Speaker
Gudmund Pammer
Description
We establish a Brenier theorem and a superposition principle for iterated Wasserstein spaces. Specifically, for a separable Hilbert space $H$ and $N\ge 1$, we construct a full-support probability $\Lambda \in P_2^{N}(H)= P_2(\ldots P_2(H)\ldots)$ that is transport regular: for all $P,Q\in P_2^{N}(H)$ with $P\ll \Lambda$, the $W_2^2$-optimal transport from $P$ to $Q$ is unique and of Monge type. Moreover, we establish the gradient structure of optimal transports between stochastic processes. As an important consequence, we obtain a fast fixed point algorithm for the construction of calibrated martingale models, which addresses a main problem in mathematical finance.