A Conditional-Density Approach Under Gaussian–Volterra Dynamics: Credit Term Structures and Finite-Rank Approximation

Conditional-density models describe the full future law of a random time as information evolves, but explicit non-Markovian specifications of the resulting survival and density surfaces are difficult to obtain. We construct such surfaces from a latent quadratic Gaussian–Volterra path functional and an independent exponential threshold. Conditional survival is represented by a Laplace-transform martingale field. Malliavin calculus and Gaussian resolvent identities yield its Brownian coefficient in closed operator form, giving explicit dynamics for the conditional survival and density surfaces, a direct mass-compatibility identity, the moving-diagonal Azéma-supermartingale decomposition, and the predictable default intensity. Under a pricing measure, the same coefficients determine the forward-hazard curve, defaultable-bond volatility, and continuously paid CDS spread volatility. A one-Brownian-factor specification implies a rank-at-most-one instantaneous covariance matrix of CDS log-spread changes. We also establish finite-rank Galerkin convergence and perturbation stability. A stylized numerical study illustrates exact initial-curve fitting, local rank diagnostics, and recovery of a Volterra amplitude from a dynamic covariance target.