A Framework of Multivariate Utility Optimization with General Benchmarks
aa r X i v : . [ m a t h . O C ] J a n A Framework of Multivariate Utility Optimization with GeneralBenchmarks
Zongxia Liang ∗ , Yang Liu † , and Litian Zhang ‡ Department of Mathematical Sciences, Tsinghua University, China
Abstract
Benchmarks in the utility function have various interpretations, including performance guarantees andrisk constraints in fund contracts and reference levels in cumulative prospect theory. In most literature,benchmarks are a deterministic constant or a fraction of the underlying wealth; as such, the utility is still aunivariate function of the wealth. In this paper, we propose a framework of multivariate utility optimiza-tion with general benchmark variables, which include stochastic reference levels as typical examples. Theutility is state-dependent and the objective is no longer distribution-invariant. We provide the optimalsolution(s) and fully investigate the issues of well-posedness, feasibility, finiteness and attainability. Thediscussion does not require many classic conditions and assumptions, e.g., the Lagrange multiplier alwaysexists. Moreover, several surprising phenomena and technical difficulties may appear: (i) non-uniquenessof the optimal solutions, (ii) various reasons for non-existence of the Lagrangian multiplier and corre-sponding results on the optimal solution, (iii) measurability issues of the concavification of a multivariateutility and the selection of the optimal solutions, and (iv) existence of an optimal solution not decreasingwith respect to the pricing kernel. These issues are thoroughly addressed, rigorously proved, completelysummarized and insightfully visualized. As an application, the framework is adopted to model and solvea constraint utility optimization problem with state-dependent performance and risk benchmarks.
Keywords: multivariate utility optimization, general benchmarks, non-unique solutions, state de-pendence, measurability, Lagrange multiplier
MSC(2020) : Primary: 49J55, 91B16; Secondary: 49K45, 91G80. ∗ Email: [email protected] † Email: [email protected] ‡ Corresponding Author. Email: [email protected] Introduction
Fix a probability space (Ω , F , P ). Traditionally, the framework of expected utility maximization inportfolio selection (cf. Merton (1969)) is given bysup X E [ u ( X )]subject to E [ ξX ] x and u ( X ) > −∞ a.s. (1)where X : Ω → R is a random variable representing the wealth and u : D → R is a differentiable and strictlyconcave utility function on the wealth level ( D ⊂ R is the domain of u ). The random variable ξ : Ω → (0 , + ∞ )is the so-called pricing kernel. The number x ∈ R represents the budget constraint upper bound. It is clearthat the objective E [ u ( X )] is invariant under the same distribution of X .This model has been challenged in many aspects. Practically, the portfolio manager’s utility is nolonger concave because of convex incentives in hedge funds; see Carpenter (2000) and Bichuch and Sturm(2014). The manager typically makes decisions based on the performance of some benchmark variable, e.g.,a minimum riskless money market value or a minimum stochastic performance constraint; see Basak et al.(2006). Theoretically, in the seminal work of Kahneman and Tversky (1979), the cumulative prospect theory(CPT) is proposed. It is claimed that in the utility function there should be a benchmark B , towards whichan individual displays different risk attitudes. These criticism of the classic model leads to two streams ofresearch. The first is to study a non-concave utility u . This is motivated by the S-shaped utility in the CPTand convex incentives in the hedge fund; see Kouwenberg and Ziemba (2007) and He and Kou (2018). Thesecond is to investigate various benchmarks. In literature, the model E [ u ( X − B )] is widely adopted and B is interpreted as the reference level of the wealth. Specifically, B represents the deterministic referencepoint in the S-shaped utility (cf. Liang and Liu (2020)), a fraction function of X (cf. Berkelaar et al. (2004))and the guarantee variable (cf. Boulier et al. (2001)). Other than deterministic ones, it is of importanceto study stochastic reference points; see Sugden (2003). In Bouchard et al. (2004), B is formulated as acontingent claim. Furthermore, Cairns et al. (2006) and Basak et al. (2007) respectively adopt the models