Stochastic PDEs for large portfolios with general mean-reverting volatility processes

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.