  We consider a structural stochastic volatility model for the loss from a
large portfolio of credit risky assets. Both the asset value and the volatility
processes are correlated through systemic Brownian motions, with default
determined by the asset value reaching a lower boundary. We prove that if our
volatility models are picked from a class of mean-reverting diffusions, the
system converges as the portfolio becomes large and, when the vol-of-vol
function satisfies certain regularity and boundedness conditions, the limit of
the empirical measure process has a density given in terms of a solution to a
stochastic initial-boundary value problem on a half-space. The problem is
defined in a special weighted Sobolev space. Regularity results are established
for solutions to this problem, and then we show that there exists a unique
solution. In contrast to the CIR volatility setting covered by the existing
literature, our results hold even when the systemic Brownian motions are taken
to be correlated.
