In this paper, we consider the (upper) semigroup envelope, i.e., the least upper bound, of a given family of linear Feller semigroups. We explicitly construct the semigroup envelope and show that, ...

We provide an abstract framework for submodular mean field games and identify verifiable sufficient conditions that allow us to prove the existence and approximation of strong mean field equilibria in models where data may not be continuous with respect to the measure parameter and common noise is allowed. The setting is general enough to encompass qualitatively different problems, such as mean field games for discrete time finite space Markov chains, singularly controlled and reflected diffusions, and mean field games of optimal timing. Our analysis hinges on Tarski’s fixed point theorem, along with technical results on lattices of flows of probability and subprobability measures.Funding: Financial support by the German Research Foundation [Collaborative Research Centre Grant 1283/2 2021–317210226] is acknowledged.

In this paper, we consider continuous‐time Markov chains with a finite state space under nonlinear expectations. We define so‐called Q‐operators as an extension of Q‐matrices or rate matrices to a nonlinear setup, where the nonlinearity is due to model uncertainty. The main result gives a full characterization of convex Q‐operators in terms of a positive maximum principle, a dual representation by means of Q‐matrices, time‐homogeneous Markov chains under convex expectations, and a class of nonlinear ordinary differential equations. This extends a classical characterization of generators of Markov chains to the case of model uncertainty in the generator. We further derive an explicit primal and dual representation of convex semigroups arising from Markov chains under convex expectations via the Fenchel–Legendre transformation of the generator. We illustrate the results with several numerical examples, where we compute price bounds for European contingent claims under model uncertainty in terms of the rate matrix.

The main result of this paper characterises the continuity from below of monotone functionals on the space $C_{b}$C b of bounded continuous functions on an arbitrary Polish space as lower semicontinuity in the mixed topology. In this particular situation, the mixed topology coincides with the Mackey topology for the dual pair $(C_{b},\mathrm{ca})$( C b , ca ) , where $\mathrm{ca}$ca denotes the space of all countably additive signed Borel measures of finite variation. Hence lower semicontinuity in the mixed topology is for convex monotone maps $C_{b}\to \mathbb{R}$C b → R equivalent to a dual representation in terms of countably additive measures. Such representations are of fundamental importance in finance, e.g. in the context of risk measures and superhedging problems. Based on the main result, regularity properties of capacities and dual representations of Choquet integrals in terms of countably additive measures for 2-alternating capacities are studied. Moreover, a well-known characterisation of star-shaped risk measures on $L^{\infty }$L ∞ is transferred to risk measures on $C_{b}$C b . In a second step, the paper provides a characterisation of equicontinuity in the mixed topology for families of convex monotone maps. As a consequence, for every convex monotone map on $C_{b}$C b taking values in a locally convex vector lattice, continuity in the mixed topology is equivalent to continuity on norm-bounded sets.