Find a multidimensional reciprocal entropy yielding the neutral Wright–Fisher diffusion as the optimal win-martingale

Determine whether there exists an alternative definition of a multidimensional reciprocal specific relative entropy for d-dimensional continuous martingales such that, when used as the loss function in the win-martingale optimization over d-dimensional martingales on the sub-probability simplex that terminate at the vertices at time 1, the multidimensional neutral Wright–Fisher diffusion is the optimal win-martingale.

Background

Using the proposed multidimensional candidate based on the von Neumann relative entropy, the authors report numerical evidence that the multidimensional neutral Wright–Fisher diffusion is not optimal for the associated win-martingale optimization problem. This contrasts with the one-dimensional case, where the neutral Wright–Fisher diffusion is the optimizer.

This motivates seeking a different multidimensional divergence that restores the desirable selection property observed in one dimension.