  In this paper, a leader-follower stochastic differential game is studied for
a linear stochastic differential equation with a quadratic cost functional. The
coefficients in the state equation and the weighting matrices in the cost
functionals are all deterministic. Closed-loop strategies are introduced, which
require to be independent of initial states; and such a nature makes it very
useful and convenient in applications. The follower first solves a stochastic
linear quadratic optimal control problem, and his optimal closed-loop strategy
is characterized by a Riccati equation, together with an adapted solution to a
linear backward stochastic differential equation. Then the leader turns to
solve a stochastic linear quadratic optimal control problem of a
forward-backward stochastic differential equation, necessary conditions for the
existence of optimal closed-loop strategies for the leader is given by the
existence of a Riccati equation. Some examples are also given.
