Lower semicontinuity of monotone functionals in the mixed topology on $C_{b}$

Abstract The main result of this paper characterises the continuity from below of monotone functionals on the space $C_{b}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>b</mml:mi> </mml:msub> </mml:math> 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 $(C_{b},\mathrm{ca})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>b</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:mi>ca</mml:mi> <mml:mo>)</mml:mo> </mml:math> , where $\mathrm{ca}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ca</mml:mi> </mml:math> 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 $C_{b}\to \mathbb{R}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>b</mml:mi> </mml:msub> <mml:mo>→</mml:mo> <mml:mi>R</mml:mi> </mml:math> 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 $L^{\infty }$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> </mml:math> is transferred to risk measures on $C_{b}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>b</mml:mi> </mml:msub> </mml:math> . 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 $C_{b}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>b</mml:mi> </mml:msub> </mml:math> taking values in a locally convex vector lattice, continuity in the mixed topology is equivalent to continuity on norm-bounded sets.