Robust expected utility maximization with medial limits
aa r X i v : . [ q -f i n . P M ] N ov Robust expected utility maximization with medial limits
Daniel Bartl ∗ Patrick Cheridito † Michael Kupper ‡ November 2018
Abstract
In this paper we study a robust expected utility maximization problem with random endowmentin discrete time. We give conditions under which an optimal strategy exists and derive a dual rep-resentation for the optimal utility. Our approach is based on a general representation result formonotone convex functionals, a functional version of Choquet’s capacitability theorem and mediallimits. The novelty is that it works under nondominated model uncertainty without any assumptionsof time-consistency. As applications, we discuss robust utility maximization problems with momentconstraints, Wasserstein constraints and Wasserstein penalties.
Keywords:
Robust expected utility maximization, convex duality, Choquet capacitability, medial limit, momentconstraints, Wasserstein distance.
MSC 2010 Subject Classification:
We consider a robust expected utility maximization problem of the form U ( X ) = sup ϑ ∈ Θ inf P ∈P ( E P u X + T X t =1 ϑ t ∆ S t ! + α ( P ) ) , (1.1)where X is a random endowment, S , S , . . . , S T the price evolution of a tradable asset, Θ the set ofpossible trading strategies, u a random utility function, P a set of probability measures and α : P → [0 , ∞ ) a penalty function. In the special case α ≡ , (1.1) reduces to U ( X ) = sup ϑ ∈ Θ inf P ∈P E P u X + T X t =1 ϑ t ∆ S t ! . (1.2)A large strand of the literature on robust utility maximization assumes that the family P is domi-nated ; see e.g. [13, 23, 12, 24, 25, 5, 22, 1]. In this case, one can, as in the classical expected utility ∗ Department of Mathematics and Statistics, University of Konstanz, 78464 Konstanz, Germany and Department ofMathematics, University of Vienna, 1090 Vienna, Austria. Financial support from the Austrian Science Fund (FWF)through grant Y00782 is gratefully acknowledged. † RiskLab, Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland. ‡ Department of Mathematics and Statistics, University of Konstanz, 78464 Konstanz, Germany i.e., all P ∈ P are absolutely continuous with respect to a common probability measure P ∗ P = { P } , apply Komlós’ theorem to construct an optimal strategy from a sequence of ap-proximately optimal strategies. The existence of optimal strategies can then be used to deduce a dualrepresentation for U . Different discrete-time versions of problem (1.2) under nondominated model un-certainty have been studied by [21, 4, 18, 2]. They all make time-consistency assumptions , which allowsthem to tackle the problem step by step backwards in time using dynamic programming arguments. Incontinuous time, nondominated problems of the form (1.2) have been investigated by [15, 17] in the case,where P consists of a time-consistent family of martingale or Lévy process laws.In this paper we study problem (1.1) without domination or time-consistency assumptions. Asa consequence, we cannot apply Komlós’ theorem or dynamic programming arguments. Instead, weuse convex duality methods, a functional version of Choquet’s capacitability theorem [9] and mediallimits. For our purposes, a medial limit is a positive linear functional lim med : l ∞ → R satisfying lim inf ≤ lim med ≤ lim sup with the following property: for any uniformly bounded sequence of univer-sally measurable functions X n : E → R on a measurable space ( E, F ) , X = lim med X n is universallymeasurable and E P X = lim med E P X n for every probability measure P on the universal completion of F .Mokobozki proved that medial limits exist under the usual axioms of ZFC together with the continuumhypothesis; see [16]. Later, Normann [19] showed that it is enough to assume ZFC and Martin’s axiom.In [20] medial limits were used to establish the existence of optimal quasi-sure superhedging strategieswith respect to general sets of martingale measures.We first derive a dual representation of U ( X ) for lower semicontinuous random endowments X onlyfrom convexity and integrability assumptions. Then we show that a suitable no-arbitrage condition andthe existence of a medial limit imply that problem (1.1) admits optimal strategies. From there we canextend the dual representation of U ( X ) from lower semicontinuous to measurable random endowments X . As sample space we consider a non-empty subset Ω of ((0 , ∞ ) × R ) T +1 endowed with the Euclideanmetric and the corresponding Borel σ -algebra. We suppose there is a money market account evolvingaccording to M t ( ω ) = ω t, and a financial asset whose price in units of M t is given by S t ( ω ) = ω t, . X : Ω → R is a Borel measurable mapping describing a random endowment in units of M T . As usual, ∆ S t denotes the increment S t − S t − . P is assumed to be a non-empty set of Borel probability measureson Ω and α : P → R + := [0 , ∞ ) a mapping with the property inf P ∈P α ( P ) = 0 . Denote by ( F t ) Tt =0 the filtration generated by ( M t , S t ) Tt =0 . The set Θ consists of all strategies ( ϑ t ) Tt =1 such that for each t , ϑ t : Ω → R is measurable with respect to the universal completion F ∗ t − of F t − and the Borel σ -algebraon R . u : Ω × R → R is a random utility function, which we assume to satisfy the following conditions:(U1) u ( ω, x ) is increasing and concave in x (U2) for every n ∈ N , u : Ω × [ − n, ∞ ) → R is continuous and bounded(U3) lim x →−∞ sup ω ∈ Ω u ( ω, x ) / | x | = −∞ . Problem (1.2) is time-consistent if the set P is stable under concatenation of transition probabilities. Conditions fortime-consistency of problems of the form (1.1) are given in e.g. [7, 8]. Recall that the universal completion F ∗ of a σ -algebra F is defined as the intersection of σ ( F ∪ N P ) over all probabilitymeasures P on F , where N P denotes the collection of P -null sets. By saying that X : E → R is universally measurable,we mean that it is measurable with respect to the universal completion F ∗ of F and the Borel σ -algebra on R , which isequivalent to saying that X is measurable with respect to F ∗ and the universal completion of the Borel σ -algebra on R . In the whole paper we understand the words “increasing” and “decreasing” in the weak sense. That is, u satisfies u ( ω, x ) ≥ u ( ω, y ) for all x ≥ y . u does not depend on ω , (1.1) measures the utility of the discounted terminal wealth X + P Tt =1 ϑ t ∆ S t . On the other hand, if u is of the form u ( ω, x ) = ˜ u ( ω T x ) for a function ˜ u : R → R , then(1.1) evaluates the undiscounted terminal wealth M T ( X + P Tt =1 ϑ t ∆ S t ) .We suppose there exists a continuous function Z : Ω → [1 , ∞ ) such that Z ≥ ∨ P Tt =0 | S t | and allsublevel sets { ω ∈ Ω : Z ( ω ) ≤ z } , z ∈ R + , are compact. Let B Z be the space of all Borel measurablefunctions X : Ω → R such that X/Z is bounded, L Z the set of all lower semicontinuous X ∈ B Z and C Z the space of all continuous X ∈ B Z . By M Z we denote the set of all Borel probability measures P on Ω satisfying E P Z < ∞ . Then E P X is well-defined for all P ∈ M Z and X ∈ B Z .To derive dual representations for U , we need P and α to satisfy the following two conditions:(A1) P is a convex subset of M Z and α : P → R + a convex mapping with σ ( M Z , C Z ) -closed sublevelsets P c := { P ∈ P : α ( P ) ≤ c } , c ∈ R + (A2) there exists an increasing function β : [1 , ∞ ) → R such that lim x →∞ β ( x ) /x = ∞ and inf P ∈P n E P u ( − β ( Z )) + α ( P ) o > −∞ . By v we denote the convex conjugate of u , given by v ( ω, y ) := sup x ∈ R { u ( ω, x ) − xy } , ( ω, y ) ∈ Ω × R + . If u satisfies (U2), u ( ω, is bounded in ω , and one has v ( ω, y ) = sup x ∈ Q { u ( ω, x ) − xy } ≥ u ( ω, . In particular, v is a Borel measurable function from Ω × R + to ( −∞ , ∞ ] that is bounded from below. Sofor q ∈ R + and a Borel probability measure Q on Ω , one can define D αv ( q Q ) := inf P ∈P { D v ( q Q k P ) + α ( P ) } , where D v ( q Q k P ) is the v -divergence between q Q and P , given by D v ( q Q k P ) := (cid:26) E P v (cid:0) qd Q /d P (cid:1) if q Q ≪ P ∞ otherwise.Let Q Z be the set of all probability measures P ∈ M Z under which ( S t ) Tt =0 is a martingale and ˆ Q Z theset of all pairs ( q, Q ) ∈ R + × M Z such that q = 0 or Q ∈ Q Z . Our first duality result is as follows: Theorem 1.1.
Assume (U1)–(U3) and (A1)–(A2) . Then U ( X ) = min ( q, Q ) ∈ ˆ Q Z n q E Q X + D αv ( q Q ) o ∈ R for all X ∈ L Z . (1.3)3o be able to derive the existence of optimal strategies and extend the duality (1.3) to Borel measur-able random endowments X , we need the following no-arbitrage condition :(NA) every P ∈ P is dominated by a P ′ ∈ P that does not admit arbitrage,where a Borel probability measure P on Ω is said to admit arbitrage if there exists a strategy ϑ ∈ Θ suchthat P [ P Tt =1 ϑ t ∆ S t > > and P [ P Tt =1 ϑ t ∆ S t ≥
0] = 1 . Theorem 1.2.
