Lower semicontinuity of monotone functionals in the mixed topology on $C_b$

TL;DR

This paper characterizes the lower semicontinuity of monotone functionals on bounded continuous functions in terms of the mixed topology, linking it to dual representations with countably additive measures, with applications in finance.

Contribution

It establishes the equivalence between lower semicontinuity in the mixed topology and dual representations for convex monotone functionals on $C_b$, and studies their regularity and continuity properties.

Findings

Lower semicontinuity in the mixed topology corresponds to dual representation with countably additive measures.

Continuity in the mixed topology is equivalent to continuity on norm bounded sets for convex monotone maps.

Regularity properties of capacities and dual representations of Choquet integrals are characterized.

Abstract

The main result of this paper characterizes the continuity from below of monotone functionals on the space $C_b$ 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 $(C_b,{\rm ca})$, where ${\rm ca}$ denotes the space of all countably additive signed Borel measures of finite variation. Hence, lower semicontinuity in the mixed topology of convex monotone maps $C_b\to \mathbb R$ is 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 super hedging 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. In a second step, the paper provides a characterization of equicontinuity in the mixed topology for families of convex monotone maps. As a consequence, for every convex monotone map on $C_b$ taking values in a locally convex vector lattice, continuity in the mixed topology is equivalent to continuity on norm bounded sets.