Convex expectations for countable-state uncertain processes with càdlàg sample paths

Abstract

This work investigates convex expectations, mainly in the setting of uncertain processes with countable state space. In the general setting it shows how, under the assumption of downward continuity, a convex expectation on a linear lattice of bounded functions can be extended to a convex expectation on the measurable extended real functions. This result is especially relevant in the setting of uncertain processes: there, an easy way to obtain a convex expectation on the linear lattice of finitary bounded functions is to combine an initial convex expectation with a convex transition semigroup. Crucially, this work presents a sufficient condition on this semigroup which guarantees that the induced convex expectation is downward continuous, so that it can be extended to the set of measurable extended real functions. To conclude, this work looks at existing results on convex transition semigroups from the point of view of the aforementioned sufficient condition, in particular to construct a sublinear Poisson process.