Assume a medial limit exists, u fulfills (U1) – (U3) and (NA) holds. Then the supremumin (1.1) is attained for every Borel measurable function X : Ω → R such that U ( X ) ∈ R . If, in addition, (A1)–(A2) are satisfied, then U ( X ) = inf ( q, Q ) ∈ ˆ Q Z n q E Q X + D αv ( q Q ) o ∈ R for all X ∈ B Z . (1.4)In the special case, where α ≡ and u is of the form u ( x ) = − exp( − λx ) for a risk-aversion parameter λ > , the dual expression (1.4) simplifies if instead of (1.2), one considers the equivalent problem W ( X ) = sup ϑ ∈ Θ inf P ∈P − λ log E P exp − λX − λ T X t =1 ϑ t ∆ S t ! . Corollary 1.3.
Assume a medial limit exists and P is a non-empty σ ( M Z , C Z ) -closed convex subset of M Z satisfying (NA) . If there exists an increasing function β : [1 , ∞ ) → R such that lim x →∞ β ( x ) /x = ∞ and sup P ∈P E P exp( β ( Z )) < ∞ , then W ( X ) = inf Q ∈Q Z (cid:26) E Q X + 1 λ H ( Q k P ) (cid:27) ∈ R for all X ∈ B Z , where H ( Q k P ) := inf P ∈P H ( Q k P ) is the robust version of the relative entropy H ( Q k P ) := ( E Q log( d Q /d P ) if Q ≪ P ∞ otherwise. In the following, we discuss three examples of robust utility maximization problems that are neitherdominated nor time-consistent but still fit in our framework.
Example 1.4.
Our first example is of the form (1.2) for a set of probability measures P given by momentconstraints. Consider a sample space of the form Ω = Ω × · · · × Ω T , where Ω = { ( a , s ) } for fixedinitial values a , s > and Ω t = [ a t , b t ] × (0 , ∞ ) for constants < a t ≤ b t , t = 1 , . . . , T . Note that Z ( ω ) = P Tt =0 ω t, ∨ ( ω t, ) − defines a continuous function Z : Ω → [1 , ∞ ) with compact sublevel sets Obviously, (NA) is weaker than the assumption that no P ∈ P admits arbitrage. On the other hand, it implies e.g.the robust no-arbitrage condition NA ( P ) of [6], which has been used in [21, 4, 18, 2] to derive the existence of optimalstrategies. Indeed, assume (NA) holds and there exists a strategy such that P [ P Tt =1 ϑ t ∆ S t ≥
0] = 1 for all P ∈ P . Theneach P ∈ P is dominated by a P ′ ∈ P that does not admit arbitrage. Hence, P [ P Tt =1 ϑ t ∆ S t >
0] = P ′ [ P Tt =1 ϑ t ∆ S t >
0] = 0 ,showing that NA ( P ) holds. Z ≤ z } , z ∈ R + , such that Z ≥ ∨ P Tt =0 | S t | . For all t = 1 , . . . , T and i = 1 , . . . , I , let c i < and d i , C it , D it > be constants such that min i c i < − and max i d i > . Assume that the set P of all Borelprobability measures on Ω satisfying the moment constraints E P [ S c i t ] ≤ C it and E P [ S d i t ] ≤ D it for all t = 1 , . . . , T and i = 1 , . . . , I, is non-empty. Then P fulfills (A1) for α ≡ . Moreover, if u : Ω × R → R is a random utility functionsatisfying (U1)–(U3) and there exists a constant q < max ≤ i ≤ I | c i | ∧ max ≤ i ≤ I | d i | such that u ( ω, x ) / (1 + | x | q ) is bounded, then (A2) holds for α ≡ . Finally, if s c i < C it and s d i < D it forall t = 1 , . . . , T and i = 1 , . . . , I , then P also satisfies (NA). Proofs are given in Appendix A.1. Example 1.5.
As a second example, we consider a problem of the form (1.2) with a set P of probabilitymeasures that are within a given Wasserstein distance of a reference measure. Let the sample space Ω be of the same form as in Example 1.4, and consider the metric d ( ω, ω ′ ) := T X t =1 e − ρκt ( | ω t, − ω ′ t, | κ + | ϕ ( ω t, ) − ϕ ( ω ′ t, ) | κ ) ! /κ , ω, ω ′ ∈ Ω , where ρ ≥ and κ ≥ are constants and the function ϕ : (0 , ∞ ) → R is given by ϕ ( x ) := ( x − if x > x ) if x ≤ . Denote ω ∗ = (( a , s ) , ( a , , . . . , ( a T , ∈ Ω . Then, Z ( ω ) = s + T + e ρT T − /κ d ( ω, ω ∗ ) is a continuousfunction Z : Ω → [1 , ∞ ) with compact sublevel sets { Z ≤ z } , z ∈ R + , such that Z ≥ ∨ P Tt =0 | S t | .Choose a reference measure P ∗ ∈ M Z satisfying E P ∗ Z p < ∞ for a given exponent p > . Fix a constant η > , and consider the ball P := { P ∈ M Z : W p ( P , P ∗ ) ≤ η } around P ∗ with respect to the p -Wasserstein distance W p , given by W p ( P , P ∗ ) := inf π (cid:18)Z Ω × Ω d ( ω, ω ′ ) p dπ ( ω, ω ′ ) (cid:19) /p , where the infimum is taken over all Borel probability measures π on Ω × Ω with marginals P and P ∗ .Then P satisfies (A1) for α ≡ as well as (NA). Moreover, if u : Ω × R → R is a random utility functionsatisfying (U1)–(U3) and there exists a constant q < p such that u ( ω, x ) / (1 + | x | q ) is bounded, then also(A2) holds for α ≡ . This is proved in Appendix A.2. Alternatively, one can consider a set P of Borel probability measures satisfying moment conditions of the form E P [( M t S t ) c i ] ≤ C it and E P [( M t S t ) d i ] ≤ D it for t = 1 , . . . , T and i = 1 , . . . , I , where M t describes the evolution of themoney market account. Then, provided that P is non-empty, (A1) is still satisfied, and (A2) holds under the same condi-tions on u . A sufficient condition for (NA) is that there exist constants e t ∈ [ a t , b t ] such that ( e t s ) c i < C it and ( e t s ) d i < D it for all t = 1 , . . . , T and i = 1 , . . . , I . xample 1.6. As our last example, we consider a problem of the form (1.1) with a Wasserstein penalty.Let the sample space Ω be of the same form as in Examples 1.4 and 1.5. Fix an exponent p > , and let Z , d , W p be as in Example 1.5. For a given constant η > and a reference measure P ∗ ∈ M Z satisfying E P ∗ Z p < ∞ , define α ( P ) := ηW p ( P , P ∗ ) p and P := { P ∈ M Z : α ( P ) < ∞} . Then (A1) and (NA) hold.Moreover, if u : Ω × R → R is a random utility function satisfying (U1)–(U3) and there exists a constant q < p such that u ( x ) / (1 + | x | q ) is bounded, then (A2) is fulfilled as well. Proofs are provided in AppendixA.3.The rest of the paper is organized as follows. In Section 2 we first establish a functional versionof Choquet’s capacitability theorem. Then we derive dual representation results for increasing convexfunctionals on different sets of real-valued functions. These results hold for general sample spaces endowedwith a perfectly normal topology and do not require the existence of a medial limit. In Section 3, wefirst prove Theorem 1.1. Then we derive some elementary properties of medial limits, before we giveproofs of Theorem 1.2 and Corollary 1.3. In the appendix we show that conditions (A1), (A2) and (NA)hold in the three Examples 1.4, 1.5 and 1.6. In this section, we first derive a functional version of Choquet’s capacitability theorem by working out aremark at the end of his paper [9]. Then we establish a dual representation result for increasing convexfunctionals defined on spaces of measurable functions.Denote by R the extended real line [ −∞ , ∞ ] . For a given non-empty set E , consider two nestedsubsets H ⊆ G ⊆ R E such that H is a non-empty lattice and G contains all suprema of increasing sequences in G as well as all infima of arbitrary sequences in G . An H -Suslin scheme is a mapping σ : S n ∈ N N n → H and an H -Suslin function an element X ∈ R E of the form X = sup γ ∈ N N inf n ∈ N σ ( γ , . . . , γ n ) , where σ is an H -Suslin scheme. We denote the set of all H -Suslin functions by S ( H ) and all infima ofsequences in H by H δ . If φ : G → R is an increasing mapping, we extend it to R E by setting ˆ φ ( X ) := inf { φ ( Y ) : X ≤ Y, Y ∈ G } , X ∈ R E with the convention inf ∅ := + ∞ . The following is a functional version of Theorem 1 in [9]:
Proposition 2.1.