E (cid:2) u (cid:0) XB (cid:1)(cid:3) and E [ u ( Xf ( X, B ))], with various specified benchmark variables and some specified function f : R × R → R . The model becomes more complicated in K˝oszegi and Rabin (2007), where a gain-loss term f ( u ( X ) − u ( B )) is introduced with another function f : R → R .In this paper, we propose a framework of multivariate utility optimization with general benchmarks:sup X E [ U ( X, B )]subject to E [ ξX ] x and U ( X, B ) > −∞ a.s. (2)Here E is a measurable space and B : (Ω , F ) → ( E, E ) is an E -valued random variable representing thebenchmark. U : R × E → R ∪ {−∞} is a multivariate utility function depending on both the wealth variable X and the benchmark variable B . 2t is worthwhile to emphasize that the framework lies on a rather general setting and the objective isno longer distribution-invariant. The benchmark B is only required to be measurable on a space E . Usually, E = R or R d and B is a deterministic function or a random variable/vector on (Ω , F , P ). Particularly, amathematically-motivated problem is that E = Ω and B = ω with ω ∈ Ω, and hence the objective becomesa state-dependent function U ( X ( ω ) , ω ); see Pliska (1986). Further, the multivariate utility function U isonly required to be non-decreasing and upper semicontinuous (hence may be discontinuous and non-concave)on X and measurable on B . If B is a deterministic constant, Problem (2) reduces to Problem (1) with aunivariate utility; see, e.g., He et al. (2020).Our contribution is to rigorously provide the optimal solution(s) and thoroughly investigate the followingissues of this new framework (2):(i) Optimality: a wealth variable X is called optimal if X solves Problem (2).(ii) Feasibility: a wealth variable X is called feasible if E [ U ( X, B )] > −∞ . Problem (2) is called feasibleif it admits a feasible solution.(iii) Finiteness: a wealth variable X is called finite if E [ U ( X, B )] < + ∞ . Problem (2) is called finite if thesupremum in (2) does not equal + ∞ .(iv) Attainability: Problem (2) is called attainable if it admits an optimal solution.(v) Uniqueness: Problem (2) is called unique if for any two optimal solutions X and ˜ X , they are equalalmost surely.For the classic framework (1), a standard approach is the duality method (cf. Karatzas and Shreve (1998)).With some assumptions and standard conditions on u , for any x at the domain of u , one can alwaysobtain a unique, finite and non-trivial optimal solution X ∗ = ( u ′ ) − ( λ ∗ ξ ) for Problem (1), where λ ∗ ∈ (0 , + ∞ ) is the Lagrange multiplier solved from the equation E [ ξ ( u ′ ) − ( λ ∗ ξ )] = x . For a non-concave utility u , the problem can be solved by the Legendre-Fenchel transformation (cf. Rockafellar (1970)) and theconcavification technique (cf. Carpenter (2000)). This technique aims to prove that the optimal solutionunder a non-concave utility is also the optimal one under its concavification (the minimal dominating concavefunction of the non-concave utility) and solve the latter problem; it generally requires the assumption of anon-atomic ξ (cf. Bichuch and Sturm (2014)). The existence of λ ∗ is a key issue. Traditionally, it is alwaysassumed a priori that the function g ( λ ) , E [ ξ ( u ′ ) − ( λξ )] is finite (i.e. g ( λ ) < + ∞ for all λ >
0) andthe Lagrange multiplier always exists (cf. Karatzas et al. (1987); Kramkov and Schachermayer (1999); Wei(2018)). To this point, Jin et al. (2008) investigate this issue in the classic framework (1) and provide acounterexample that g ( λ ) equals to + ∞ for small λ , and hence the Lagrange multiplier does not exist. In thenon-concave univariate setting, Reichlin (2013) proves the existence of λ ∗ when the optimal solution exists. For simplicity, the concept “finite” in (iii) only refers to + ∞ . Therefore, if Problem (2) or a wealth variable X is infeasible,or there is no wealth variable satisfying all constraints in Problem (2), we still call it finite; see also Assumption 2. These casesare trivial and can be easily recognized in our formulation. g is continuous and strictly decreasing on its domain. Therefore, the existence of λ ∗ for a proper x is guaranteed by the intermediate value theorem. It is also known that X ∗ is a decreasing function of ξ ; seeLiang and Liu (2019) for a detailed economic discussion.In this paper, unless specified, we do not always require all of the