A Set Optimization Approach to Utility Maximization Under Transaction Costs

A set optimization approach to utility maximization under transaction costs

Andreas H. Hamel,∗ Sophie Qingzhen Wang†

Abstract A set optimization approach to multi-utility maximization is presented, and duality results are obtained for discrete market models with proportional transaction costs. The novel approach admits to obtain results for non-complete preferences, where the formulas derived closely resemble but generalize the scalar case.

Keywords. utility maximization, non-complete preference, multi-utility representation, set optimization, duality theory, transaction costs

JEL classiﬁcation. C61, G11

1 Introduction

In this note, we propose a set-valued approach to utility maximization for market models with transaction costs. For finite probability spaces and a one-period set-up, we derive results which resemble very closely the scalar case as discussed in [4, Theorem 3.2.1]. This is far beyond other approaches in which only scalar utility functions are used as, for example, in [1, 3], where a complete preference for multivariate position is assumed. As far as we are aware of, there is no argument justifying such strong assumption, and it does not seem appropriate for market models with transaction costs. On the other hand, recent results on multi-utility representations as given, among others, in [5] lead to the question how to formulate and solve an “expected multi-utility” maximization problem. The following optimistic goal formulated by Bosi and Herden in [2] does not seem achievable since, in particular, there is no satisfactory multi-objective duality which matches the power of the scalar version: ‘Moreover, as it reduces finding the maximal elements in a given subset of X with respect to to a multi-objective optimization problem (cf. Evren and Ok, 2011), in applications, this approach is likely to be more useful than the Richter-Peleg approach.’ The question remains how duality could work in a multi-utility framework. Using a simple model with finitely many utility functions as an exemplary case, our answer is that it works almost in the same way as in the scalar case if set optimization methods are used.

∗Free University Bozen, Faculty of Economics and Management, [email protected] †Harvard University Department of Economics, [email protected]

1 Our approach is different, and it has several advantages. First, we demonstrate that the “usual” duality can be established if one allows for a set-valued extension of the problem. This means that the vector-valued problem is replaced by a problem where the objective function maps into a well-defined complete lattice of sets. Secondly, the problem formulation allows to separate the market model, namely the time dependent solvency cone, from the the preference relation expressed by a (vector-valued) utility function and provides a link to no-arbitrage type results for market models with transaction costs as, for example, given in [16]. Our model can be interpreted as follows: The decision maker has a complete preference for random positions of each asset in the market separately which is representable by a utility function (the classic von Neumann-Morgenstern set-up). These preferences work “component-wise,’ i.e. they are determined independently of the other assets. This may seem very restrictive and is indeed not the most general situation. However, there are two arguments in favor of studying it. First, it admits non-complete preferences which goes beyond the above quoted references and actually provides a path for a solution of the corresponding (multi-)utility maximization problem. Secondly, via the market model, an exchange mechanism is built into the model which makes the components of the portfolio dependent on each other when one looks for best alternatives. The underlying idea is that one can exchange assets, but cannot exchange utility of one asset for utility of another. The main tool is a set-valued Lagrangian, to which recent duality results for set-valued functions from [10] are applied (compare also the survey [8]). The resulting formulas look very much like the scalar ones, a feature that is nearly impossible to achieve if one would only apply multi-objective optimization duality. The results are based on the senior thesis of the second author written in 2011 at Princeton University and supervised by the first author and B. Rudloff (now at Vienna University for Economics and Business).

2 Problem formulation

Let (Ω,P ) be a ﬁnite probability space where the σ-algebra is assumed to be the power set of Ω = {ω1, ω2, . . . , ωN } with pn := P ({ωn}) > 0 for all n ∈ {1,...,N}.