Let φ : G → R be an increasing mapping with the following two properties: (C1) lim n φ ( X n ) = φ (lim n X n ) for every decreasing sequence ( X n ) in H (C2) lim n φ ( X n ) = φ (lim n X n ) for every increasing sequence ( X n ) in G . in particular, for metrizable sample spaces We call a sequence ( X n ) in G increasing if X n +1 ≥ X n for all n and decreasing if X n +1 ≤ X n for all n . that is, φ ( X ) ≥ φ ( Y ) for all X, Y ∈ G such that X ≥ Y hen, ˆ φ ( X ) = sup { φ ( Y ) : Y ≤ X, Y ∈ H δ } for all X ∈ S ( H ) .Proof. Denote F = E × R , and let A be the collection of subsets of F of the form S x ∈ E { x } × A x , wherefor each x , A x = [ −∞ , a x ) or A x = [ −∞ , a x ] for some a x ∈ R . Then A is stable under intersections andunions. For A ∈ A , define X A : E → R by X A ( x ) := a x . Then for any family of subsets ( A α ) ⊆ A , onehas X T α A α = inf α X A α and X S α A α = sup α X A α . In particular, H δ := { A ∈ A : X A ∈ H δ } is stableunder finite unions and countable intersections. It is clear that the set function ˜ φ : 2 F → R , given by ˜ φ ( B ) := inf { ˆ φ ( X A ) : B ⊆ A, A ∈ A} , is increasing and satisfies lim n ˜ φ ( B n ) = ˜ φ ( T n B n ) for decreasing sequences ( B n ) in H δ . Moreover,if ( B n ) is an increasing sequence of subsets of F such that lim n ˜ φ ( B n ) < + ∞ , there exist A n ∈ A and Y n ∈ G such that B n ⊆ A n , X A n ≤ Y n and φ ( Y n ) ≤ ˜ φ ( B n ) + 1 /n (or φ ( Y n ) ≤ − n in case ˜ φ ( B n ) = −∞ ). The sequences ˜ A n = T m ≥ n A m and ˜ Y n = inf m ≥ n Y m are increasing, and one has S n B n ⊆ A := S n ˜ A n ∈ A as well as X A ≤ Y := sup n ˜ Y n ∈ G . So ˜ φ ( S n B n ) ≤ ˆ φ ( X A ) ≤ φ ( Y ) = lim n φ ( ˜ Y n ) ≤ lim n ˜ φ ( B n ) . This shows that ˜ φ is an abstract capacity on ( F, H δ ) according to [9]. For an H -Suslin function of theform X = sup γ ∈ N N inf n ∈ N σ ( γ , . . . , γ n ) , define ˜ σ : S n ∈ N N n → H δ by ˜ σ ( · ) := S x ∈ E { x } × [ −∞ , σ ( · )( x )] .Then A = [ γ ∈ N N \ n ∈ N ˜ σ ( γ , . . . , γ n ) is a Suslin set generated by H δ satisfying X A = X . So one obtains from Theorem 1 of [9] that ˆ φ ( X ) = ˜ φ ( A ) = sup { ˜ φ ( B ) : B ⊆ A, B ∈ H δ } = sup { φ ( Y ) : Y ≤ X, Y ∈ H δ } . In the following, let E be a perfectly normal topological space and V : E → R + \ { } a continuousfunction. Denote by B V the set of all Borel measurable functions X : E → R such that X/V is boundedand by C V and U V the subsets consisting of all continuous and upper semicontinuous functions in B V ,respectively. If ( X n ) is an increasing (decreasing) sequence of real-valued functions on E that convergespointwise to a real-valued function X on E , we write X n ↑ X ( X n ↓ X ). Let ca + V be the set of allBorel measures µ on E satisfying h V, µ i < + ∞ . For a real-valued mapping φ defined on a subset of B V containing C V , we define φ ∗ C V ( µ ) := sup X ∈ C V {h X, µ i − φ ( X ) } , µ ∈ ca + V . (2.1)Then the following holds: that is, ˜ φ ( B ) ≥ ˜ φ ( C ) for all B, C ∈ F such that B ⊇ C that is, B n +1 ⊆ B n for all n that is, B n +1 ⊇ B n for all n In particular, this covers all metric spaces. heorem 2.2. If φ : C V → R is an increasing convex functional satisfying (R1) φ ( X n ) ↓ φ (0) for every sequence ( X n ) in C V such that X n ↓ ,then φ ( X ) = max µ ∈ ca + V (cid:8) h X, µ i − φ ∗ C V ( µ ) (cid:9) for each X ∈ C V , (2.2) and all sublevel sets { µ ∈ ca + V : φ ∗ C V ( µ ) ≤ c } , c ∈ R , are σ ( ca + V , C V ) -compact.Moreover, every increasing convex functional φ : U V → R with the property (R2) φ ( X n ) ↓ φ ( X ) for each sequence ( X n ) in C V such that X n ↓ X for some X ∈ U V ,has a representation of the form φ ( X ) = max µ ∈ ca + V (cid:8) h X, µ i − φ ∗ C V ( µ ) (cid:9) , X ∈ U V , (2.3) and every increasing convex functional φ : B V → R satisfying (R2) together with (R3) φ ( X n ) ↑ φ ( X ) for each sequence ( X n ) in B V such that X n ↑ X for some X ∈ B V ,can be written as φ ( X ) = sup µ ∈ ca + V (cid:8) h X, µ i − φ ∗ C V ( µ ) (cid:9) , X ∈ B V . (2.4) Proof.
First, let φ : C V → R be an increasing convex functional satisfying (R1). It is clear from thedefinition of φ ∗ C V that for fixed X ∈ C V , φ ( X ) ≥ h X, µ i − φ ∗ C V ( µ ) for all µ ∈ ca + V . (2.5)Moreover, it follows from the Hahn–Banach extension theorem that there exists a positive linear functional ψ : C V → R such that ψ ( Y ) ≤ φ ( X + Y ) − φ ( X ) for all Y ∈ C V . Now, consider a sequence ( X n ) of functions in C V such that X n ↓ . Then, one has for all λ ∈ (0 , , φ ( X + X n ) ≤ λφ (cid:16) Xλ (cid:17) + (1 − λ ) φ (cid:16) X n − λ (cid:17) . (2.6)Since y φ ( yX ) is a convex function from R to R , it is continuous. Therefore, for λ close to , λφ ( X/λ ) is close to φ ( X ) . By (R1), one has (1 − λ ) φ ( X n / (1 − λ )) ↓ (1 − λ ) φ (0) . It follows that φ ( X + X n ) ↓ φ ( X ) ,and consequently, ψ ( X n ) ↓ for n → + ∞ . Since on a perfectly normal space, the Borel σ -algebracoincides with the σ -algebra generated by all continuous real-valued functions (see [26]), one obtainsfrom the Daniell–Stone theorem that there exists a µ ∈ ca + V such that ψ ( Y ) = h Y, µ i for all Y ∈ C V .Hence, h X + Y, µ i − φ ( X + Y ) ≤ h X, µ i − φ ( X ) for all Y ∈ C V . In particular, φ ∗ C V ( µ ) = h X, µ i − φ ( X ) , which together with (2.5), proves (2.2).8ext, we show that the sublevel sets Λ c := { µ ∈ ca + V : φ ∗ C V ( µ ) ≤ c } , c ∈ R , are σ ( ca + V , C V ) -compact. Note that C V equipped with the norm k X k V := sup x | X ( x ) /V ( x ) | is a Banachspace. We extend φ ∗ C V to the positive cone C ∗ , + V in the topological dual C ∗ V of C V using definition (2.1).Then the set ˜Λ c := { µ ∈ C ∗ , + V : φ ∗ C V ( µ ) ≤ c } is σ ( C ∗ V , C V ) -closed. Moreover, since φ is real-valued,the increasing convex function ϕ : R + → ( −∞ , ∞ ] , given by ϕ ( y ) := sup x ∈ R + { xy − φ ( xV ) } , satisfies lim y → + ∞ ϕ ( y ) /y = ∞ . As a consequence, the right-continuous inverse ϕ − : R → R + has the property lim x → + ∞ ϕ − ( x ) /x = 0 . Since φ ∗ C V ( µ ) ≥ sup x ∈ R + {h xV, µ i − φ ( xV ) } = ϕ ( h V, µ i ) , one obtains for µ ∈ ˜Λ c , k µ k C ∗ V = h V, µ i ≤ ϕ − ( φ ∗ C V ( µ )) ≤ ϕ − ( c ) < ∞ . So it follows from the Banach–Alaoglu theorem that ˜Λ a is σ ( C ∗ V , C V ) -compact. Now, choose a µ ∈ C ∗ , + V with φ ∗ C V ( µ ) < ∞ and let ( X n ) be a sequence in C V such that X n ↓ . Then, for every constant y > , φ ∗ C V ( µ ) ≥ h yX n , µ i − φ ( yX n ) , and therefore, h X n , µ i ≤ φ ( yX n ) y + φ ∗ C V ( µ ) y . By (R1), one obtains h X n , µ i ↓ , and it follows from the Daniell–Stone theorem that µ is in ca + V .This shows that φ ∗ C V ( µ ) = ∞ for all µ ∈ C ∗ , + V \ ca + V . In particular, Λ c is equal to ˜Λ c and therefore, σ ( ca + V , C V ) -compact.Now, assume φ : U V → R is an increasing convex functional with the property (R2). To show that thedual representation (2.2) extends from C V to U V , we use that on a perfectly normal space, every uppersemicontinuous function is the pointwise limit of a decreasing sequence of continuous functions (see [26]).As an easy consequence, every X ∈ U V can be written as the pointwise limit of a decreasing sequence ( X n ) in C V . It follows from (R2) and the definition of φ ∗ C V that φ ( X ) = lim n φ ( X n ) ≥ lim n h X n , µ i − φ ∗ C V ( µ ) ≥ h X, µ i − φ ∗ C V ( µ ) for all µ ∈ ca + V . (2.7)On the other hand, one obtains from (2.2) that φ ( X ) ≤ φ ( X n ) = max µ ∈ ca + V (cid:8) h X n , µ i − φ ∗ C V ( µ ) (cid:9) for every n. Since h X n , µ i − φ ∗ C V ( µ ) ≤ h X , µ i − φ ∗ C V ( µ ) ≤ k X k V k µ k C ∗ V − φ ∗ C V ( µ ) ≤ k X k V ϕ − ( φ ∗ C V ( µ )) − φ ∗ C V ( µ ) , this implies that there exists a level c ∈ R such that φ ( X n ) = max µ ∈ Λ c (cid:8) h X n , µ i − φ ∗ C V ( µ ) (cid:9) for all n .9ote that h X n , µ i − φ ∗ C V ( µ ) is decreasing in n as well as σ ( ca + V , C V ) -upper semicontinuous and concavein µ . So it follows from the minimax result, Theorem 2 of [10], and the monotone convergence theoremthat φ ( X ) = inf n φ ( X n ) = inf n max µ ∈ Λ a (cid:8) h X n , µ i − φ ∗ C V ( µ ) (cid:9) = max µ ∈ Λ a inf n (cid:8) h X n , µ i − φ ∗ C V ( µ ) (cid:9) = max µ ∈ Λ a (cid:8) h X, µ i − φ ∗ C V ( µ ) (cid:9) , which together with (2.7), proves (2.3).The last part of Theorem 2.2 follows from Proposition 2.1. Indeed, if φ : B V → R is an increasingconvex functional satisfying (R2)–(R3), we fix a constant r > and let G be the set of X ∈ B V satisfying | X | ≤ r | V | . Then, φ , G and H = C V ∩ G satisfy the assumptions of Proposition 2.1. Moreover, H δ = U V ∩ G . So it follows from Proposition 2.1 and (2.3) that φ ( X ) = sup Y ≤ X, Y ∈ U V ∩ G φ ( Y ) = sup Y ≤ X, Y ∈ U V ∩ G max µ ∈ ca + V (cid:8) h Y, µ i − φ ∗ C V ( µ ) (cid:9) for all X ∈ G ∩ S ( H ) . Since for fixed µ ∈ ca + V , the mapping X
7→ h
X, µ i together with G and H also satisfies the assumptionsof Proposition 2.1, one has φ ( X ) = sup µ ∈ ca + V sup Y ≤ X, Y ∈ U V ∩ G (cid:8) h Y, µ i − φ ∗ C V ( µ ) (cid:9) = sup µ ∈ ca + V (cid:8) h X, µ i − φ ∗ C V ( µ ) (cid:9) for all X ∈ G ∩ S ( H ) . So, if we can show that G ⊆ S ( H ) , the representation (2.4) holds for all X ∈ B V since r was arbitrary.To prove G ⊆ S ( H ) , we note that a function X ∈ G can be written as X = sup q (cid:8) qV { X ≥ qV } − rV { X An increasing convex functional φ : U V → R with the property (R1) satisfies (R2) ifand only if φ ∗ C V ( µ ) = φ ∗ U V ( µ ) := sup X ∈ U V {h X, µ i − φ ( X ) } for all µ ∈ ca + V . (2.8)10 roof. First, let us assume φ satisfies (R2). For a given X ∈ U Z , there exists a sequence ( X n ) in C V such that X n ↓ X (see [26]). By the monotone convergence theorem and (R2), one has h X n , µ i − φ ( X n ) → h X, µ i − φ ( X ) . This shows that φ ∗ C V ( µ ) = φ ∗ U V ( µ ) for all µ ∈ ca + V .Now, assume φ satisfies (R1) together with (2.8) and let ( X n ) be a sequence in C V such that X n ↓ X ∈ U V . It is immediate from the definition of φ ∗ U V and (2.8) that φ ( X ) ≥ sup µ ∈ ca + V (cid:8) h X, µ i − φ ∗ U V ( µ ) (cid:9) = sup µ ∈ ca + V (cid:8) h X, µ i − φ ∗ C V ( µ ) (cid:9) . On the other hand, it follows from the arguments in the proof of Theorem 2.2 that there exists a σ ( ca + V , C V ) -compact convex subset Λ of ca + V such that φ ( X n ) = max µ ∈ Λ ( h X n , µ i − φ ∗ C V ( µ )) for all n . Anapplication of the minimax result, Theorem 2 of [10], and the monotone convergence theorem gives lim n φ ( X n ) = inf n max µ ∈ Λ (cid:8) h X n , µ i − φ ∗ C V ( µ ) (cid:9) = max µ ∈ Λ inf n (cid:8) h X n , µ i − φ ∗ C V ( µ ) (cid:9) = max µ ∈ Λ (cid:8) h X, µ i − φ ∗ C V ( µ ) (cid:9) . In particular, φ ( X n ) ↓ φ ( X ) . Remark 2.4. Assume E is a Polish space and denote by S V the set of all Suslin functions X : E → R generated by C V such that X/V is bounded. Then S V equals the set of all upper semianalytic functions X : E → R such that X/V is bounded (see Proposition 7.41 of [3]), and every upper semianalytic functionis measurable with respect to the universal completion of the Borel σ -algebra on E (see Corollary 7.42.1of [3]). Since every Borel measure on E has a unique extension to the universal completion of the Borel σ -algebra, h X, µ i is well-defined for all X ∈ S V and µ ∈ ca + V . So if φ : S V → R is an increasing convexfunctional satisfying (R2) and φ ( X n ) ↑ φ ( X ) for every sequence ( X n ) in S V such that X n ↑ X for some X ∈ S V , it follows exactly as in the proof of Theorem 2.2 that φ ( X ) = sup µ ∈ ca + V (cid:8) h X, µ i − φ ∗ C V ( µ ) (cid:9) for all X ∈ S V . For the proof of Theorem 1.1 we need the following lemmas: Lemma 3.1. If u satisfies (U2)–(U3) , then sup ( ω,y ) ∈ Ω × [0 ,n ] | v ( ω, y ) | < ∞ for every n ∈ N .Proof. Fix n ∈ N . By (U3), there exists a constant x ≤ such that sup ω u ( ω, x ) ≤ ( n + 1) x for all x ≤ x . On the other hand, it follows from (U2) that c := sup ω sup x ≥ x | u ( ω, x ) | ∈ R . Now, let x ∈ R and y ∈ [0 , n ] . Then u ( ω, x ) − xy ≤ ( n + 1) x − xn ≤ if x ≤ x u ( ω, x ) − xy ≤ c − x n if x ≥ x . This shows that v ( ω, y ) = sup x ∈ R ( u ( ω, x ) − xy ) ≤ c − x n . On the other hand, v ( ω, y ) ≥ u ( ω, ≥ − c ,and the proof is complete. 11 emma 3.2. If u satisfies (U1)–(U2) , then there exists a constant c ∈ R such that q E Q Y − ≤ q E Q X + − E P u ( X + Y ) + E P v (cid:18) q d Q d P (cid:19) + c for all Borel measurable functions X, Y : Ω → R , every q ∈ R + and every pair of Borel probabilitymeasures P and Q on Ω such that qd Q ≪ d P .Proof. Since u satisfies (U1)–(U2), c := sup ( ω,x ) ∈ Ω × R + { u ( ω, x ) − u ( ω, } is finite and satisfies E P u ( X + Y ) ≤ E P u (cid:0) − ( X + Y ) − (cid:1) + c. Moreover, it follows from the definition of v that ( X + Y ) − q d Q d P ≤ − u (cid:0) − ( X + Y ) − (cid:1) + v (cid:18) q d Q d P (cid:19) . Hence, q E Q Y − ≤ q E Q X + + q E Q ( X + Y ) − ≤ q E Q X + − E P u (cid:0) − ( X + Y ) − (cid:1) + E P v (cid:18) q d Q d P (cid:19) ≤ q E Q X + − E P u ( X + Y ) + E P v (cid:18) q d Q d P (cid:19) + c. Lemma 3.3. If u satisfies (U1)–(U3) , then the functional D ( X ) := inf ( q, Q ) ∈ ˆ Q Z n q E Q X + D αv ( q Q ) o satisfies U ( X ) ≤ D ( X ) < ∞ for all X ∈ B Z .Proof. By Lemma 3.1, one has D ( X ) ≤ D αv (0) = inf P ∈P (cid:8) E P v (0) + α ( P ) (cid:9) < ∞ for all X ∈ B Z .Now, consider P ∈ P , X ∈ B Z and ϑ ∈ Θ such that E P u (cid:16) X + P Tt =1 ϑ t ∆ S t (cid:17) > −∞ . It is immediatefrom the definition of v that E P u X + T X t =1 ϑ t ∆ S t ! ≤ E P v (0) . Moreover, for q ∈ (0 , ∞ ) and Q ∈ Q Z such that q Q ≪ P