We study a class of dynamically consistent risk measures that robustify a time-homogeneous Markovian reference model by allowing for distributional uncertainty in its transition laws. We start from one-step convex risk evaluations in which ambiguity is captured by penalized worst-case expectations over alternative transition laws. Imposing time consistency then yields a convex monotone semigroup on bounded continuous payoff functions, and this semigroup represents the associated dynamic risk measure. The semigroup is uniquely characterized by its risk generator. Under a lower bound on the family of penalties in terms of suitable optimal transport costs relative to the reference laws, we identify the generator on smooth test functions. For optimal transport bounds with linear small-time scaling, this produces a first-order, drift-type correction given by a convex Hamiltonian acting on the gradient. Under martingale transport constraints and a different scaling, however, the leading correction is genuinely of second order and is described by a convex monotone functional acting on the Hessian.\ We illustrate both regimes for Wasserstein and martingale Wasserstein penalizations and derive explicit formulas via convex conjugates of the underlying transport costs. The associated dynamic risk measures admit stochastic control representations in which the control acts on the drift in the first-order case and on the volatility in the second-order case.

The talk is based on joint work with Sven Fuhrmann and Michael Kupper.

After a brief introduction to martingale optimal transport, I will present the martingale Schrödinger bridge problem introduced by Nutz and Wiesel, which consists in finding a canonical coupling among all martingale couplings between two probability measures in convex order. I will then discuss several equivalent characterizations of this object: in particular, I will explain how the continuous continuous couterpart of the martingale Schrödinger bridge relates to the Föllmer process.

Time permitting, I will conclude with numerical illustrations, applications, and comparisons with other canonical constructions in martingale transport. The talk is based on joint work with Julio Backhoff, Mathias Beiglböck, and Giorgia Bifronte.

Joint work with Thang Nhat Le (Mulhouse). We present a new static transport problem in the same spirit as martingale optimal transport. Its solutions are described by using an appropriate shadow parametrization.

We establish a Brenier theorem for iterated Wasserstein spaces. Specifically, for a separable Hilbert space $H$ and $N\ge 1$, we construct a full-support probability $\Lambda\in \mathcal{P}_2^{N}(H)= \mathcal{P}_2(\ldots \mathcal{P}_2(H)\ldots)$ that is transport regular: for all $P,Q\in \mathcal{P}_2^{N}(H)$ with $P\ll \Lambda$, the $\mathcal{W}_2^2$-optimal transport from $P$ to $Q$ is unique and of Monge type. In the first non-classical case $N=2$ we show that optimal transports are given as the push-forward by the $\mathcal{W}_2$-gradient (or Lions' derivative) of an $L$-convex function. To establish the result for general $N$ we develop new adapted notions of Lions' lift, $L$-convexity and Lions' derivative. A key idea is a new identification between optimal-transport $c$-conjugation (with $c$ given by maximal covariance) and classical convex conjugation on the lift.

A primary motivation comes from the adapted Wasserstein distance $\mathcal{AW}_2$: our results yield a first Brenier theorem for $\mathcal{AW}_2$ and characterize $\mathcal{AW}_2^2$-optimal couplings through convex functionals on the space of $L_2$-processes.