In the dual L Φ * of a ∆ 2 -Orlicz space L Φ , that we call a dual Orlicz space, we show that a proper (resp. finite) convex function is lower semicontinuous (resp. continuous) for the Mackey topology τ, it is lower semicontinuous (resp. continuous) for the topology of convergence in probability. For this purpose, we provide the following Komlós type result: every norm bounded sequence (ξ n ) n in L Φ * admits a sequence of forward convex combinationsξ n ∈ conv(ξ n , ξ n+1 , ...) such that sup n |ξ n | ∈ L Φ * andξ n converges a.s.

Given a monotone convex function on the space of essentially bounded random variables with the Lebesgue property (order continuity), we consider its extension preserving the Lebesgue property to as big solid vector space of random variables as possible. We show that there exists a maximum such extension, with explicit construction, where the maximum domain of extension is obtained as a (possibly proper) subspace of a natural Orlicz-type space, characterized by a certain uniform integrability property. As an application, we provide a characterization of the Lebesgue property of monotone convex function on arbitrary solid spaces of random variables in terms of uniform integrability and a "nice" dual representation of the function. arXiv:1304.7934v2 [math.FA] 19 Jan 2014 2 K. OwariAs a first (trivial) example, we briefly see what happens when ϕ 0 is linear.Example 1.2. Let ϕ 0 be a positive (monotone) linear functional on L ∞ . Then it is finite-valued and identified with a finitely additive measure ν 0 (A) := ϕ 0 (1 A ) as ϕ 0 (X) = Ω Xdν 0 , while (1.2) is equivalent to saying that ν 0 is σ-additive. If the latter is the case, the "usual" integralφ(X) := Ω Xdν 0 defines a Lebesgue-preserving extension of ϕ 0 to L 1 (ν 0 ) := {X ∈ L 0 : Ω |X|dν 0 < ∞}. On the other hand, if ϕ is a monotone convex function on a solid space X ⊂ L 0 with (1.1) and ϕ| L ∞ = ϕ 0 , it is easy that ϕ must be positive, linear and finite on X . Then |X|dν 0 = lim nφ (|X| ∧ n) = lim n ϕ 0 (|X| ∧ n) = lim n ϕ(|X| ∧ n) = ϕ(|X|) < ∞ if X ∈ X , hence X ⊂ L 1 (ν 0 ), where the first equality follows from the monotone convergence theorem, and the fourth from the Lebesgue property of ϕ on X . Similarly, but with X1 {|X|≤n} instead of |X| ∧ n, we see also that ϕ =φ| X . Namely, (φ, L 1 (ν 0 )) is the maximum Lebesgue-preserving extension of ϕ 0 . ♦

We study the convex duality method for robust utility maximization in the presence of a random endowment. When the underlying price process is a locally bounded semimartingale, we show that the fundamental duality relation holds true for a wide class of utility functions on the whole real line and unbounded random endowment. To obtain this duality, we prove a robust version of Rockafellar's theorem on convex integral functionals and apply Fenchel's general duality theorem.

We discuss the problem of exponential hedging in the presence of model uncertainty expressed by a set of probability measures. This is a robust utility maximization problem with a contingent claim. We first consider the dual problem which is the minimization of penalized relative entropy over a product set of probability measures, showing the existence and variational characterizations of the solution. These results are applied to the primal problem. Then we consider the robust version of exponential utility indifference valuation, giving the representation of indifference price using a duality result.

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