and E Q v ( qd Q /d P ) < ∞ , one obtains fromLemma 3.2 that there exists a constant c ∈ R such that q E Q T X t =1 ϑ t ∆ S t ! − ≤ q E Q X + − E P u X + T X t =1 ϑ t ∆ S t ! + E P v (cid:18) q d Q d P (cid:19) + c < ∞ . So it follows from Theorems 1 and 2 in [14] that (cid:0)P ts =1 ϑ ns ∆ S s (cid:1) Tt =0 is a Q -martingale, and therefore, E Q P Tt =1 ϑ t ∆ S t = 0 . By the definition of v , one has u X + T X t =1 ϑ t ∆ S t ! ≤ X + T X t =1 ϑ t ∆ S t ! q d Q d P + v (cid:18) q d Q d P (cid:19) . E P u X + T X t =1 ϑ t ∆ S t ! ≤ q E Q " X + T X t =1 ϑ t ∆ S t + E P v (cid:16) q d Q d P (cid:17) = q E Q X + E P v (cid:16) q d Q d P (cid:17) . Now, first taking the infimum in E P u X + T X t =1 ϑ t ∆ S t ! + α ( P ) ≤ q E Q X + E P v (cid:16) q d Q d P (cid:17) + α ( P ) over all P ∈ P and ( q, Q ) ∈ ˆ Q Z such that q Q ≪ P and then the supremum over all ϑ ∈ Θ , yields U ( X ) ≤ D ( X ) . Lemma 3.4. If u satisfies (U1)–(U2) , then D v ( q Q k P ) = sup X ∈ C Z n E P u ( X ) − q E Q X o for all q ∈ R + and P , Q ∈ M Z .Proof. First, note that if q ∈ R + and P , Q ∈ M Z are such that q Q is not absolutely continuous withrespect to P , there exists a Borel set A ⊆ Ω such that q Q [ A ] > and P [ A ] = 0 . Since Q is a regularmeasure, there is a closed set K ⊆ A such that q Q [ K ] > and P [ K ] = 0 . For every m ∈ N , thereexists a sequence of bounded continuous functions X n : Ω → R such that X n ↓ m K . It follows from themonotone convergence theorem that E P u ( − X n ) + q E Q X n → E P u ( − m K ) + q E Q [ m K ] = E P u (0) + qm Q [ K ] , and as a consequence, sup X ∈ C Z n E P u ( X ) − q E Q X o = ∞ = D v ( q Q k P ) . Next, assume that q Q is absolutely continuous with respect to P . Then, E P u ( X ) − q E Q X = E P (cid:20) u ( X ) − q d Q d P X (cid:21) ≤ E P v (cid:18) q d Q d P (cid:19) for all X ∈ C Z . On the other hand, there exists a sequence of simple random variables ( Y n ) such that E P (cid:20) u ( Y n ) − q d Q d P Y n (cid:21) → E P v (cid:18) q d Q d P (cid:19) , from which it follows that there exists a sequence ( X n ) in C Z such that E P (cid:20) u ( X n ) − q d Q d P X n (cid:21) → E P v (cid:18) q d Q d P (cid:19) = D v ( q Q k P ) . This completes the proof of the lemma. 13 emma 3.5. Assume (U1)–(U3) and (A2) hold. Then, for every constant m ∈ R + , there exists a c ∈ R + such that inf P ∈P n E P u ( X ) + α ( P ) o = inf P ∈P c n E P u ( X ) + α ( P ) o for all Borel measurable functions X : Ω → R satisfying X ≥ − mZ .Proof. Fix m ∈ R + . It follows from (U2)–(U3) and (A2) that ϕ ( x ) := inf P ∈P n x E P u ( − mZ ) + α ( P ) o is finite for all x ∈ R + . So, the function ψ : R → ( ∞ , ∞ ] , given by ψ ( y ) := sup x ≥ { xy + ϕ ( x ) } , is increasing and satisfies lim y →∞ ψ ( y ) /y → ∞ . As a consequence, the right-continuous inverse ψ − ( y ) := inf { x ∈ R : ψ ( x ) > y } has the property lim y →∞ ψ − ( y ) /y = 0 . Since α ( P ) ≥ ϕ ( x ) − x E P u ( − mZ ) for all x ∈ R + , one has α ( P ) ≥ ψ ( − E P u ( − mZ )) , and therefore, E P u ( − mZ ) ≥ − ψ − ( α ( P )) . By (U1), one has for all X ≥ − mZ , inf P ∈P n E P u ( X ) + α ( P ) o ≤ E P u ( ∞ ) < ∞ and E P u ( X ) + α ( P ) ≥ E P u ( − mZ ) + α ( P ) ≥ − ψ − ( α ( P )) + α ( P ) . Since lim c →∞ c − ψ − ( c ) = ∞ , this shows that there exists a c ∈ R such that inf P ∈P n E P u ( X ) + α ( P ) o = inf P ∈P c n E P u ( X ) + α ( P ) o for all X ≥ − mZ .Next, note that if u satisfies (U2), then for every continuous function γ : [1 , ∞ ) → R , Z γ := 1 ∨ ( − u ( − γ ( Z ))) defines a continuous function from Ω to [1 , ∞ ) . 14 emma 3.6. Assume (U2)–(U3) and (A1)–(A2) hold. Then, there exists a continuous increasing function γ : [0 , ∞ ) → R such that lim x →∞ γ ( x ) /x = ∞ , and for all c ∈ R + , P c is a σ ( M Z γ , C Z γ ) -compact subsetof M Z γ .Proof. By (A2), there exists an increasing function β : [1 , ∞ ) → R such that lim x →∞ β ( x ) /x = ∞ and inf P ∈P (cid:8) E P u ( − β ( Z )) + α ( P ) (cid:9) > −∞ . So one can construct a continuous increasing function γ : [1 , ∞ ) → R such that lim x →∞ γ ( x ) /x = lim x →∞ β ( x ) /γ ( x ) = ∞ . It follows from (U3) that there exists a z ∈ R such that u ( − γ ( Z )) ≤ − Z on { Z > z } . This shows that C Z ⊆ C Z γ and M Z γ ⊆ M Z . Since for given c ∈ R + , one has inf P ∈P c E P u ( − β ( Z )) + c ≥ inf P ∈P c n E P u ( − β ( Z )) + α ( P ) o > −∞ , one obtains lim z →∞ sup P ∈P c E P [ Z γ { Z>z } ] = 0 . Moreover, it follows from (U2) that Z γ is bounded on the sets { Z ≤ z } . Hence, P c is contained in M Z γ ,and since by (A1), it is σ ( M Z , C Z ) -closed, it is also σ ( M Z γ , C Z γ ) -closed. Note that P Z γ d P transforms P c into a subset ˜ P c of the finite Borel measures M on Ω . Since the sets { Z ≤ z } are compact, it followsfrom Prokhorov’s theorem that ˜ P c is σ ( M , C b ) -compact, where C b are all bounded continuous functionson Ω . But this is equivalent to P c being σ ( M Z γ , C Z γ ) -compact.Next, let us denote by ˜Θ the set of all strategies ϑ ∈ Θ such that ϑ t is continuous and bounded forall t = 1 , . . . , T , and define ˜ U ( X ) := sup ϑ ∈ ˜Θ inf P ∈P ( E P u X + T X t =1 ϑ t ∆ S t ! + α ( P ) ) , X ∈ B Z , as well as ˜ U ∗ C Z ( q Q ) := sup X ∈ C Z n ˜ U ( X ) − q E Q X o , for q ∈ R + and Q ∈ M Z . Then the following holds: Lemma 3.7. If (U1)–(U3) and (A1)–(A2) hold, then ˜ U is an increasing concave mapping from B Z to R satisfying ˜ U ( X n ) ↑ ˜ U ( X ) for every sequence ( X n ) in C Z such that X n ↑ X for some X ∈ L Z (3.1) and ˜ U ∗ C Z ( q Q ) = ( D αv ( q Q ) if q = 0 or Q ∈ Q Z ∞ if q > and Q ∈ M Z \ Q Z . (3.2) Proof. It is straight-forward to check that ˜ U is an increasing concave mapping from B Z to R . To show(3.2), we note that for given q ∈ R + and Q ∈ M Z , ˜ U ∗ C Z ( q Q ) = sup X ∈ C Z sup ϑ ∈ ˜Θ inf P ∈P ( E P u X + T X t =1 ϑ t ∆ S t ! + α ( P ) − q E Q X ) = sup X ∈ C Z sup ϑ ∈ ˜Θ inf P ∈P ( E P u ( X ) + α ( P ) − q E Q X + q E Q T X t =1 ϑ t ∆ S t ) . E Q P Tt =1 ϑ t ∆ S t = 0 for all ϑ ∈ ˜Θ if and only if S is a Q -martingale, one has ˜ U ∗ C Z ( q Q ) = ∞ for q > and Q ∈ M Z \ Q Z . On the other hand, if q = 0 or Q ∈ Q Z , then ˜ U ∗ C Z ( q Q ) = sup X ∈ C Z inf P ∈P n E P u ( X ) + α ( P ) − q E Q X o . So, it follows from Lemma 3.4 that ˜ U ∗ C Z ( q Q ) ≤ inf P ∈P sup X ∈ C Z n E P u ( X ) + α ( P ) − q E Q X o = inf P ∈P { D v ( q Q k P ) + α ( P ) } . (3.3)By Lemma 3.6, there exists a continuous increasing function γ : [1 , ∞ ) → R such that lim x →∞ γ ( x ) /x = ∞ , and for all c ∈ R + , P c is a σ ( M Z γ , C Z γ ) -compact subset of M Z γ . For a given constant m ∈ R + ,denote C mZ := { X ∈ C Z : X ≥ − mZ } and D mv ( q Q k P ) := sup X ∈ C mZ n E P u ( X ) − q E Q X o . By Lemma 3.5, there exists an a ∈ R + such that sup X ∈ C