The xVA BSDE decomposition provides a unified mathematical framework for computing all valuation adjustments — CVA, DVA, FVA, and ColVA — as components of a single backward stochastic differential equation. Following the framework of Biagini et al. (2021), the full portfolio value V decomposes as V = clean value - XVA, where the XVA process satisfies an F-BSDE under the reduced (default-free) filtration. This reduction from the enlarged filtration G (which includes default information) to F is enabled by the immersion hypothesis, which ensures that F-martingales remain G-martingales.
The XVA admits the additive decomposition XVA = -CVA + DVA + FVA + ColVA. The credit valuation adjustment (CVA) captures the expected loss from counterparty default, weighted by the loss-given-default and default intensity. The debt valuation adjustment (DVA) symmetrically captures the benefit from the bank’s own default. The funding valuation adjustment (FVA) accounts for the spread between unsecured funding rates and the risk-free rate on the uncollateralised portion of the portfolio. The collateral valuation adjustment (ColVA) captures the cost of posted and received collateral when collateral rates differ from the risk-free rate.
A crucial structural distinction is that CVA, DVA, and ColVA (when collateral depends only on clean values) are non-recursive: they can be computed as conditional expectations given the clean value paths. FVA, however, is recursive because the funding cost depends on the full value V = clean value - XVA, creating a feedback loop where the adjustment appears in its own driver. This distinction drives the algorithmic design in Deep xVA Solver - A Neural Network Based Counterparty Credit Risk Management Framework, where non-recursive adjustments use outer Monte Carlo (Algorithm 2 - Deep xVA for non-recursive adjustments) and recursive adjustments require a second BSDE solve (Algorithm 3 - Deep xVA Solver).
Key Details
- The F-BSDE driver f-bar encodes default intensities (lambda^{B,Q}, lambda^{C,Q}), recovery rates (R^B, R^C), funding spreads (r^{f,l} - r, r^{f,b} - r), and collateral rates (r^{c,l}, r^{c,b})
- The terminal condition is XVA_T = 0, reflecting that all adjustments vanish at portfolio maturity
- The discount rate in the XVA BSDE is r-tilde = r + lambda^{C,Q} + lambda^{B,Q}, incorporating survival probabilities
- Collateral C is assumed to be a Lipschitz function of the clean portfolio value, ensuring well-posedness
- Existence and uniqueness follows from standard BSDE theory under boundedness of rates and intensities