  A Dynkin game is considered for stochastic differential equations with random
coefficients. We first apply Qiu and Tang's maximum principle for backward
stochastic partial differential equations to generalize Krylov estimate for the
distribution of a Markov process to that of a non-Markov process, and establish
a generalized It\^o-Kunita-Wentzell's formula allowing the test function to be
a random field of It\^o's type which takes values in a suitable Sobolev space.
We then prove the verification theorem that the Nash equilibrium point and the
value of the Dynkin game are characterized by the strong solution of the
associated Hamilton-Jacobi-Bellman-Isaacs equation, which is currently a
backward stochastic partial differential variational inequality (BSPDVI, for
short) with two obstacles. We obtain the existence and uniqueness result and a
comparison theorem for strong solution of the BSPDVI. Moreover, we study the
monotonicity on the strong solution of the BSPDVI by the comparison theorem for
BSPDVI and define the free boundaries. Finally, we identify the counterparts
for an optimal stopping time problem as a special Dynkin game.