A one-period conical model for a market with d assets is given by a pair (K0,KT = KT (ω)) d d d d of ﬁnitely generated convex cones satisfying IR+ ⊆ K0 6= IR and IR+ ⊆ KT (ω) 6= IR for all ω ∈ Ω. The cones K0, KT are called solvency cones and may arise, for example, due to explicit proportional transaction costs (see [13]) or bid-ask spreads (see [16]). d d 0 The linear space of IR -valued random variables X :Ω → IR is denoted by Ld := 0 dN Ld(Ω,P ) which can be identiﬁed with IR . Further, denote

0 0 Ld (KT ) = X ∈ Ld | ∀ω ∈ Ω: X (ω) ∈ KT (ω) ,

0 0 d 0 and Ld + = Ld IR+ . Finally, 1I ∈ Ld stands for the function with 1I(ω) = 1 for all ω ∈ Ω. In this note, it is assumed that d utility functions Ui : IR → IR ∪ {−∞}, i = 1, . . . , d are given which are concave and non-decreasing. We deﬁne a vector-valued function U : IRd →

2 IRd ∪ {−∞} by ( (U (x ) ,U (x ) ,...,U (x ))T : ∀i ∈ {1, 2, . . . , d} : x ∈ dom U U (x) = 1 1 2 2 d d i i −∞ : otherwise This can be interpreted as follows. The investor has a scalar utility function for each of the d assets which is independent from holdings in the others, and a portfolio is not acceptable for the investor if one asset produces a utility −∞. Of course, this means that the investor has a complete preference for random positions of each asset which admits a von Neumann- Morgenstern representation. We note that even this might be a strong assumption and refer to [6, Chap. 2] for further details. However, this assumption is in some respect much less strong than the existence of a function Uˆ : IRd → IR ∪ {−∞} which represents a complete preference (a total order) on the set of all multivariate positions. This is the starting point e.g. in the recent [1, 3] d Since the Ui’s are concave and non-decreasing, the function U is IR+-concave, that is 0 d 0 0 d ∀t ∈ (0, 1) , ∀x, x ∈ IR : U (tx + (1 − t) x ) ∈ tU (x) + U ((1 − t) x ) − IR+, d and U is IR+-monotone increasing, that is 0 d 0 d x ∈ x + IR+ ⇒ U (x) ∈ U (x ) + IR+ where we agree upon x + (−∞) = (−∞) + x = −∞, t (−∞) = −∞ for all t > 0 and −∞ ≤ x for all x ∈ IRd. The basic problem of this note is

0 maximize IE[U (X)] subject to X ∈ x1I − K01I − Ld (KT ) where x ∈ IRd is the given initial endowment and the expected value is understood component- wise with IE[−∞] = −∞. 0 The set x1I − K01I − Ld (KT ) is precisely the set of (random) portfolios which can be obtained by trading at time t = 0 and time t = T starting with the initial endowment x ∈ IRd. The first question is how the maximization is understood. In contrast to (more popular) vector approaches, we will understand the above problem as a set-valued one. The advantage of this approach is that the notions of supremum and infimum make sense, and consequently, that a complete duality theory is available. One may compare the appendix and the survey [8] for details. We define the set

d d n d d o G(IR , −IR+) = A ⊆ IR | A = cl co A − IR+

d d and introduce an addition on G IR , −IR+ by ( cl {a1 + a2 | a1 ∈ A1, a2 ∈ A2} : A1,A2 6= ∅ A1 ⊕ A2 = , ∅ : otherwise and a multiplication · with non-negative real numbers by  −IRd : t = 0  + t · A = {ta | a ∈ A} : t > 0,A 6= ∅  ∅ : t > 0,A = ∅

3 d d On G IR , −IR+ , the inclusion ⊆ is a partial order which is compatible with the two d d algebraic operations just defined, thus the quadruple G IR , −IR+ , ⊕, ·, ⊆ is an ordered conlinear space in the sense of [7, 8]. We shall denote this structure by GO and drop the · for multiplication in most cases. The infimum and supremum of a set A ⊆ GO are given by [ \ sup A = cl co A, inf A = A A∈A A∈A which are clearly elements of GO, thus GO even is a complete lattice (see [7, 8]). Here, we agree upon sup A = ∅ in GO for A = ∅. Our basic problem now is to find, in GO,

n d 0 o sup IE[U (X)] − IR+ | X ∈ x1I − K01I − Ld (KT ) (UMAX)

d where we use the convention IE[U (X)] − IR+ = ∅ whenever IE[U (X)] = −∞. The plan of the remainder of the paper is as follows. In the next section, we will give a dual characterization of the constraint of (UMAX). Then, we introduce a Lagrangian function, deﬁne primal and dual value functions and establish a strong duality theorem which provides the essentials for a solution of (UMAX). The reader may compare [4, Chapter 3] for a concise presentation for the scalar case d = 1. It is the main purpose of this note to demonstrate that the vector-valued case d > 1 can be dealt with by means of set-valued duality as proposed in [7, 10], and that in doing so one obtains meaningful analogs to the well-known scalar formulas. Such a theory is not really possible in terms of the vector order since the supremum (or inﬁmum) with respect to a vector order usually does not make sense since it produces “utopia” solutions which are not feasible in general.

