Analysis of stochastic PDEs arising from large portfolios of stochastic volatility models - Kolliopoulos, N (2018)
The aim of this thesis is to study a large market model of defaultable assets in which the asset price processes are modelled as stochastic volatility models with volatility mean-reversion and default upon hitting a lower boundary. We assume that both the asset prices and their volatilities are correlated through systemic Brownian motions. In the first part of the thesis, 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. Next, we employ Malliavin calculus to establish the existence of a regular density for the volatility component of the SPDE, 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 and uniqueness for solutions, except in the CIR case where uniqueness remains an open problem. In the second part of the thesis, we study the convergence of the total mass of a solution to this stochastic initial-boundary value problem when the mean-reversion coefficients of the volatilities are multiples of a parameter which tends to infinity. When the coefficients of the noises of the volatilities are multiples of the square root of the same parameter, the convergence is extremely weak. On the other hand, when the noise coefficients are independent of this exploding parameter, the volatilities converge to their means and we can have much better approximations. Such approximations can be used to improve the accuracy of certain risk-management methods in markets where fast volatility mean-reversion is observed.