Jonas Blessing

Abstract: Motivated by model uncertainty and stochastic control problems, we develop a systematic theory for convex monotone semigroups on spaces of continuous functions. The present approach is self-contained and does, in particular, not rely on the theory of viscosity solutions. Instead, we provide a comparison principle for semigroups related to Hamilton-Jacobi-Bellman equations which uniquely determines the semigroup by its infinitesimal generator evaluated at smooth functions. While the statement itself resembles the classical analogue for linear semigroups, the proof requires the introduction of several novel analytical concepts such as the Lipschitz set and the $\Gamma$-generator. Furthermore, starting with a generating family $(I(t))_{t\geq 0}$ of operators, we show that the limit  $S(t)f:=\lim_{n\to\infty}I(\frac{t}{n})^n f$ defines a semigroup which is uniquely determined by the time derivative $I’(0)f$ for smooth functions $f$. We identify explicit conditions for the generating family that are transferred to the semigroup and can easily be verified in applications. The abstract results are illustrated by emphasizing the structural link between approximation schemes for convex monotone semigroups and law of large numbers and central limit theorem type results for convex expectations. Furthermore, the limit can be represented as a stochastic control problem.