3 Constraints transformation

Defining 0 C = K01I+ Ld (KT ) we want to describe C by means of dual variables. Note that in our setting C is always closed since it can be identified with a finitely generated cone in IRdN (see [15, Theorem 19.1]). In a more general situation, the closedness of the set of all terminal positions follows from a no-arbitrage type condition, see for example [16, Theorem 2.1]. T 0 0 0 Using the duality pairing (X,Y ) 7→ IE Y X on Ld × Ld, Ld is turned into a Hilbert space which can be identified with its dual, and for this topological setting, we obtain

0 + 0 + 0 + Ld (KT ) = Ld KT = Y ∈ Ld | ∀ω ∈ Ω: Y (ω) ∈ KT (ω) where + denotes the (positive) dual cone (see [17, p. 7]). Compare [9, Lemma 3.1].

d 0 Lemma 3.1 For x ∈ IR , X ∈ Ld, T T x1I − X ∈ C ⇔ ∀ (Y, v) ∈ Yd : IE Y X ≤ v x where n 0 d + 0 + o Yd = (Y, v) ∈ Ld × IR | v ∈ K0 \{0},Y ∈ Ld KT , IE[Y ] = v .

4 Proof. By a separation argument, one may see that the dual cone of C is

+ 0 0 + + C = Y ∈ Ld | Y ∈ Ld KT ∧ IE[Y ] ∈ K0 .

Since C is a closed convex cone the bipolar theorem produces

x1I − X ∈ C ⇔ ∀Y ∈ C+ : IE Y T (x1I − X) = xT IE[Y ] − IE Y T X ≥ 0.

+ 0 + The set Yd is in one-to-one relationship with C \{0}. Indeed, Y ∈ Ld KT implies Y ∈ 0 d + d Ld + and this in turn IE[Y ] ≥ 0 since IR+ ⊆ KT (ω), hence KT (ω) ⊆ IR+ for all ω ∈ Ω. Thus, v = IE[Y ] = 0 implies Y = 0. The rest is obvious.

By means of Lemma 3.4 in [9] it can be shown that the set Yd also is in one-to-one relationship with the set dQ W = (Q, w) ∈ MP × IRd | w ∈ K+\{0}, diag (w) ∈ L0 K+ . d 1,d 0 dP d T

dQ P This can be arranged by Y = diag (w) dP , IE[Y ] = w. Here, M1,d is the set of all vector probability measures with components which are absolutely continuous with respect to P . Thus, we can write the utility maximization problem as

n d T Q T o sup IE[U (X)] − IR+ | ∀ (Q, w) ∈ Wd : w IE [X] ≤ w x .

Under our assumptions, we can reduce the number of constraints in the above problem to a

finite number. This can be done due to the fact that the cones K0 and KT (ω) are assumed to be finitely generated. Therefore, the cone C can be identified with a finitely generated cone in IRdN . Hence (see [15, Corollary 19.2.2]) the cone C+ in IRdN is finitely generated by, 1 2 M 1 1 M M say, Y ,Y ,...,Y ∈ Yd. Let Q , w ,..., Q , w ∈ Wd be the pairs corresponding to Y 1, IE Y 1 ,..., Y M , IE Y M . Then

m x1I − X ∈ C ⇔ ∀m ∈ {1,...,M} :(wm)T IEQ [X] ≤ (wm)T x.

