Mild to classical solutions for XVA equations under stochastic volatility

TL;DR

This paper extends XVA valuation models to stochastic volatility settings, characterizing pre-default values through BSDEs and PDEs, and broadening the applicable market information and default time frameworks.

Contribution

It introduces a unified approach to valuing contingent claims with default and funding considerations under stochastic volatility, using martingale, BSDE, and PDE methods.

Findings

Pre-default values characterized by BSDEs with path-dependent coefficients.

Existence and uniqueness conditions for mild solutions to PDEs under stochastic volatility.

Broader class of default times and market information models considered.

Abstract

We extend the valuation of contingent claims in presence of default, collateral and funding to a random functional setting and characterise pre-default value processes by martingales. Pre-default value semimartingales can also be described by BSDEs with random path-dependent coefficients and martingales as drivers. En route, we generalise previous settings by relaxing conditions on the available market information, allowing for an arbitrary default-free filtration and constructing a broad class of default times. Moreover, under stochastic volatility, we characterise pre-default value processes via mild solutions to parabolic semilinear PDEs and give sufficient conditions for mild solutions to exist uniquely and to be classical.