following classic assumptions:(I) the utility satisfies the asymptotic elasticity condition (cf. Kramkov and Schachermayer (1999)), Inadaconditions (Example 1(c)) and other conditions;(II) the Lagrange multiplier always exists; see Case 1 in Section 5;(III) the probability space (Ω , F , P ) or the pricing kernel ξ is non-atomic ;(IV) Problem (2) is finite; see Assumption 5.Through investigation, we will demonstrate some surprising phenomena in the new framework (2):(i) The optimal solution may be non-unique, i.e., there may be a “random set”, denoted by X B ( λ ∗ ξ ) in(9); see Section 3. It is roughly because the “conjugate point” in the Legendre-Fenchel transformationis no longer always unique for our U .(ii) Define X B ( λξ (cid:1) = inf X B ( λξ ) in (20). The analogue g ( λ ) = E (cid:2) ξX B (cid:0) λξ (cid:1)(cid:3) defined in (21) may also equalto + ∞ as in Jin et al. (2008); see Figure 2 (iii)(v)(vi). Moreover, it may even be discontinuous on itsdomain; see Figure 2 (ii)(iv)(vi). Hence, the intermediate value theorem can not be applied directly toguarantee the existence of the Lagrange multiplier; see Section 5.(iii) In light of (i), to find λ ∗ , we need to make a measurable selection from the set X B ( λ ∗ ξ ). The measur-ability issue also arises when applying concavification to multivariate utility functions; see Sections 3and 5.(iv) The optimal solution X ∗ may not be decreasing with respect to the pricing kernel ξ , which is verydifferent from the results of the quantile formulation approach (cf. Jin and Zhou (2008); He and Zhou(2011); Xu (2016)); see Remark 4.Technically, together with the situation where g may not be finite, the optimal solution(s) and the aboveissues are fully discussed in Theorems 1-4 and summarized in Table 1. In Theorems 1-2, we overcome themeasurability difficulties, apply the variational method to obtain the optimal solution(s), and hence give amultivariate version of the concavification theorem. In Theorems 3-4, we give a complete characterization onfeasibility, finiteness, attainability and uniqueness of optimal solution(s), including situations where g maybe infinite. The insights of some proofs are illustrated by Figures 1-2. In a word, the main reason of these A measure µ is called non-atomic, if for any measurable set A with a positive measure, there exists a measurable subset B ⊂ A satisfying µ [ A ] > µ [ B ] >
0. A random variable X is called non-atomic, if its distribution measure is non-atomic. Theprobability space (Ω , F , P ) is called non-atomic, if its probability measure P is non-atomic, which is equivalent to the existenceof a continuous distribution; see Proposition A.31 in F¨ollmer and Schied (2016). B .Moreover, the benchmark is also motivated to serve as a risk management constraint. The first exampleis that a so-called liquidation boundary is set as the benchmark process, which the wealth is required to bealways above; see Hodder and Jackwerth (2007). The second example is the constraints on default probabilityand Value-at-Risk to mitigate excessive risk taking; see Chen and Hieber (2016), Dong and Zheng (2020) andNguyen and Stadje (2020). These constraint problems can be also solved through the benchmark problemby Lagrangian duality arguments. We will give a concrete example and motivate an application in Section 8.The rest of this paper is organized as follows. Section 2 establishes the model settings of Problem(2). The optimal solution(s) are obtained in Section 3. The issues of feasibility, finiteness, attainability anduniqueness are respectively investigated in Sections 4-6. Section 7 provides a complete result to connect withthe univariate framework. Section 8 presents a concrete application for our framework. Section 9 concludesthe paper. Some tedious and less important proofs are left in the Appendix. In this section, we specify the required settings of a multivariate utility function U : R × E → R ∪{−∞} .A key requirement is the existence of a “good” concavification (Assumption 3(C)(D)). In Subsection 2.1, wewill propose our abstract but general settings with some assumptions. In Subsection 2.2, we will show someproperties of the concavification in Lemma 1 and prove the generality of our settings in Proposition 1. First, we require the following standing assumptions throughout the paper. A sufficient condition ofAssumption 2 will be given in Section 6.