We can further simplify the problem by observing that, for (Q, w) ∈ Wd,

N T Q T X T T n T w IE [X] ≤ w x ⇔ w qˆ(n)ξ ≤ w x n=1

n d where ξ = X (ωn) ∈ IR , qin = Qi ({ωn}) for i = 1, . . . , d, n = 1,...,N and   q1n 0 ··· 0  . .  d×d qˆ(n) =  . .  ∈ IR .   0 ··· 0 qdn Using this notation, the utility maximization problem can be reformulated as

N X n d sup pnU (ξ ) − IR+ n=1 N X m T n m T subject to (w ) qˆ(n)ξ ≤ (w ) x, m = 1, 2,...,M. n=1

5 The M linear constraints can be seen as a generalization of the budget constraints in the scalar case: The wm’s are frictionless “price systems” at initial time, thus the initial endowment x is evaluated according to all (generating) frictionless initial price systems which are possible without admitting arbitrage. The same applies for the future prices of the claim X expressed via equivalent (vector) martingale measures Qm. The reader may compare the problem above to the scalar problem in [4, Section 3.2, p. 41]. We deﬁne the GO-valued “budget function” u: IRd → GO associated with the primal problem by

( N N ) X n d X m T n m T u (x) = sup pnU (ξ ) − IR+ | (w ) qˆ(n)ξ ≤ (w ) x, m = 1, 2,...,M n=1 n=1 where the supremum is understood in GO. The budget function u produces the maximal utility which can be generated given the initial endowment (budget), hence the name. In the ﬁnance community, the scalar counterpart of this function is usually called the value function of the utility maximization problem, see [4, p. 34]. We call it budget function since this function is not the value function which is usually associated with a (constrained) optimization problem (for example, in the sense of [15] or [17, Section 2.6]).

4 Lagrangian duality and main theorem

4.1 The Lagrangian approach

In this section, we shall propose a Lagrangian approach which is inspired by [10]. Adapting the constructions from [10] to the present concave setting we deﬁne the Lagrangian for the O dN M d O above G -valued problem as the function L: IR × IR+ × IR+\{0} → G given by

N N ! X n X n L (ξ, λ, η) = pnU (ξ ) + S(λ,η) A(n)ξ − W x n=1 n=1

1 1 1 N N N T dN where ξ = ξ1 , ξ2 , . . . , ξd, . . . , ξ1 , ξ2 , . . . , ξd ∈ IR ,

 1 1   1T 1  w1 ··· wd w qˆ(n)  . .  M×d  .  M×d W =  . .  ∈ IR ,A(n) =  .  ∈ IR , n = 1, 2,...,N,     M M M T M w1 ··· wd w qˆ(n)

n d T T o and S(λ,η) (x) = z ∈ IR | λ x + η z ≤ 0 . The set-valued functions x 7→ S(λ,η) (x) are additive and positively homogeneous (see [7, Proposition 6]). They serve as replacements for continuous linear function in a set-valued duality theory. Compare [7, 8] for motivation and further properties. The additional dual variable η ranges in the dual cone of the ordering + d d d cone which here is IR+ = IR+. The case η = 0 ∈ IR is excluded because the function S(λ,0) is improper.

6 Since the function S(λ,η) is additive we can write

N ! N X n X n S(λ,η) A(n)ξ − W x = S(λ,η) A(n)ξ + S(λ,η) (−W x) n=1 n=1 N X n = S T (ξ ) + S(λ,η) (−W x) . A(n)λ,η n=1 We may rewrite the Lagrangian as follows

N X h n n i L (ξ, λ, η) = pn U (ξ ) + S 1 A ,η (ξ ) + S(W T λ,η) (−x) . (4.1) ( pn (n) ) n=1 The ﬁrst result shows that the original problem can be reconstructed from the Lagrangian, i.e. there is no loss of information by passing to a Lagrangian version of the original problem.

Proposition 4.1 It holds

 N P n d PN n  pnU (ξ ) − IR+ : n=1 A(n)ξ ≤IRM W x, inf L (ξ, λ, η) = n=1 + λ∈IRM \{0}, η∈IRd \{0} + +  ∅ : otherwise

PN n M Proof. First, assume n=1 A(n)ξ − W x ∈ −IR+ . Then

N ! M T X n ∀λ ∈ IR+ \{0}: λ A(n)ξ − W x ≤ 0, n=1 hence N ! M X n ∀λ ∈ IR+ \{0}: S(λ,η) A(n)ξ − W x ⊇ H (η) n=1 which gives

( N ) N X n X n d inf L (ξ, λ, η) ⊇ inf pnU (ξ ) + H (η) = pnU (ξ ) − IR+. λ∈IRM \{0}, η∈IRd \{0} η∈IRd \{0} + + + n=1 n=1

