  We introduce a generalized Dynkin game problem with non linear conditional
expectation ${\cal E}$ induced by a Backward Stochastic Differential Equation
(BSDE) with jumps. Let $\xi, \zeta$ be two RCLL adapted processes with $\xi
\leq \zeta$. The criterium is given by \begin{equation*}
  {\cal J}_{\tau, \sigma}= {\cal E}_{0, \tau \wedge \sigma }
\left(\xi_{\tau}\textbf{1}_{\{ \tau \leq
\sigma\}}+\zeta_{\sigma}\textbf{1}_{\{\sigma<\tau\}}\right)
  \end{equation*} where $\tau$ and $ \sigma$ are stopping times valued in
$[0,T]$. Under Mokobodski's condition, we establish the existence of a value
function for this game, i.e. $\inf_{\sigma}\sup_{\tau} {\cal J}_{\tau, \sigma}
= \sup_{\tau} \inf_{\sigma} {\cal J}_{\tau, \sigma}$. This value can be
characterized via a doubly reflected BSDE. Using this characterization, we
provide some new results on these equations, such as comparison theorems and a
priori estimates. When $\xi$ and $\zeta$ are left upper semicontinuous along
stopping times, we prove the existence of a saddle point. We also study a
generalized mixed game problem when the players have two actions: continuous
control and stopping. We then address the generalized Dynkin game in a
Markovian framework and its links with parabolic partial integro-differential
variational inequalities with two obstacles.
