Wishart Schrödinger Bridges: A Normative Benchmark for Covariance Dynamics

Validating learned or assumed covariance dynamics requires a principled null model against which realized paths can be checked, but standard covariance interpolations (linear, geometric-mean, BW) are geometric choices that lack this justification. We use the Schrödinger bridge of Bru's Wishart diffusion not as a solver for the interpolation but as a structure-preserving null model, casting covariance interpolation as a reference-dependent minimal-distortion benchmark relative to a chosen reference process, and score realized paths by the deviation $\\mathcal{S}\_t = d\_{\\mathrm{BW}}(\\hat{\\Sigma}\_t, \\Sigma^{\\star}\_t)^{2}$. The benchmark's principled character rests on its dynamical origin and a classification. The Freidlin–Wentzell metric of an isotropic, driftless Wishart diffusion equals the Bures–Wasserstein (BW) metric, and classifying the deformation family generated by the isotropic-frame generator $\\beta$ shows that the family's metric is always BW; the driftless slice therefore coincides with Generalized-BW (GBW), and the departure from GBW lies not in a competing metric but in the drift. We further characterize the null distribution of $\\mathcal{S}_{t}$, turning it into a calibrated model diagnostic, and show under ground truth that $\\mathcal{S}$ detects inefficiency rather than displacement. Through a preliminary application to real-market data (the Fama–French 49 industry portfolios), we provide a reference-dependent benchmark for model-checking learned or assumed covariance dynamics.