mZ inf P ∈P n E P u ( X ) + α ( P ) − q E Q X o = sup X ∈ C mZ inf P ∈P a n E P u ( X ) + α ( P ) − q E Q X o . So, since E P u ( X ) + α ( P ) − q E Q X is concave in X ∈ C mZ as well as convex and σ ( M Z γ , C Z γ ) -lowersemicontinuous in P ∈ P c , it follows from the minimax result, Theorem 2 of [10], that sup X ∈ C mZ inf P ∈P n E P u ( X ) + α ( P ) − q E Q X o = inf P ∈P a sup X ∈ C mZ n E P u ( X ) + α ( P ) − q E Q X o ≥ inf P ∈P { D mv ( q Q k P ) + α ( P ) } . Now, note that −∞ < inf ω ∈ Ω u ( ω, ≤ D mv ( q Q k P ) ≤ sup ( ω,x ) ∈ Ω × R + u ( ω, x ) + qm E Q Z < ∞ for all P ∈ P . Moreover, D mv ( q Q k P ) + α ( P ) is increasing in m ∈ R + as well as convex and σ ( M Z γ , C Z γ ) -lower semi-continuous in P ∈ P . So, if there exists a b ∈ R + such that sup m ∈ R + inf P ∈P { D mv ( q Q k P ) + α ( P ) } = sup m ∈ R + inf P ∈P b { D mv ( q Q k P ) + α ( P ) } , (3.4)another application of Theorem 2 in [10] yields sup m ∈ R + inf P ∈P { D mv ( q Q k P ) + α ( P ) } = inf P ∈P b sup m ∈ R + { D mv ( q Q k P ) + α ( P ) }≥ inf P ∈P { D v ( q Q k P ) + α ( P ) } . On the other hand, if (3.4) does not hold for any b ∈ R + , there exists a sequence ( b n ) in R + such that b n → ∞ and sup m ∈ R + inf P ∈P { D mv ( q Q k P ) + α ( P ) } ≥ lim n →∞ (cid:26) inf ω ∈ Ω u ( ω, 0) + b n (cid:27) = ∞ . ˜ U ∗ C Z ( q Q ) = sup m ∈ R + sup X ∈ C mZ inf P ∈P n E P u ( X ) + α ( P ) − q E Q X o ≥ inf P ∈P { D v ( q Q k P ) + α ( P ) } , which, together with (3.3), implies (3.2).Next, consider a sequence ( X n ) in C Z such that X n ↑ X for some X ∈ C Z . Since X ∈ C Z , one has X ≥ − mZ for some m ∈ R + . So, by Lemma 3.5, there exists a c ∈ R + such that inf P ∈P n E P u ( X n ) + α ( P ) o = inf P ∈P c n E P u ( X n ) + α ( P ) o for all n. Using Theorem 2 of [10] once more, we obtain sup n inf P ∈P n E P u ( X n ) + α ( P ) o = inf P ∈P c sup n n E P u ( X n ) + α ( P ) o ≥ inf P ∈P n E P u ( X ) + α ( P ) o , which by monotonicity, gives sup n inf P ∈P n E P u ( X n ) + α ( P ) o = inf P ∈P n E P u ( X ) + α ( P ) o . (3.5)Since, for a given strategy ϑ ∈ ˜Θ , P Tt =1 ϑ t ∆ S t belongs to C Z , we get from (3.5) that sup n inf P ∈P E P u X n + T X t =1 ϑ t ∆ S t ! = inf P ∈P E P u X + T X t =1 ϑ t ∆ S t ! , which, due to ˜ U ( X ) = sup ϑ ∈ ˜Θ inf P ∈P E P u (cid:16) X + P Tt =1 ϑ t ∆ S t (cid:17) , implies that ˜ U satisfies (3.1) for X ∈ C Z .In particular, φ ( X ) = − ˜ U ( − X ) is an increasing convex mapping from B Z to R satisfying condition (R1)of Theorem 2.2. Moreover, φ ∗ C Z ( q Q ) = ˜ U ∗ C Z ( q Q ) ≤ φ ∗ U Z ( q Q ) = ˜ U ∗ L Z ( q Q ) := sup X ∈ L Z inf P ∈P n E P u ( X ) + α ( P ) − q E Q X o ≤ inf P ∈P { D v ( q Q k P ) + α ( P ) } = ˜ U ∗ C Z ( q Q ) for all ( q, Q ) ∈ ˆ Q Z . So it follows from Proposition 2.3 that φ satisfies condition (R2) of Theorem 2.2,which means that ˜ U satisfies (3.1).Now, we are ready to prove Theorem 1.1. Proof of Theorem 1.1 It follows from Lemma 3.7 and Theorem 2.2 that ˜ U ( X ) = min ( q, Q ) ∈ ˆ Q Z n q E Q X + D αv ( q Q ) o for all X ∈ L Z . Since, by Lemma 3.3, ˜ U ( X ) ≤ U ( X ) ≤ inf ( q, Q ) ∈ ˆ Q Z n q E Q X + D αv ( q Q ) o for all X ∈ L Z , this proves the theorem. 17 .2 Medial limits To prove Theorem 1.2 and Corollary 1.3, we need the concept of a medial limit, which for our purposes,is a positive linear functional, lim med : l ∞ → R , satisfying lim inf ≤ lim med ≤ lim sup such that forany uniformly bounded sequence X n : M → R of universally measurable functions on a measurable space ( M, F ) , X = lim med n X n is universally measurable and E P X = lim med n E P X n for every probabilitymeasure P on the universal completion F ∗ of F . It was originally shown by Mokobozki that medial limitsexist under the usual ZFC axioms and the continuum hypothesis; see [16]. Later, Normann [19] showedthat it is enough to assume ZFC and Martin’s axiom. If a medial limit exists, we extend it to R N bysetting lim med n x n := sup k ∈ N inf m ∈ N lim med n ( − m ) ∨ ( x n ∧ k ) . (3.6) Lemma 3.8. Assume a medial limit exists. Then the following hold: (i) The set L of sequences ( x n ) in R N satisfying lim med n | x n | < ∞ is a linear space. (ii) lim med : L → R is a positive linear functional. (iii) ϕ (lim med n x n ) ≤ lim med n ϕ ( x n ) for every convex function ϕ : R → R and ( x n ) ∈ L . (iv) lim med n X n is universally measurable for every sequence of universally measurable functions X n : Ω → R . (v) E P lim med n X n ≤ lim med n E P X n for each probability measure P on F ∗ and every sequence ofuniversally measurable functions X n : Ω → R such that X n ≥ c for all n and a constant c ∈ R .Proof. (i) and (ii) are simple consequences of (3.6). To show (iii), we note that by the Fenchel–Moreauxtheorem, ϕ can be written as ϕ ( x ) = sup y ∈ R xy − ϕ ∗ ( y ) for the convex conjugate ϕ ∗ of ϕ . Moreover,since lim inf ≤ lim med ≤ lim sup , one has lim med n ( x n ) = c for constant sequences x n ≡ c . So, since lim med is linear on L , one obtains ϕ (lim med n x n ) = sup y ∈ R (cid:0) lim med n x n y − ϕ ∗ ( y ) (cid:1) ≤ lim med n (cid:0) sup y ∈ R x n y − ϕ ∗ ( y ) (cid:1) = lim med n ϕ ( x n ) . (iv) follows from (3.6) since lim med n X n is universally measurable for any uniformly bounded sequenceof universally measurable functions X n : Ω → R .(v): For every k ∈ N , E P lim med n ( X n ∧ k ) = lim med n E P ( X n ∧ k ) ≤ lim med n E P X n , and therefore, by (3.6) and the monotone convergence theorem, E P lim med n X n ≤ lim med n E P X n . Lemma 3.9. Assume a medial limit exists, u fulfills (U1)–(U3) and P satisfies (NA) . Let X n : Ω → R bea sequence of Borel measurable functions decreasing pointwise to a Borel measurable function X : Ω → R such that U ( X ) ∈ R . Then U ( X n ) decreases to U ( X ) , and there exists a strategy ϑ ∗ ∈ Θ such that U ( X ) = inf P ∈P ( E P u X + T X t =1 ϑ ∗ t ∆ S t ! + α ( P ) ) . roof. Since U is bounded from above, there exists for each n , a ϑ n ∈ Θ such that inf P ∈P ( E P u X n + T X t =1 ϑ nt ∆ S t ! + α ( P ) ) ≥ U ( X n ) − n . Denote A ± t := { ω ∈ Ω : lim med n ( ϑ nt ( ω )) ± = ∞} and define ϑ ∗ t ( ω ) := ( lim med n ϑ nt ( ω ) if ω / ∈ A + t ∪ A − t otherwise.We want to show that P h lim med n | ϑ nt ∆ S t | < ∞ i = 1 for all t = 1 , . . . , T and P ∈ P . (3.7)To do that, we note that by (NA), every P ∈ P is dominated by a P ′ ∈ P that does not admit arbitrage.By the fundamental theorem of asset pricing, there exists a martingale measure Q equivalent to P ′ suchthat E Q X +1 < ∞ and d Q /d P ′ is bounded . If we can show that lim med n | ϑ nt ∆ S t | < ∞ Q -almost surely (3.8)for all t = 1 , . . . , T , (3.7) follows since Q dominates P . To prove (3.8), we set ϑ n = 0 and use an inductionargument. Fix t ≥ , and assume that (3.8) holds for