Assumption 1 (Utility) . U ( x, · ) is measurable on ( E, E ) for any x ∈ R and U ( · , b ) is nondecreasing andupper semicontinuous on R for any b ∈ E . For every b ∈ E , there exists x ∈ R such that U ( x, b ) > −∞ . Assumption 2 (Well-posedness) . Problem (2) is well-posed , i.e., for every random variable X satisfying E [ ξX ] x and U ( X, B ) > −∞ a.s. , the expectation E [ U ( X, B )] is well-defined, i.e., E (cid:2) U ( X, B ) + (cid:3) < + ∞ or E (cid:2) U ( X, B ) − (cid:3) < + ∞ . (3)Assumption 1 is basic. Based on Assumption 2 and the fact that U ( · , b ) is nondecreasing, it is equivalentto study Problem (2) with the budget constraint E [ ξX ] = x . If not, we can replace X by some X ′ = X + c { ξ
0, which will increase both E [ ξX ] and E [ U ( X, B )]. For the second coordinate Similar to the concept “finite”, if Problem (2) is infeasible, or there is no wealth variable satisfying all constraints in Problem(2), we still call it well-posed (but meaningless). ∈ E , as we only require the measurability, we will refer to the first coordinate x when discussing the otherproperties of U , such as concavity, differentiability, etc.In many non-concave contexts, we require Assumption 3, which covers Assumption 1. Assumption 3 (Existence of a “good” concavification) . U ( x, · ) is measurable on ( E, E ) for any x ∈ R and U ( · , b ) ∈ H for any b ∈ E , where H is the set of all of the function h : R → R ∪ {−∞} satisfying:(A) h is nondecreasing and upper semicontinuous on R ;(B) There exists x ∈ R such that h ( x ) > −∞ ;(C) h admits a concavification ˜ h , inf { g : R → R ∪ {−∞} | g is concave and g > h } .(D) For any n ∈ N + , define H n ( t ) = sup (cid:26) x t : ˜ h ( x ) h ( x ) + 1 n (cid:27) , G n ( t ) = inf (cid:26) x > t : ˜ h ( x ) h ( x ) + 1 n (cid:27) , t ∈ R . (4)For any n ∈ N + and t ∈ R , it holds that H n ( t ) > −∞ and G n ( t ) < + ∞ .If (C) and (D) hold, we call the function ˜ h is a good concavification . It means that the concavification ˜ h exists and has a relatively close distance with h . A counterexample to (C) is the S-shaped utility (Example4) without a finite left endpoint; it does not admit a (finite) concavification (here ˜ h = + ∞ ). Moreover, if(D) holds, for any t ∈ R satisfying h ( t ) < ˜ h ( t ), we have h ( H n ( t )) > −∞ . A counterexample to (D) is h ( x ) = − ( − x ) / { x< } + 2 { x > } , x ∈ R with its concavification ˜ h ≡
2; here H n ( t ) = −∞ and h ( H n ( t )) = −∞ forany n ∈ N + and t < U satisfying Assumption 3, we get a “family” of multivariate concavifications: ˜ U ( · , b ) for all b ∈ E .We further define H n ( t, b ) and G n ( t, b ) as the multivariate version of H n ( t ) and G n ( t ): H n ( t, b ) = sup (cid:26) x t : ˜ U ( x, b ) U ( x, b ) + 1 n (cid:27) , G n ( t, b ) = inf (cid:26) x > t : ˜ U ( x, b ) U ( x, b ) + 1 n (cid:27) . (5)Further, we define ˆ H ( n ) b , inf { t ∈ R : H n ( t, b ) < t } and propose Assumption 4, which is required in the proof of the concavification theorem 2.
