• optimal transport
• probability
• reinsurance

In this work, initially, we present a preliminary chapter on Optimal Transport, with a focus on the one-dimensional case.
The second chapter introduces the optimal reinsurance problem, beginning with a classical existence result for optimal treaties. Then, we analyze the case of a finite number of inequality constraints using a convex linearization approach. By identifying a convex set where the directional (right) derivatives of the considered functionals are convex linear, we derive key properties of the measures defining optimal contracts. Under additional assumptions, we further characterize the support points of these measures, providing practical insights into their implementation.
Afterwards, we broaden the scope of constraints and address the problem using optimal transport. We reformulate the problem into a nested minimization, which, in the one-dimensional case reduces to a minimization problem over deterministic functions.
Finally, the practical applications of both approaches are explored in the concluding chapter. The convex linearization method proves particularly effective in addressing constraints such as the Value at Risk, while the optimal transport framework offers a novel perspective for tackling classical and new problems in reinsurance.