Mild to classical solutions for XVA equations under stochastic volatility

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.

References in corpus (5)

• Number of paths versus number of basis functions in American option pricing
• Classical solutions to reaction-diffusion systems for hedging problems with interacting Ito and point processes
• Arbitrage-Free XVA
• On the support of solutions of stochastic differential equations with path-dependent coefficients
• Markovian Integral Equations

Cited by in corpus (1)

• Stability, uniqueness and existence of solutions to McKean-Vlasov SDEs: a multidimensional Yamada-Watanabe approach