Assumption 4 (Integrability) . ∀ n ∈ N + , E h ξ ˆ H ( n ) − B i < + ∞ .Indeed, our setting involves a rather abstract set H . We point out that the most essential condition inour setting is Assumption 3(C)(D); we only require that a “good” concavification exists for the multivariateutility. We will show that our setting of Assumptions 3-4 includes and is much larger than the classic settingsincluding Inada conditions (Example 1(c)) in Proposition 1. Moreover, we illustrate the meanings of H n ( t, b )and G n ( t, b ) in Figure 1; they are key elements in the proof of our main results. Throughout, we denote x + = max { x, } and x − = − min { x, } for any x ∈ R . G , H , G n and H n Example 1 (A reduced concrete setting) . For each b ∈ E , we define x b , inf { x ∈ R : U ( x, b ) > −∞} as thelower bound of the wealth, which is allowed to vary with the benchmark value b . We denote by a randomvariable x B = inf { x ∈ R : U ( x, B ) > −∞} the (state-dependent) lower bound relying on the benchmark value B ( ω ) at each state ω ∈ Ω. Suppose:(a) ∀ b ∈ E, −∞ < x b < + ∞ , and E [ ξx B ] > −∞ ;(b) ∀ b ∈ E, U ( · , b ) is upper semicontinuous on [ x b , + ∞ );(c) Inada condition: ∀ b ∈ E, lim sup x → + ∞ U ( x,b ) x = 0.We give some explanations on these conditions. Actually, all of them coincide with the classic theory.Condition (a) means that for any variable X under consideration in Problem (2), one should have X > x B (or sometimes X > x B ). Condition (a) is consistent with the classic settings, as we embed the restrictionon the lower bound of the return X into the utility function U . For the instance of a (univariate) CRRAutility u (cf. Merton (1969)), the wealth X is required to be nonnegative, and the domain of the utility u is [0 , + ∞ ). Here, the domain is extended to R and the value of the left tail should be −∞ ; in this case, wehave x = 0. For an S-shaped utility u (cf. Liang and Liu (2020)), the wealth X is bounded from below by adeterministic liquidation level L ∈ R , and u should be truncated at the finite left endpoint L and the valueon the left tail is −∞ ; in this case, x = L . Moreover, as ξ is the pricing kernel, E [ ξx B ] > −∞ practicallymeans that the worst return is affordable. The assumption is easy to satisfy, as in the classic case we oftenassume x B = 0. Condition (b) looses the widely-used concavity and differentiability restriction on utilityfunctions, which include S-shaped functions and step functions. Upper semicontinuity is indeed equivalent7o right-continuity when the nondecreasing property in Assumption 1 holds. It leads to two possible casesat the endpoint x b :(I) U ( x b , b ) > −∞ with U ( · , b ) being right continuous at x b , and U ( x, b ) = −∞ for x < x b , that is, atruncation occurs at x b ;(II) U ( x b , b ) = −∞ , and lim x → x b + U ( x, b ) = −∞ .Hence, Condition (b) contains both power utilities (type I) and logarithm utilities (type II). Condition (c)is required to ensure the finiteness of the optimization problem if the utility function is not differentiable. Itis slightly weaker than the classic Inada condition ( u ′ (+ ∞ ) = 0) and can be interpreted as the diminishingmarginal utility.We proceed to prove in Proposition 1 that the range of H includes that of Example 1. Proposition 1. If U satisfies (a)-(c) and Assumption 1, then Assumptions 3-4 hold. Example 1 already includes many classic cases in literature, but the range of our H is much larger. Asin Example 1(c), the slope of U near + ∞ has the limit 0. However, the slope is allowed to be any positivenumber in H . For example, if the utility is affine, then Example 1(c) is not satisfied; see Example 5 fordetails. Moreover, for u ∈ H , we do not require a finite left endpoint x , inf { x ∈ R : u ( x ) > −∞} (Example1(a)). A counterexample to (a) and (c) is u ( x, b ) = x { x }∪{ x > } , x ∈ R , b ∈ E (here x b = −∞ for any b ∈ E ), while Assumptions 3-4 can be verified under some simple conditions.Finally, we define the bliss point (cf. Binger and Hoffman (1998)) x b , inf { x ∈ R : U ( x, b ) = U (+ ∞ , b ) } . (6)In most literature, x b = + ∞ for each b ∈ E . In our model, x b ∈ ( −∞ , + ∞ ] and is allowed to be finite. Inthis light, a wealth variable X is called a bliss solution, if X > x B , i.e., U ( X ( ω ) , B ( ω )) attains its maximumat every state point ω without any risk; see Remark 3 later for more details. Lemma 1 demonstrates some properties of the concavification.
Lemma 1.
For a nondecreasing and upper semicontinuous function h : R → R ∪ {−∞} , suppose that thereexists x ∈ R such that h ( x ) > −∞ , and that h admits a concavification ˜ h : R → R ∪ {−∞} . We have:(1) ˜ h = h on ( −∞ , x ) , and ˜ h is continuous on ( x, + ∞ ) . Moreover, ˜ h is nondecreasing on R .(2) For a, b ∈ R and a < b , if ˜ h > h on ( a, b ) , then ˜ h is affine on ( a, b ) .(3) If x > −∞ , then ˜ h ( x ) = h ( x ) , and ˜ h is right continuous at x . Moreover, if h ( x ) = −∞ , then thereexists { x n } n > ⊂ ( x, + ∞ ) with x n ↓ x and ˜ h ( x n ) = h ( x n ) (This means that the function H n ( t ) defined in (4) is always finite, and for ˜ h ( t ) > h ( t ) we have h ( H n ( t )) > −∞ ).