all s ≤ t − .Since E P ′ u X n + T X t =1 ϑ nt ∆ S t ! + α ( P ′ ) ≥ U ( X n ) ≥ U ( X ) ∈ R , one obtains from Lemmas 3.1 and 3.2 that there exist constants c, c ′ ∈ R such that E Q T X t =1 ϑ nt ∆ S t ! − ≤ E Q X +1 − E P ′ u X n + T X t =1 ϑ nt ∆ S t ! + E P ′ v (cid:18) q d Q d P ′ (cid:19) + c ≤ E Q X +1 + α ( P ′ ) − U ( X ) + E P ′ v (cid:18) q d Q d P ′ (cid:19) + c = c ′ for all n . So it follows from Theorems 1 and 2 in [14] that P ts =1 ϑ ns ∆ S s is a Q -martingale. Consequently, (cid:0)P ts =1 ϑ ns ∆ S s (cid:1) − is a Q -submartingale, and therefore, E Q t X s =1 ϑ ns ∆ S s ! − ≤ E Q T X s =1 ϑ ns ∆ S s ! − ≤ c ′ . To see this, note that d ˜ P /d P ′ = (1 / X +1 ) / E P ′ (1 / X +1 ) defines a measure ˜ P equivalent to P ′ such that E ˜ P X +1 < ∞ . ˜ P still does not admit arbitrage. Therefore, there exists a martingale measure Q with bounded density d Q /d ˜ P ; see e.g.Theorem 5.17 in [11]. Q is equivalent to P ′ such that E Q X +1 < ∞ and d Q /d P ′ is bounded. lim med n (cid:0)P ts =1 ϑ ns ∆ S s (cid:1) − is Q -almost surely finite. Butsince ( ϑ nt ∆ S t ) − ≤ t − X s =1 ϑ ns ∆ S s ! + + t X s =1 ϑ ns ∆ S s ! − , we get from the induction hypothesis that lim med n ( ϑ nt ) ± (∆ S t ) ∓ is Q -almost surely finite. Since lim med n ( ϑ nt ) + (∆ S t ) − = ∞ on A + t ∩ { ∆ S t < } , one has Q [ A + t ∩ { ∆ S t < } ] = 0 . By the martingale property, this implies Q [ A + t ∩ { ∆ S t = 0 } ] = 0 . The same argument applied to A − t gives Q [ A − t ∩ { ∆ S t = 0 } ] = 0 . It followsthat lim med n | ϑ nt ∆ S t | < ∞ Q -almost surely, which implies (3.7).As a result, one has lim med n P Tt =1 ϑ nt ∆ S t = P Tt =1 ϑ ∗ t ∆ S t P -almost surely for all P ∈ P . Since u isincreasing, concave and bounded from above, an application of (iii) and (v) of Lemma 3.8 to − u gives U ( X ) ≥ inf P ∈P ( E P u X + T X t =1 ϑ ∗ t ∆ S t ! + α ( P ) ) ≥ inf P ∈P ( E P lim med n u X n + T X t =1 ϑ nt ∆ S t ! + α ( P ) ) ≥ inf P ∈P ( lim med n E P u X n + T X t =1 ϑ nt ∆ S t ! + α ( P ) ) ≥ lim med n inf P ∈P ( E P u X n + T X t =1 ϑ nt ∆ S t ! + α ( P ) ) = inf n U ( X n ) . By monotonicity, U ( X n ) ↓ U ( X ) and U ( X ) = inf P ∈P n E P u (cid:16) X + P Tt =1 ϑ ∗ t ∆ S t (cid:17) + α ( P ) o . Proof of Theorem 1.2 Assume a medial limit exists, u satisfies (U1)–(U3) and P fulfills (NA). Then an application of Lemma3.9 with X n = X yields that the supremum in (1.1) is attained for every Borel measurable function X : Ω → R satisfying U ( X ) ∈ R .If in addition, (A1)–(A2) hold, we know from the proof of Theorem 1.1 that φ ( X ) = − U ( − X ) is anincreasing convex mapping from B Z to R satisfying condition (R2) of Theorem 2.2 and φ ∗ C Z ( q Q ) = U ∗ C Z ( q Q ) = (cid:26) D v ( q Q , P ) if q = 0 or Q ∈ Q Z ∞ if q > and Q ∈ M Z \ Q Z . Moreover, by Lemma 3.9, φ fulfills (R3). Hence, it follows from Theorem 2.2 that U ( X ) = − φ ( − X ) = inf ( q, Q ) ∈ R + ×M Z n q E Q X + U ∗ C Z ( q Q ) o = inf ( q, Q ) ∈ ˆ Q Z n q E Q X + U ∗ C Z ( q Q ) o for all X ∈ B Z . Proof of Corollary 1.3 Note that W ( X ) = − λ log( − U ( X )) U ( X ) = sup ϑ ∈ Θ inf P ∈P E P u X + T X t =1 ϑ t ∆ S t ! and u ( x ) = − exp( − λx ) . (3.9)Clearly, u satisfies (U1)–(U3), and under the assumptions of the corollary, P together with the trivialfunction α ≡ fulfill (A1)–(A2). Therefore, it follows from Theorem 1.2 that the supremum in (3.9) isattained for all X ∈ B Z . In particular, U ( X ) ∈ ( −∞ , , and therefore, W ( X ) ∈ R for all X ∈ B Z .Furthermore, v ( y ) = sup x ∈ R { u ( x ) − xy } = yλ (cid:16) log yλ − (cid:17) , from which it follows that inf P ∈P D v ( q Q k P ) = qλ H ( Q k P ) + qλ (cid:16) log qλ − (cid:17) . So, by Theorem 1.2, W ( X ) = − λ log( − U ( X )) = − λ log (cid:16) − inf ( q, Q ) ∈ ˆ Q Z (cid:18) q E Q X + inf P ∈P D v ( q Q k P ) (cid:19) (cid:17) = − λ log (cid:16) − inf q ∈ R + (cid:16) q (cid:16) inf Q ∈Q Z ( E Q X + 1 λ H ( Q k P )) (cid:17) + qλ (cid:16) log qλ − (cid:17) (cid:17)(cid:17) . Solving for the minimizing q gives W ( X ) = inf Q ∈Q Z (cid:8) E Q X + λ H ( Q k P ) (cid:9) . A Appendix A.1 Properties of Example 1.4 Clearly, P is a convex subset of M Z . So to prove that it satisfies (A1) for α ≡ , it is enough to showthat it is σ ( M Z , C Z ) -closed. To do that, let ( P n ) be a sequence in P converging in σ ( M Z , C Z ) to a Borelprobability measure P . Then, E P [ S c i t ∧ m ] = lim n E P n [ S c i t ∧ m ] ≤ C it and E P [ S d i t ∧ m ] = lim n E P n [ S d i t ∧ m ] ≤ D it for all t = 1 , . . . , T , i = 1 , . . . , I and m ∈ N , from which it follows by monotone convergence that E P [ S c i t ] ≤ C it and E P [ S d i t ] ≤ D it for all t and i .Hence, P is σ ( M Z , C Z ) -closed.Moreover, if u : Ω × R → R is a random utility function satisfying (U1)–(U3) and there exists aconstant q < p := max ≤ i ≤ I | c i | ∧ max ≤ i ≤ I | d i | such that u ( ω, x ) / (1 + | x | q ) is bounded, then (A2) holds for α ≡ and β ( x ) = x p/q .Now, let us assume that s c i < C it and s d i < D it for all t and i . To show that P satisfies (NA), weassume for notational simplicity that T = 2 and a t = b t = 1 for t = 1 , . Then Ω can be identified with21 , ∞ ) × (0 , ∞ ) . The general case follows from similar arguments. Choose a P ∈ P and disintegrate itas P = P ⊗ K , where P is the first marginal distribution (corresponding to the distribution of S ) and K is a transition probability kernel (corresponding to the conditional distribution of S given S ). Forevery ε ∈ (0 , , denote by P ε and K ε the measure and kernel given by P ε := δ (1 − ε ) s + δ (1+ ε ) s and K εx := δ (1 − ε ) x + δ (1+ ε ) x . Then, the measure P ε := ( ε P + (1 − ε ) P ε ) ⊗ ( εK + (1 − ε ) K ε ) dominates P and does not admit arbitrage. It remains to show that P ε belongs to P for some ε > .First, note that for m = c i or d i , E P ε S m = ε E P S m + (1 − ε ) (1 − ε ) m + (1 + ε ) m s m → s m as ε → . This shows that the moment conditions for S under P ε are satisfied as soon as ε > is sufficiently small.Moreover, for m = c i or d i , one has E P ε S m = ε E P S m + ε (1 − ε ) E P ⊗ K ε S m + ε (1 − ε ) E P ε ⊗ K S m + (1 − ε ) E P ε ⊗ K ε S m . (A.1)The term E P ε ⊗ K ε S m converges to s m for ε → . So if we can show that the other expectations in (A.1)are bounded in ε , it follows that S satisfies the moment constraints for ε > small enough. The firstexpectation E P S m is independent of ε and finite since P belongs to P . The second expectation satisfies E P ⊗ K ε S m = E P (1 − ε ) m S m + E P (1 + ε ) m S m → E P S m ≤ C i for ε → . Finally, note that one can change K on a P -zero set and still have P = P ⊗ K . Therefore, one can assumethat K (1 ± ε n ) s = δ s for a sequence of positive numbers ( ε n ) converging to . Then E P εn ⊗ K S m = s m .Hence, the moment