This paper investigates Nash games for a class of linear stochastic systems governed by Itoˆ’s differential equation with Markovian jump parameters both in finite-time horizon and infinite-time horizon. First, stochastic Nash games are formulated by applying the results of indefinite stochastic linear quadratic (LQ) control problems. Second, in order to obtain Nash equilibrium strategies, crosscoupled stochastic Riccati differential (algebraic) equations (CSRDEs and CSRAEs) are derived. Moreover, in order to demonstrate the validity of the obtained results, stochastic H₂/H control with state- and control-dependent noise is discussed as an immediate application. Finally, a numerical example is provided.