4) If lim sup x → + ∞ h ( x ) x = 0 , then the function G n ( t ) defined in (4) is always finite.(5) If h ∈ H and ˜ h ( t ) > h ( t ) + n for some n ∈ N + , t ∈ R , then H n ( t ) < t < G n ( t ) , and −∞ < ˜ h ( H n ( t )) h ( H n ( t )) + n , ˜ h ( G n ( t )) h ( G n ( t )) + n .(6) If h ∈ H , then ˜ h ( x ) = sup ( a,b ) ∈ R : a x,b > x,a = b ( x − a ) h ( b )+( b − x ) h ( a ) b − a .Proof. The proof of Lemma 1 is tedious, and we leave it to Appendix A.
Proof of Proposition 1.
We first prove that Assumption 3 holds, that is, we prove that U ( · , b ) satisfies (A)-(D)for any b ∈ E . (A) and (B) are obvious based on (a) and (b). For (C), noting that lim sup x → + ∞ U ( x,b ) x = 0 for anygiven b , there exist α ∈ R and β ∈ R (depending on b ) such that U ( x, b ) < αx + β for x > x b > −∞ . As such, U ( · , b ) is dominated by a concave function. Hence U ( · , b ) admits a concavification ˜ U ( · , b ) : R → R ∪ {−∞} .Using Lemma 1 (3) and (4) we know that (D) holds. Then we prove that Assumption 4 holds. In fact, basedon the definition of ˆ H ( n ) b we know ˆ H ( n ) b > x b . Hence E h ξ ˆ H ( n ) − B i E (cid:2) ξx − B (cid:3) < + ∞ . In this section, we give one of our main results on optimal solution(s) in Theorems 1-2 in Subsection3.1. The detailed proofs are respectively given in Subsections 3.2-3.3.
To begin with, we develop the multivariate Legendre-Fenchel transformation. For b ∈ E and 0 y < + ∞ , we define the conjugate function V b and the conjugate set X b of U ( · , b ) as V b ( y ) = sup x ∈ R (cid:2) U ( x, b ) − yx (cid:3) ∈ R ∪ { + ∞} , (7) X b ( y ) = arg sup x ∈ R ∪{±∞} (cid:2) U ( x, b ) − yx (cid:3) . (8)As U is upper semicontinuous, if x n → x ∈ R with U ( x n , b ) − yx n → V b ( y ), then we have V b ( y ) = lim n → + ∞ ( U ( x n , b ) − yx n ) U ( x, b ) − yx V b ( y ) . Hence x ∈ X b ( y ). In this light, we can define the term X b ( y ) as following: • If x ∈ R , then x ∈ X b ( y ) if and only if U ( x, b ) − yx = V b ( y ). • If x ∈ {±∞} , then x ∈ X b ( y ) if and only if there exists a sequence { x n } ↑ (or ↓ ) x .9nd we define X b (+ ∞ ) = { x b } ; see also Lemma 5(iv). Then X b ( y ) is non-empty for y ∈ [0 , + ∞ ]. For λ ∈ [0 , + ∞ ], we define a “random set” X B ( λξ ) : Ω → R ∪{±∞} , ω
7→ X B ( ω ) ( λξ ( ω )) . (9)Compared to the classic results, one may conjecture that X B ( λξ ) is the optimal solution to Problem(2). However, it is worth pointing out that X B ( λξ ) here is defined in terms of a set, as it may become nolonger unique when U ( · , b ) is non-concave for some b ∈ E . Hence, we have to study the random set X B ( λξ )for the optimal solutions. Indeed, Theorem 2 shows that the optimal solution still locates in X B ( λξ ), butdifferent from the classic case, we need extra work to find out a measurable selection; see Section 5 later. Inthe following content, we will use the notation X Ub and x Ub instead of X b and x b in case of possible confusion.Now we present our main results, some of which require Assumption 5; details on the finiteness issuewill be studied in Sections 5-6. Assumption 5 (Finiteness of the problem) . Problem (2) is finite, i.e.,sup X : E [ ξX ] x U ( X,B ) > −∞ E [ U ( X, B )] < + ∞ . We first investigate the concave utility function in Theorem 1. If U ( · , b ) is concave for any b ∈ E , thenwe define U ′ + ( x, b ) = lim h ↓ U ( x + h, b ) − U ( x, b ) h , U ′− ( x, b ) = lim h ↓ U ( x, b ) − U ( x − h, b ) h , x ∈ R , b ∈ E. Hence, we have the following characterization of (9): X ∈ X B ( λξ ) ⇐⇒ U ′ + ( x B , B ) λξ on { X = x B } ,U ′ + ( X, B ) λξ U ′− ( X, B ) on { X > x B } . (10) Theorem 1.
