This paper exploits the weak approximation method to study a zero-sum linear
differential game under mixed strategies. The stochastic nature of mixed
strategies poses challenges in evaluating the game value and deriving the
optimal strategies. To overcome these challenges, we first define the mixed
strategy based on time discretization given the control period δ. Then,
we design a stochastic differential equation (SDE) to approximate the
discretized game dynamic with a small approximation error of scale
O(δ2) in the weak sense. Moreover, we prove that the game
payoff is also approximated in the same order of accuracy. Next, we solve the
optimal mixed strategies and game values for the linear quadratic differential
games. The effect of the control period is explicitly analyzed when the payoff
is a terminal cost. Our results provide the first implementable form of the
optimal mixed strategies for a zero-sum linear differential game. Finally, we
provide numerical examples to illustrate and elaborate on our results