  This paper is concerned with the problem of budget control in a large
particle system modeled by stochastic differential equations involving hitting
times, which arises from considerations of systemic risk in a regional
financial network. Motivated by Tang and Tsai (Ann. Probab., 46(2018), pp.
1597{1650), we focus on the number or proportion of surviving entities that
never default to measure the systemic robustness. First we show that both the
mean-field particle system and its limiting McKean-Vlasov equation are
well-posed by virtue of the notion of minimal solutions. We then establish a
connection between the proportion of surviving entities in the large particle
system and the probability of default in the limiting McKean-Vlasov equation as
the size of the interacting particle system N tends to infinity. Finally, we
study the asymptotic efficiency of budget control in different economy regimes:
the expected number of surviving entities is of constant order in a negative
economy; it is of order of the square root of N in a neutral economy; and it is
of order N in a positive economy where the budget's effect is negligible.
