Michael Kupper (University of Konstanz)

We discuss convex risk measures with weak optimal transport
penalties and show that these risk measures admit an explicit
representation via a nonlinear transform of the loss function. We
discuss several examples, including classical optimal transport
penalties and martingale constraints. In the second part of the talk,
we focus on the composition of related functionals. We consider a
stochastic version of the Hopf–Lax formula, where the Hopf–Lax operator
is composed with the transition kernel of a Lévy process. We show that,
depending on the order of composition, one obtains upper and lower
bounds for the value function of a stochastic optimal control problem
associated with drift-controlled Lévy dynamics. The value function of
the control problem is approximated both from above and below as the
number of iterations tends to infinity, and we provide explicit
convergence rates for the approximation procedure.
The talk is based on joint work with Max Nendel and Alessandro
Sgarabottolo.