Optimal payoff under Bregman-Wasserstein divergence constraints

We study optimal payoff choice for an expected utility maximizer under the constraint that their payoff is not allowed to deviate ``too much'' from a given benchmark. We solve this problem when the deviation is assessed via a Bregman-Wasserstein (BW) divergence, generated by a convex function $\phi$. Unlike the Wasserstein distance (i.e., when $\phi(x)=x^2$) the inherent asymmetry of the BW divergence makes it possible to penalize positive deviations different than negative ones. As a main contribution, we provide the optimal payoff in this setting. Numerical examples illustrate that the choice of $\phi$ allow to better align the payoff choice with the objectives of investors.