This paper studies a systemic risk control problem by the central bank, which
dynamically plans monetary supply for the interbank system with borrowing and
lending activities. Facing both heterogeneity among banks and the common noise,
the central bank aims to find an optimal strategy to minimize the average
distance between log-monetary reserves and some prescribed capital levels for
all banks. A relaxed control approach is adopted, and an optimal randomized
control can be obtained in the system with finite banks by applying Ekeland's
variational principle. As the number of banks grows large, we further prove the
convergence of optimal strategies using the Gamma-convergence arguments, which
yields an optimal relaxed control in the mean field model. It is shown that the
limiting optimal relaxed control is linked to a solution of a stochastic
Fokker-Planck-Kolmogorov (FPK) equation. The uniqueness of the solution to the
stochastic FPK equation is also established under some mild conditions.Comment: Keywords: Systemic risk; interbank system; relaxed control; mean
field model; stochastic FPK equation; Gamma-convergenc