  We study the system of heterogeneous interbank lending and borrowing based on
the relative average of log-capitalization given by the linear combination of
the average within groups and the ensemble average and describe the evolution
of log-capitalization by a system of coupled diffusions. The model incorporates
a game feature with homogeneity within groups and heterogeneity between groups
where banks search for the optimal lending or borrowing strategies through
minimizing the heterogeneous linear quadratic costs in order to avoid to
approach the default barrier. Due to the complicity of the lending and
borrowing system, the closed-loop Nash equilibria and the open-loop Nash
equilibria are both driven by the coupled Riccati equations. The existence of
the equilibria in the two-group case where the number of banks are sufficiently
large is guaranteed by the solvability for the coupled Riccati equations as the
number of banks goes to infinity in each group. The equilibria are consisted of
the mean-reverting term identical to the one group game and the group average
owing to heterogeneity. In addition, the corresponding heterogeneous mean filed
game with the arbitrary number of groups is also discussed. The existence of
the $\epsilon$-Nash equilibrium in the general $d$ heterogeneous groups is also
verified. Finally, in the financial implication, we observe the Nash equilibria
governed by the mean-reverting term and the linear combination of the ensemble
averages of individual groups and study the influence of the relative
parameters on the liquidity rate through the numerical analysis.
