Mild to Classical Solutions for XVA Equations under Stochastic Volatility

Brigo, Damiano; Graceffa, Federico and Kalinin, Alexander (2024): Mild to Classical Solutions for XVA Equations under Stochastic Volatility. In: SIAM Journal on Financial Mathematics, Vol. 15, No. 1: pp. 215-254

Abstract

We extend the valuation of contingent claims in the presence of default, collateral, and funding to a random functional setting and characterize 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 relax conditions on the available market information and construct a broad class of default times. Moreover, under stochastic volatility, we characterize 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.

• Item Type: Journal article
• Faculties: Mathematics, Computer Science and Statistics > Mathematics > Workgroup Financial Mathematics
• Subjects: 500 Science > 510 Mathematics
• ISSN: 1945-497X
• Language: English