conditions for S under P ε n hold too for ε n close enough to , showing that P fulfills(NA). A.2 Properties of Example 1.5 Obviously, W ( · , P ∗ ) p , and consequently also P , are convex. Moreover, by Kantorovich duality (see e.g.Theorem 5.10 in [27]), one has W p ( P , P ∗ ) p = sup f ∈ C b (Ω) , g ∈ C b (Ω) , f + g ≤ d p (cid:16) E P f + E P ∗ g (cid:17) , from which it is easy to see that P is σ ( M Z , P Z ) -closed. This shows that (A1) holds for α ≡ .Next, note that Z ( ω ) = s + T + e ρT T − /κ d ( ω, ω ∗ ) defines a continuous function Z : Ω → [1 , ∞ ) withcompact sublevel sets { Z ≤ z } , z ∈ R + , such that Z ( ω ) ≥ s + T + T X t =1 | ϕ ( ω t, ) − ϕ ( ω ∗ t, ) | ≥ ∨ s + T X t =1 ω t, ! = 1 ∨ T X t =0 | S t | . c ∈ R + such that Z ( ω ) p ≤ c (1 + d ( ω, ω ∗ ) p ) , W p satisfies the triangleinequality (see e.g. Chapter 6 of [27]) and E P d ( · , ω ∗ ) p = W p ( P , δ ω ∗ ) p , one has E P Z p ≤ c (1 + W p ( P , δ ω ∗ ) p ) ≤ p − c (1 + W p ( P , P ∗ ) p + W p ( P ∗ , δ ω ∗ ) p ) ≤ p − c (1 + W p ( P , P ∗ ) p + E P ∗ Z p ) for all P ∈ M Z . In particular, if u ( ω, x ) / (1 + | x | q ) is bounded for a constant q < p , then E P u ( − β ( Z )) ≥ − c (1 + W p ( P , P ∗ ) p ) for β ( x ) = x p/q , a new constant c ∈ R + and all P ∈ P , showing that (A2) holds for α ≡ .To prove that P satisfies (NA), we again assume T = 2 and a t = b t = 1 for t = 1 , . The general casefollows analogously. Choose a P ∈ P and disintegrate it as P = P ⊗ K . Similarly, write P ∗ = P ∗ ⊗ K ∗ ,and define for λ ∈ (0 , , P λ := λ δ (1 − λ ) s + δ (1+ λ ) s − λ ) P ∗ and K λx := λ δ (1 − λ ) x + δ (1+ λ ) x − λ ) K ∗ x . Then the measure P λ := P λ ⊗ K λ does not admit arbitrage. Moreover, there exists a λ ∈ (0 , suchthat E P λ d ( · , ω ∗ ) p < ∞ and W p ( P λ , P ∗ ) ≤ η/ . Now choose a measure ˜ P equivalent to P and a tran-sition probability kernel ˜ K such that for all x > , ˜ K x is equivalent to K x and the three expectations E ˜ P ⊗ ˜ K d ( · , ω ∗ ) p , E ˜ P ⊗ K λ d ( · , ω ∗ ) p and E P λ ⊗ ˜ K d ( · , ω ∗ ) p are finite. For each ε ∈ (0 , , the measure P ε,λ := ( ε ˜ P + (1 − ε ) P λ ) ⊗ ( ε ˜ K + (1 − ε ) K λ ) does not admit arbitrage and P ≪ ˜ P ⊗ ˜ K ≪ P ε,λ . Since W p ( · , P ∗ ) p is convex, W p ( P ε,λ , P ∗ ) p is dominatedby ε W p (˜ P ⊗ ˜ K, P ∗ ) p + ε (1 − ε ) W p (˜ P ⊗ K λ , P ∗ ) p + (1 − ε ) εW p ( P λ ⊗ ˜ K, P ∗ ) p + (1 − ε ) W p ( P λ , P ∗ ) p . Due to E P ∗ d ( · , ω ∗ ) p < ∞ , one obtains from the triangle inequality that W p (˜ P ⊗ ˜ K, P ∗ ) ≤ W p (˜ P ⊗ ˜ K, δ ω ∗ ) + W p ( δ ω ∗ , P ∗ ) = (cid:16) E ˜ P ⊗ ˜ K d ( · , ω ∗ ) p (cid:17) /p + (cid:16) E P ∗ d ( · , ω ∗ )) p (cid:17) /p < ∞ , and similarly, W p (˜ P ⊗ K λ , P ∗ ) < ∞ as well as W p ( P λ ⊗ ˜ K, P ∗ ) < ∞ . This shows that W p ( P ε,λ , P ∗ ) ≤ η for ε > small enough, proving that P satisfies (NA). A.3 Properties of Example 1.6 It is easy to see that P and α are convex. Moreover, it follows from the arguments in Appendix A.2that all sublevel sets P c , c ∈ R + , are σ ( M Z , C Z ) -closed, and for each c > , P c satisfies (NA). So (A1)holds and P fulfills (NA). Finally, if u ( ω, x ) / (1 + | x | q ) is bounded for a constant q < p , one obtains as inAppendix A.2 that E P u ( − β ( Z )) ≥ − c (cid:16) W ( p + q ) / ( P , P ∗ ) ( p + q ) / (cid:17) ≥ − c (cid:16) W p ( P , P ∗ ) ( p + q ) / (cid:17) (A.2)for β ( x ) = x ( p + q ) / q , a constant c ∈ R + and all P ∈ P . This shows that inf P ∈P n E P u ( − β ( Z )) + ηW p ( P , P ∗ ) p o > −∞ , and (A2) holds. 23 eferences [1] J. Backhoff and J. Fontbona (2016). Robust utility maximization without model compactness. SIAMJ. Fin. Math. 7(1), 70–103.[2] D. Bartl (2018). Exponential utility maximization under model uncertainty for unbounded endow-ments. Forthcoming in Ann. Appl. Prob.[3] D. P. Bertsekas and S. E. Shreve (1978). Stochastic Optimal Control: The Discrete Time Case.Academic Press New York.[4] R. Blanchard and L. Carassus (2018). Robust optimal investment in discrete time for unboundedutility function. Ann. Appl. Proba. 28(3), 1856-1892.[5] G. Bordigoni, A. Matoussi and M. Schweizer (2007). A stochastic control approach to a robust utilitymaximization problem. in Stochastic Analysis and Applications, Abel Symposium 2, 125–151.[6] B. Bouchard, M. Nutz (2015). Arbitrage and duality in nondominated discrete-time models. Ann.Appl. Proba. 25(2), 823–859.[7] P. Cheridito, F. Delbaen and M. Kupper (2006). Dynamic monetary risk measures for boundeddiscrete-time processes. Electr. J. Probab. 11, 57–106.[8] P. Cheridito and M. Kupper (2006). Composition of time-consistent dynamic monetary risk measuresin discrete time. Int. J. Theor. Appl. Fin. 14(1), 137–162.[9] G. Choquet (1959). Forme abstraite du théorème de capacitabilité. Annales de L’Institut Fourier 9,83–89.[10] K. Fan (1953). Minimax theorems. Proc. Nat. Academy of Sciences 39(1), 42–47.[11] H. Föllmer and A. Schied (2004). Stochastic Finance. An Introduction in Discrete Time. Walter deGruyter GmbH & Co., Berlin, New York.[12] A. Gundel (2005). Robust utility maximization for complete and incomplete market models. Financeand Stochastics 9(2), 151–176.[13] L.P. Hansen and T.J. Sargent (2001) Robust control and model uncertainty. Am. Econ. Rev. 91,60–66.[14] J. Jacod and A. N. Shiryaev (1998). Local martingales and the fundamental asset pricing theoremsin the discrete-time case. Finance and Stochastics 2(3), 259–273.[15] A. Matoussi, D. Possamaï and C. Zhou (2015). Robust utility maximization in nondominated modelswith 2BSDE: the uncertain volatility model. Math. Fin. 25(2), 258–287.[16] P. A. Meyer (1973). Limites médiales d’après Mokobozki. Séminaire de Probabilités 7, 198–204.[17] A. Neufeld and M. Nutz (2018). Robust utility maximization with Lévy processes. Math. Fin. 28 (1),82–105. 2418] A. Neufeld and M. Šikić (2018). Robust utility maximization in discrete-time markets with friction. SIAM J. Contr. Optim. 56(3), 1912–1937.[19] D. Normann (1976). Martin’s axiom and medial functions. Math. Scand. 38, 167–176.[20] M. Nutz (2014). Superreplication under model uncertainty in discrete time. Fin. Stoch. 18(4), 791–803.[21] M. Nutz (2016). Utility maximization under model uncertainty in discrete time. Math. Fin. 26(2),252–268.[22] K. Owari (2011). Robust utility maximization with unbounded random endowment. Adv. Math. Econ.14, 147–181.[23] M. C. Quenez (2004). Optimal portfolio in a multiple-priors model. Sem. Stoch. Analysis, RandomFields and Appl. IV. Birkhäuser, Basel.[24] A. Schied (2005). Optimal investments for robust utility functionals in complete market models. Math. Oper. Res. 30(2), 750–764.[25] A. Schied (2007). Optimal investments for risk- and ambiguity-averse preferences: a duality approach. Fin. Stoch. 11, 107–129.[26] H. Tong (1952).