Recovered-Coefficient Discontinuous Galerkin Approximation of an Averaged-State Credit-Rating Migration PDE

Abstract

We develop and analyze a discontinuous Galerkin (DG) method for an averaged-state regularization of a credit-rating migration equation, in which the volatility switch is smoothed at scale $\eps$ and evaluated at a mollified state of width $\ell$. The diffusion operator is written in conservative form and discretized by symmetric interior penalty, and the effective drift by upwinding. The nonlinear coefficients are evaluated from a recovered state obtained by Oswald averaging, the model mollifier, and an explicit norm cutoff calibrated by a data-computable maximum bound. The recovered coefficients are single-valued and uniformly $W^{2,\infty }$, so the standard upwind energy identity holds, and the recovery is Lipschitz from $L^2$ and compatible with the projected exact state at order $h^{p+1}$. For fixed $(\eps,\ell)$ we prove well-posedness and a maximum bound for the continuous problem, an $O({\ell }^2)$ distance to the pointwise-regularized model, stability of the frozen-coefficient backward Euler DG scheme, and an $O(h^p+\Delta t)$ error estimate in the DG energy norm; an exact Nitsche--lifting identity covers nonhomogeneous Dirichlet data. Under local transversality, the regularized migration boundary is recovered at a proved, suboptimal rate. Computations with the analyzed scheme confirm the predicted rates and report the quantities on which the analysis rests.