Linear quadratic stochastic Nash differential games for Markov jump linear systems are studied. By utilizing some
results of stochastic optimal control for Markov jump linear systems, the existence condition of finite horizon stochastic Nash
games is equivalent to the solvability of the associated differential Riccati equations, and that of infinite horizon stochastic
Nash games is equivalent to the solvability of the associated algebraic Riccati equations. Moreover, explicit expressions of
the optimal strategies are constructed. The results are applied to the mixed H2/H∞ control problem for Markov jump linear
systems. Finally, a numeric example is given to show the feasibility of the proposed method.