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

## TL;DR

This paper develops a stochastic PDE framework for modeling large credit portfolios with mean-reverting stochastic volatility, allowing for correlated systemic factors and providing existence, uniqueness, and regularity results.

## Contribution

It introduces a new stochastic PDE model for large portfolios with correlated systemic Brownian motions and general mean-reverting volatility processes, extending previous CIR-based models.

## Key findings

- Convergence of the empirical measure as portfolio size grows.
- Existence and uniqueness of solutions to the stochastic PDE.
- Regularity results in weighted Sobolev spaces.

## Abstract

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.