Stochastic PDEs for large portfolios with general mean-reverting volatility processes
In this article we consider a large structural market model of defaultable assets, where the asset value processes are modelled by using stochastic volatility models with default upon hitting a lower boundary. The volatility processes are picked from a class of general mean-reverting diffusions satisfying certain regularity assumptions. The value processes and the volatility processes are all correlated through systemic Brownian motions. We prove that our system converges as the portfolio becomes large, and the limit of the empirical measure process has a density which is the unique solution to an SPDE in the two-dimensional half-space with a Dirichlet boundary condition. We use Malliavin calculus to establish the existence of a regular density for the volatility component of the density satisfying our stochastic initial-boundary value problem, and we improve an existing kernel smoothing technique to obtain higher regularity and uniqueness results.