Risk measures based on weak optimal transport and approximation of drift control problems
Friday, 4.4.25, 12:00-13:30, Seminarraum 404
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.
General Diffusions on Metric Graphs as limits of Time-Space Markov Chains
Friday, 25.7.25, 12:00-13:00, Seminarraum 404
We introduce the Space-Time Markov Chain Approximation (STMCA) for a general diffusion process on a finite metric graph \(\Gamma\). The STMCA is a doubly asymmetric (in both time and space) random walk defined on a subdivisions of \(\Gamma\), with transition probabilities and conditional transition times that match, in expectation, those of the target diffusion. We derive bounds on the \(p\)-Wasserstein distances between the diffusion and its STMCA in terms of a thinness quantifier of the subdivision. Additionally, (i) we present a method for constructing numerically efficient approximations with optimal convergence rates and (ii) provide explicit analytical formulas for transition probabilities and times, enabling practical implementation of the STMCA. Numerical experiments illustrate our results.