On the other hand, using e = (1, 1,..., 1)T ∈ IRM we obtain

inf L (ξ, λ, η) ⊆ inf L (ξ, εe, η) M d d λ∈IR+ \{0}, η∈IR+\{0} ε>0, η∈IR+\{0} ( N N ! ) X n T X n T = inf z + pnU (ξ ) | εe A(n)ξ − W x + η z ≤ 0 η∈IRd \{0}, ε>0 + n=1 n=1 ( N ) N X n X n d = inf pnU (ξ ) + H (η) = pnU (ξ ) − IR+. η∈IRd \{0} + n=1 n=1

PN n M Secondly, assume n=1 A(n)ξ −W x 6∈ −IR+ . Then, by a standard separation argument, there exists λˆ ∈ IRM \{0} such that

N ! M ˆT ˆT X n ∀η ∈ IR+ : λ η ≥ 0 and λ A(n)ξ − W x > 0. n=1

7 ˆ M ˆT PN n These conditions imply kλ ∈ IR+ \{0} for all k ∈ {1, 2, 3,...} and lim kλ n=1 A(n)ξ − W x = k→∞ +∞. Consequently,

" N N !# \ X n X n inf L (ξ, λ, η) ⊆ pnU (ξ ) + S kλ,ηˆ A(n)ξ − W x = ∅ λ∈IRM \{0}, zη∈IRd \{0} ( ) + + d n=1 n=1 k∈IN\{0}, η∈IR+\{0}

d since for each η ∈ IR+\{0}

N ! ( N ! ) \ X n \ d ˆT X n T S(kλ,ηˆ ) A(n)ξ − W x = z ∈ IR | kλ A(n)ξ − W x + η z ≤ 0 = ∅. k∈IN\{0} n=1 k∈IN\{0} n=1

This completes the proof. We note that λ = 0 can be dropped from the set of dual variable. This seems to constitute only a minor difference to the general result in [10], but will become important later on. The dual problem is to minimize the supremum of the Lagrange function, the latter taken M d over the primal variables, thus the dual objective becomes the function K : IR+ ×IR+\{0} → GO given by [ K (λ, η) = sup L (ξ, λ, η) = cl L (ξ, λ, η) . ξ∈IRdN ξ∈IRdN Since the function ξ 7→ L (ξ, λ, η) is concave, the convex hull in the right hand side can be S dropped as one easily checks the convexity of the set ξ∈IRdN L (ξ, λ, η) for fixed λ, η. Using (4.1) we obtain (compare the appendix for a definition of U ∗)

N ! [ X n n K (λ, η) = cl p U (ξ ) + S (−ξ ) + S T (−x) n − 1 AT λ,η (W λ,η) pn (n) ξ1,...,ξN ∈IRd n=1 N X [ n n = p cl U (ξ ) + S (−ξ ) + S T (−x) n − 1 AT λ,η (W λ,η) pn (n) n=1 ξn∈IRd N X ∗ 1 T = p (−U ) − A λ, η + S T (−x) . n p (n) (W λ,η) n=1 n The dual problem becomes

M d minimize K (λ, η) subject to λ ∈ IR+ \{0}, η ∈ IR+\{0}.

In fact, we are looking for \ n o inf K (λ, η) = K (λ, η) | λ ∈ IRM \{0}, η ∈ IRd \{0} , d M + + η∈IR+\{0}, λ∈IR+ \{0}

M d and for a set ∆ ⊆ IR+ \{0} × IR+\{0} in which this inﬁmum is attained. The principal duality results read as follows. Deﬁning a function v∗ : IRd 7→ GO by

v∗ (x) = inf K (λ, η) d M η∈IR+\{0}, λ∈IR+ \{0} we start with weak duality (the star in v∗ will be explained later).

8 Proposition 4.2 For all x ∈ IRd, u (x) ⊆ v∗ (x).

dN M d Proof. For all ξ ∈ IR , for all (λ, η) ∈ IR+ \{0} × IR+\{0}, we certainly have \ [ L (ξ, λ0, η0) ⊆ cl L (ξ0, λ, η) ,

0 0 M d 0 dN (λ ,η )∈IR+ \{0}×IR+\{0} ξ ∈IR hence [ \ \ [ cl L (ξ, λ, η) ⊆ cl L (ξ, λ, η) . dN M d M d dN ξ∈IR (λ,η)∈IR+ \{0}×IR+\{0} (λ,η)∈IR+ \{0}×IR+\{0} ξ∈IR

The left hand side of this inclusion is u (x) according to Proposition 4.1, and its right hand ∗ side is v (x) by deﬁnition. The strong duality result reads as follows.

