systems to give some explicit necessary and sufficient algebraic conditions on the controllability of the Nash equilibrium. In contrast to the classical controllability theory on forward differential equations, we now need to investigate the controllability of the associated forward-backward stochastic differential equations, which involves a more complicated investigation. Moreover, in the current stochastic case, which is more complicated than the deterministic case, we need some deep understanding of forward-backward stochastic differential equations.