A technique for handling rare default events in deep BSDE solvers applied to structural credit models. When default probabilities are very low, Monte Carlo sampling under the original measure produces too few default trajectories for the neural network to learn the correct control behaviour near default boundaries. The change-of-measure approach addresses this by tilting the drift of the forward SDE to make defaults more likely during training, while preserving the correctness of the BSDE solution.
The method works as follows. Given the original forward-backward SDE system under measure P, a drift perturbation q(t,x) is subtracted from the forward SDE’s drift b(t,x), producing a tilted process X^q. By Girsanov’s theorem, the tilted process X^q under P has the same law as the original process X under a new measure P^q defined via the exponential martingale of the Girsanov kernel theta_t = q(t,X^q_t) / sigma(t,X^q_t). The crucial insight is that the PDE solution v(t,x) associated with the BSDE remains the same — only the forward trajectories along which it is evaluated change. This means the BSDE generator acquires an additional term <q/sigma, Z^q> that absorbs the measure change, but no Radon-Nikodym reweighting of samples is needed.
This distinguishes the approach from standard importance sampling. In importance sampling, the BSDE is solved under the original measure and each sample is reweighted by the likelihood ratio, which can become numerically unstable when the ratio is large. Here, the measure change is integrated directly into both the forward SDE drift and the BSDE driver, avoiding large likelihood-ratio effects entirely. In practice, the variational problem sums terminal condition errors over K different drift perturbations q_1, …, q_K (typically K=2, with q_1=0 being the original measure and q_2 chosen to boost defaults). All share the same initial condition y_0 and the same underlying functional form for Z.
Key Details
- The drift perturbation q must satisfy the same Lipschitz and linear growth conditions as the original drift b (Assumption 3.3)
- For CVA, q is chosen to increase the counterparty’s default probability; for DVA, to increase the bank’s; for ColVA and MVA, to increase both
- Typical choices in numerical experiments: q = (0,…,0, 0.2, 0.35)^T or similar, targeting only the last two (defaultable) components of the state process
- Without the measure change, Z^{ColVA} remains approximately zero at all times because the neural network never “sees” default scenarios, leading to poor path-wise accuracy even when terminal errors appear acceptable
- Under stress testing (counterparty default probability approximately 0.6%), the measure-change version produces terminal error distributions that are tighter and more symmetric around zero
- The technique enables training on a single neural network that is valid under the original measure, since the PDE solution is measure-invariant