Theorem 4.3 For all x ∈ int (dom U), strong duality holds, that is u (x) = v∗ (x). More- d d M over, for each η ∈ IR+\{0} with u(x) ⊕ H(η) 6= IR there is λ(η) ∈ IR+ such that

u (x) ⊕ H(η) = K (λ(η), η) .

n d 0o d If u(x) ⊕ H(η) 6= sup IE[U (X)] − IR+ | X ∈ Ld ⊕ H(η) whenever u(x) ⊕ H(η) 6= IR , then λ(η) 6= 0.

Proof. In order to apply the Lagrange duality theorem [10, Theorem 6.1] we have to verify the Slater condition, i.e. we need to ﬁndx ¯ ∈ dom U ⊆ IRd satisfying

N X M A(n)x¯ − W x ∈ −int IR+ . n=1

If x ∈ int (dom U) then x−εe ∈ dom U whenever ε > 0 is small enough and e = (1, 1,..., 1)T ∈ d m + d m T IR . For all m ∈ {1,...,M}, w ∈ K0 \{0} ⊆ IR+\{0} and hence (w ) (εe) < 0. Deﬁning x¯ = x − εe we obtain

m ∀m ∈ {1,...,M} :(wm)T IEQ [¯x1I]= (wm)T x¯ < (wm)T x

PN M which is equivalent to n=1 A(n)x¯ − W x ∈ −int IR+ . Now, the result follows from Theorem 6.1 in [10]. n d 0o The last claim follows from K (0, η) = sup IE[U (X)] − IR+ | X ∈ Ld ⊕ H(η). The condition that excludes λ(η) = 0 means of course that the constraints inﬂuence the optimal value. If it is satisﬁed we call the constraints relevant (for the problem).

4.2 Transformation of the Lagrange multipliers

As it is custom in the scalar case, we shall carry out a transformation of variables. This will be done by means of the following result.

9 M Lemma 4.4 Let λ ∈ IR+ \{0} be given. Then, there is a pair (Q, y) ∈ Wd such that

dQ 1 T ∀n ∈ {1,...,N} : diag (y) (ωn) = An λ. (4.2) dP pn M Conversely, if (Q, y) ∈ Wd then there is λ ∈ IR+ \{0} such that (4.2) is satisﬁed.

M M T P m Proof. If λ ∈ IR+ \{0} is given set y = W λ = λmw and m=1

 M 1 P m m  λmw q : yi > 0 yi i in qin = m=1  pn : otherwise for i = 1, . . . , d, n = 1,...,N. One directly checks formula (4.2). Since y is a non-negative m + linear combination of the w ’s, we have y ∈ K0 . A little algebra shows

M dQ X dQm diag (y) (ω ) = λ diag (wm) (ω ) dP n m dP n m=1 + which is a non-negative linear combination of elements of KT (ωn). m dQm The converse follows from the fact that the vectors diag (w ) dP (ωn), m = 1,...,M, generate the cone C+, and every element of C+ is a non-negative linear combination of these vectors. Since dQm IE diag (wm) = wm ∈ K+ dP 0 the result follows.

By means of this lemma we can re-write the Lagrangian in terms of (Q, y) ∈ Wd. In view of (4.1) and by a slight abuse of notation we just re-deﬁne

N X h n i L (ξ, Q, y, η) = pn U (ξ ) + S dQ (ξn) + S(y,η) (−x) (4.3) (diag(y) dP (ωn),η) n=1

dN d for ξ ∈ IR ,(Q, y) ∈ Wd, η ∈ IR+\{0}. Similarly, we obtain a new expression for the dual objective in terms of the (Q, y)’s: [ K (Q, y, η) = sup L (ξ, Q, y, η) = cl L (ξ, Q, y, η) . ξ∈IRdN ξ∈IRdN Using the Lagrangian in the form given in (4.3) we obtain

N ! [ X h n n i K (Q, y, η) = cl pn U (ξ ) + S dQ (ξ ) + S(y,η) (−x) (diag(y) dP (ωn),η) ξ1,...,ξN ∈IRd n=1 N X [ h n n i = pn cl U (ξ ) + S dQ (−ξ ) + S(y,η) (−x) (−diag(y) dP (ωn),η) n=1 ξn∈IRd N X dQ = p (−U ∗) −diag (y) (ω ) , η + S (−x) n dP n (y,η) n=1 dQ = IE (−U ∗) −diag (y) , η + S (−x) . dP (y,η)

10 The dual problem becomes

d minimize K (Q, y, η) subject to (Q, y) ∈ Wd, η ∈ IR+\{0}.

