# Stochastic evolution equations for large portfolios of stochastic volatility models

## TL;DR

This paper develops a stochastic partial differential equation framework for modeling large portfolios of defaultable assets with stochastic volatility, analyzing their limiting behavior and density properties.

## Contribution

It introduces a measure-valued process limit for large portfolios with correlated stochastic volatility and default risk, using advanced calculus and smoothing techniques.

## Key findings

- Existence of a large portfolio limit as a measure-valued process
- Derivation of a stochastic PDE with a density solution
- Regularity results for the density, but open problem of uniqueness

## 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.