Title: Stochastic evolution equations for large portfolios of stochastic volatility models,
Abstract: We consider a large market model of defaultable assets in which the asset
price processes are modelled as Heston-type stochastic volatility models with
default upon hitting a lower boundary. We assume that both the asset prices and
their volatilities are correlated through systemic Brownian motions. We are
interested in the loss process that arises in this setting and we prove the
existence of a large portfolio limit for the empirical measure process of this
system. This limit evolves as a measure valued process and we show that it will
have a density given in terms of a solution to a stochastic partial
differential equation of filtering type in the two-dimensional half-space, with
a Dirichlet boundary condition. We employ Malliavin calculus to establish the
existence of a regular density for the volatility component, and an
approximation by models of piecewise constant volatilities combined with a
kernel smoothing technique to obtain existence and regularity for the full
two-dimensional filtering problem. We are able to establish good regularity
properties for solutions, however uniqueness remains an open problem.