  We consider an SPDE description of a large portfolio limit model where the
underlying asset prices evolve according to certain stochastic volatility
models with default upon hitting a lower barrier. The asset prices and their
volatilities are correlated via systemic Brownian motions, and the resulting
SPDE is defined on the positive half-space with Dirichlet boundary conditions.
We study the convergence of the loss from the system, a function of the total
mass of a solution to this stochastic initial-boundary value problem under fast
mean reversion of the volatility. We consider two cases. In the first case the
volatility converges to a limiting distribution and the convergence of the
system is in the sense of weak convergence. On the other hand, when only the
mean reversion of the volatility goes to infinity we see a stronger form of
convergence of the system to its limit. Our results show that in a fast
mean-reverting volatility environment we can accurately estimate the
distribution of the loss from a large portfolio by using an approximate
constant volatility model which is easier to handle.