Suppose that Assumptions 1-2 and 5 hold and U ( · , b ) is concave for any b ∈ E . We have that X is an optimal solution of Problem (2) if and only if X ∈ X B ( λξ ) a.s. for some λ ∈ [0 , + ∞ ] satisfying thebudget constraint E [ ξX ] = x and U ( X, B ) > −∞ . For the non-concave case, to give a concavification theorem under the multivariate setting, there aremany technical issues such as measurability to be addressed. Hence we need to investigate in detail theproperties of the concavification of functions satisfying certain conditions. In the following Theorem 2, wegive results on general utility functions with the assumption of a non-atomic probability measure.
Theorem 2.
Suppose that (Ω , F , P ) is non-atomic and Assumptions 2-3 hold. i) The concavification problem sup X : E [ ξX ] x U ( X,B ) > −∞ E [ ˜ U ( X, B )] is well-posed, and we have sup X : E [ ξX ] x U ( X,B ) > −∞ E [ U ( X, B )] = sup X : E [ ξX ] x U ( X,B ) > −∞ E [ ˜ U ( X, B )] , (11) where ˜ U ( · , b ) is the concavification of U ( · , b ) for any b ∈ E ;(ii) Suppose further that Assumption 5 holds. Then X is an optimal solution if and only if X ∈ X UB ( λξ ) a.s. for some λ ∈ [0 , + ∞ ] with E [ ξX ] = x and U ( X, B ) > −∞ . In the first part of Theorem 2, we rigorously prove the availability of the concavification technique for amultivariate utility function, while in the second part we give a necessary and sufficient condition on optimalsolutions, and we do not assume a priori that a Lagrange multiplier exists. Conversely, our results indicatethat, the existence of a Lagrange multiplier is necessary for Problem (2) to be solvable.
Remark . From the proof of Theorems 1-2, we will see that the “if” part in Theorems 1-2 only needsAssumption 2, but not Assumptions 3-5. That is, if we have found some X ∈ X B ( λξ ) with some λ ∈ [0 , + ∞ ]satisfying the budget constraint E [ ξX ] = x and U ( X, B ) > −∞ , then X is an optimal solution to Problem(2) no matter whether it is finite; see also Theorem 4. To prove Theorem 1, we need the following lemma:
Lemma 2.
