Identify a multidimensional analogue of the reciprocal specific relative entropy

Identify a well-defined multidimensional analogue of the reciprocal specific relative entropy for d-dimensional continuous martingale laws on C([0,1]; R^d), extending the one-dimensional functional (1/2) E_Q[∫_0^1 (Σ_t log Σ_t + 1 − Σ_t) dt] that depends on the scalar instantaneous quadratic variation Σ_t, to a divergence that depends on the matrix-valued quadratic variation density d⟨X⟩_t/dt of a d-dimensional martingale.

Background

In one dimension the reciprocal specific relative entropy between a martingale law Q and the Wiener measure is defined as (1/2) E_Q[∫_01 (Σ_t log(Σ_t) + 1 − Σ_t) dt], where Σ_t is the density of the quadratic variation of the martingale. This functional plays a central role in the paper’s win-martingale optimization and selects the neutral Wright–Fisher diffusion as the optimizer.

The authors discuss a possible multidimensional generalization via the quantum (von Neumann) relative entropy between positive semidefinite matrix-valued measures, but they do not assert this is the correct or canonical extension.

References

It is not immediately clear how to identify the multidimensional analogue of the reciprocal specific relative entropy.

— Reciprocal Specific Relative Entropy between Continuous Martingales  (2602.14776 - Backhoff et al., 16 Feb 2026) in Remark (label eq:multidim), end of Section 3