In fact, we are looking for

\ n d o inf K (Q, y, η) = K (Q, y, η) | (Q, y) ∈ Wd, η ∈ IR \{0} d + η∈IR+\{0}, (Q,y)∈Wd

d and for a set ∆ ⊆ Wd × IR+\{0} which generates this inﬁmum. Next, we transform the duality results into the new form. Deﬁne the convex cone D+ = nPM m o d m=1 smw | s1, . . . , sM ≥ 0 and for each y ∈ IR the set

d d MP (y) = Q ∈ MP | (Q, y) ∈ Wd .

d d O Finally, we introduce the function (−v): IR × IR+\{0} → G by

( h ∗ dQ i + infQ∈Md (y) IE (−U ) diag (y) , η : y ∈ D \{0} (−v)(y, η) = P dP IRd : otherwise where we agree that the inﬁmum over the empty set is IRd. Now, we can re-write the dual problem as

( ) ∗ dQ inf inf IE (−U ) −diag (y) , η + S(y,η) (−x) d + d η∈IR+\{0}, y∈D \{0} Q∈MP (y) dP ∗ = inf (−v)(−y, η) + S(−y,η) (x) = v (x) . (4.4) d + η∈IR+\{0}, y∈D \{0}

The quantity v∗ (x) is the dual optimal value (in the sense of convex analysis). Recall that u (x) is the primal optimal value. The strong duality result now reads as follows.

Corollary 4.5 Let the conditions of Theorem 4.3 be satisﬁed including the relevance. Then, ∗ d for each x ∈ int (dom U), u (x) = v (x). Moreover, for each η ∈ IR+\{0} satisfying u (x) ⊕ d ˆ d + H (η) 6= IR there is Q (η) , yˆ(η) ∈ MP (ˆy (η)) × D \{0} such that

u (x) ⊕ H(η) = (−v)(−yˆ(η) , η) ⊕ S(ˆy(η),η) (−x) , (4.5) " !# dQˆ (η) (−v)(−yˆ(η) , η) = IE (−U ∗) diag (−yˆ(η)) , η . (4.6) dP

d d ˆ Proof. Fix η ∈ IR+\{0} satisfying u (x) ⊕ H(η) 6= IR . Theorem 4.3 provides λ = ˆ M λ(η) ∈ IR+ \{0} such that u (x) ⊕ H (η) = K λ,ˆ η .

ˆ Transforming λ into a pair Q,ˆ yˆ ∈ Wd and using the (Q, y)-version of K produces

" !# dQˆ (η) u (x)⊕H(η) = K Qˆ (η) , yˆ(η) , η = IE (−U ∗) −diag (ˆy (η)) , η +S (−x) . dP (ˆy(η),η)

11 On the other hand, weak duality and (4.4) imply

dQ ∀Q ∈ Md (ˆy (η)) : u (x) ⊕ H(η) ⊆ IE (−U ∗) −diag (ˆy (η)) , η + S (−x) . P dP (ˆy(η),η)

The last two relationships imply

" !# dQˆ (η) u (x) ⊕ S (x) = IE (−U ∗) −diag (ˆy (η)) , η (ˆy(η),η) dP dQ = inf IE (−U ∗) −diag (ˆy (η)) , η = (−v) (ˆy (η) , η) d Q∈MP (ˆy) dP which produces (4.5) as well as (4.6).

Corollary 4.6 Assume x ∈ int (dom U), η ∈ IRd\{0} and u (x) ⊕ H(η) 6= IRd. Moreover, let yˆ(η) be as in Corollary 4.5. Then (−u∗)(−yˆ(η) , η) = (−v)(−yˆ(η) , η).

Proof. Using (4.5) we obtain

∗ 0 0 (−u )(−yˆ(η) , η) = sup u (x ) + S(ˆy(η),η) (x ) x0∈IRd

⊇ u (x) + S(ˆy(η),η) (x) = (−v)(−yˆ(η) , η) .

On the other hand, the deﬁnition of −u∗ and weak duality produce

∗ 0 0 (−u )(−yˆ(η) , η) = sup u (x ) + S(ˆy(η),η) (x ) x0∈IRd 0 0 0 0 ⊆ sup inf (−v)(−y , η ) + S(y0,η0) (−x ) + S(ˆy(η),η) (x ) d x0∈IRd y0∈D+\{0}, η0∈IR \{0} ⊆ sup {(−v) (ˆy (η) , η) + H(η)} = (−v) (ˆy (η) , η) x0∈IRd where the last inclusion is obtained by picking y0 =y ˆ(η), η0 = η in the penultimate line. Altogether, this produces the desired result.

Remark 4.7 In the math ﬁnance/economics sense [4, Chapter 3], the function u is the ”primal” and −v corresponds to the ”dual value function.” The previous corollaries tell us that they are conjugate to each other. Moreover, they are also linked via the (set-valued) conjugacy relation between U and −U ∗. We would like to emphasize the point that this could only be achieved via the set optimization approach as surveyed in [8] (compare the appendix).

d + d Remark 4.8 The set of all (Q, y, z) ∈ MP (y) × D × IR+ satisfying

u (x) ⊕ H (z) 6= IRd dQ u (x) ⊕ H (z) = K (Q, y, η) = IE (−U ∗) diag(−y) , η + S (−x) . dP (y,η) forms a solution of the dual problem in the sense of [10, Corollary 6.1]. Under the strong duality assumptions, this set is non-empty.

12 Remark 4.9 As a consequence of (4.5) and Corollary 4.6 we get

∗ u (x) ⊕ H(η) = (−u )(−yˆ(η) , η) ⊕ S(ˆy(η),η) (−x) which very much looks like the Young-Fenchel inequality satisfied as an equation. It is well- known in the scalar theory that, provided that u is proper, closed and concave, this is equivalent to two (equivalent) subdifferential conditions, one for u and one for u∗. A similar conclusion can be drawn in the set-valued case where the subdifferentials have to be understood as in [11].

5 Conclusions

It has been shown that via a set optimization approach one can lift the scalar (duality) theory for utility maximization to market models with transaction costs in which the multivariate positions X directly represent random future portfolios, i.e. the components of X represent “physical units” (see [13]) of the assets in the portfolio, not their values denoted in units of a currency. An extension to a multi-period market model now is a mere exercise and shows that the dual variables obtained through the set optimization approach via a transformation correspond to consistent price processes which turn up in no-arbitrage and super-hedging results for markets with transaction costs [16, 13]. Generalization to more general non-complete preference relations are highly desirable. A path is opened to other questions like the following: Can no-arbitrage be characterized via the existence of a solution for the set-valued utility maximization problem?

Appendix k O d d A function F : IR → G = G IR , −IR+ , ⊆ is concave if

∀s ∈ (0, 1), ∀x1, x2 ∈ dom F : F (sx1 + (1 − s)x2) ⊇ sF (x1) + (1 − s)F (x2).

The following elements of a “concave” Fenchel conjugation theory can be obtained from ∗ d d the “convex” theory in [7, 8]: the (negative) concave conjugate −F : IR × IR+\{0} → d d G IR , −IR+ of F is deﬁned by

∗ [ (−F )(y, w) = cl F (x) + S(y,w) (−x) x∈IRk

∗∗ k d d and its biconjugate F : R → G IR , −IR+ by

∗∗ \ ∗ F (x) = (−F )(y, w) + S(y,w) (x) . k d y∈IR , w∈IR+\{0}

Recall n d T T o S(y,w) (x) = z ∈ IR | y x + w z ≤ 0 ,

13 d d d and x 7→ S(y,w) (x) maps into G IR , −IR if, and only if, w ∈ IR+. The corresponding version of the Fenchel-Moreau theorem (see [8, Theorem 5.8] taken ∗∗ d d d from [7]) states that F = F if F : IR → G IR , −IR+ is proper, closed and concave. n o Closedness refers to closedness of the graph, i.e. the set graph F = (x, z) ∈ IRk × IRd | z ∈ F (x) .

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