Suppose that
Z, Z and Z are finite random variables with Z > , Z Z a.s., and A ∈ F is a set with positive measure. If E [ Z Y + − Z Y − ] holds for any bounded random variable Y satisfyingthat Y is supported on A , random variables ZY, Z Y + and Z Y − are integrable and E [ ZY ] = 0 , then thereexists a real number λ such that Z λZ Z a.s. on A .Proof of Lemma 2. Step 1:
We consider the case when
Z, Z and Z are bounded. First we show bycontradiction that Z > λZ holds almost surely on A for some λ ∈ R . Suppose that for any λ ∈ R , the set A = { Z < λZ } ∩ A has positive measure. Then we can take λ such that A = { Z > λZ } ∩ A also haspositive measure. As P [ { Z < λ ′ Z } ∩ A ] > λ ′ , we can define Y = Y + = 1 , on A , − Y − = − E [ Z A ] E [ Z { Z <λ ′ Z }∩ A ] { Z <λ ′ Z }∩ A , on A . It is verified that Y is bounded and ZY, Z Y + and Z Y − are integrable with E [ ZY ] = 0. We have E [ Z A ] = E [ Z Y + ] E [ Z Y − ] = E (cid:20) Z E [ Z A ] E [ Z { Z <λ ′ Z }∩ A ] { Z <λ ′ Z }∩ A (cid:21) λ ′ E [ Z A ] , E [ Z A ] because λ ′ can be any real number and E [ Z A ] >
0. As aresult, there exists λ such that Z > λZ holds almost surely on A . Step 2:
Now we show that we can choose λ ∈ R such that Z > λ Z > Z holds almost surely on A .Let λ = sup { λ : Z > λZ a.s. on A } ∈ R , then Z > λ Z a.s. on A . We prove by contradiction that forevery λ > λ , Z λZ a.s. on A .If P [ { Z > λZ } ∩ A ] > λ > λ , denote A = { Z > λZ } ∩ A . By definition of λ we have A = { Z < λZ }∩ A with P [ A ] >
0. As Z > Z , we also have A ∩ A = ∅ . We can take Y = Y + A − Y − A = 0a.s. on A ∪ A with E [ ZY ] = 0, then it holds that E [ Z Y + ] > E [ λZY + ] = E [ λZY − ] > E [ Z Y − ], whichcontradicts to the condition that E [ Z Y + − Z Y − ]
0. Therefore, for every λ > λ , P [ { Z > λZ } ∩ A ] = 0,which means that Z λ Z Z a.s. on A . Step 3:
For the unbounded case, define A n = A ∩ { Z, | Z | , | Z | < n } . Applying the results abovewe have λ n such that Z λ n Z Z a.s. on A n . By finiteness of Z, Z , Z we know −∞ < lim inf n → + ∞ λ n lim sup n → + ∞ λ n < + ∞ , and we can find a convergent subsequence λ n i → λ , then Z λZ Z a.s. on A . Proof of Theorem 1.
For the “if” statement, we will prove it in the second part of Theorem 2 under a moregeneral setting.For the “only if” statement, a basic idea is to apply the variational method to derive an inequality (12)appeared in Lemma 2 and we use the lemma to find a constant λ . Above all, for any optimal solution X , weknow that E [ ξx B ] E [ ξX ] x . Next we prove the “only if” part in the following two cases: • If E [ ξx B ] = x , then X = x B a.s. , that is, X ∈ X B ( λξ ) with λ = + ∞ . • If E [ ξx B ] < x , then the set A = { X > x B } satisfies P [ A ] >
0. We are going to prove that U ′ + ( X, B ) λξ U ′− ( X, B ) on A . Define A n = (cid:26) ξ n, X − x B > n , U ′ + (cid:18) X − n , B (cid:19) < n (cid:27) . We have P [ A n ] > n . Suppose that Y is a bounded random variable supported on A n satisfying E [ ξY ] = 0. Noting that U is concave, we have for t sufficiently small: (cid:12)(cid:12)(cid:12)(cid:12) U ( X + tY, B ) − U ( X, B ) t (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) U ′ + (cid:18) X − n , B (cid:19) Y (cid:12)(cid:12)(cid:12)(cid:12) is bounded , then 0 > lim t → E [ U ( X + tY, B )] − E [ U ( X, B )] t = E [ U ′ + ( X, B ) Y + − U ′− ( X, B ) Y − ] . (12)Using Lemma 2, we obtain U ′ + ( X, B ) λξ U ′− ( X, B ) on A n (and also on A by letting n → + ∞ , aswe have done in Step 3 when proving Lemma 2) for some λ > P [ X = x B ] > P [ A c ] > U ′ + ( X, B ) λξ a.s. on A c . For a bounded random variable Y satisfying E [ ξY ] = 0, Y > A c , Y A n ,12nd Y = 0 on A ∩ A cn , we have for t sufficiently small:0 > lim t → E [ U ( X + tY, B )] − E [ U ( X, B )] t = E [ U ′ + ( x B , B ) Y A c + U ′− ( X, B ) Y A n ] . Noting that we already have U ′− ( X, B ) > λ n ξ on A n , where λ n := sup { λ : λξ U ′− ( X, B ) a.s. on A n } ∈ R , we attempt to show U ′ + ( x B , B ) λξ a.s. on A c for any λ > λ n . Otherwise for some λ > λ n we have P [ U ′ + ( x B , B ) A c > λξ ] >
0, then we can take Y = Y + > C = { U ′ + ( x B , B ) > λξ } ∩ A c ