Geometric and Arithmetic Likelihood Aggregation for Diffusions with Heterogeneous Volatility

We study how to combine diffusion models that disagree about drift and covariance. Candidate-first relative-entropy minimization gives geometric pooling, whereas expert-first minimization gives the arithmetic mixture associated with weighted logarithmic wealth. Different quadratic variations can make path-space entropy infinite, and the arithmetic mixture need not be a Markov diffusion. We therefore specify a local criterion combining drift information, normalized by the second argument's covariance, with quadratic transport between Gaussian shocks in a fixed state metric. A Gaussian identity and an Euler convergence estimate justify this chosen criterion. The expert-first projection has posterior-mean drift and an inverse-covariance penalty for drift dispersion; in one dimension this penalty increases volatility. For Ornstein--Uhlenbeck experts with a common mean-reversion rate, coefficient regularity holds on the full horizon for common volatility and away from the initial time for heterogeneous volatilities. The candidate-first problem has a Hamilton--Jacobi--Bellman characterization. Its matrix covariance selector reduces by congruence to a Bures--Wasserstein barycenter. The condition $H+λM\succ0$, with value Hessian $H$ and state metric $M$, is sharp for finiteness of the unrestricted local covariance problem; compact constraints keep that problem finite. A covariance-disagreement budget interprets the penalty parameter. Linear--quadratic, exact-transition, and financial examples distinguish dynamic volatility reduction, drift-dispersion inflation, and martingale restrictions.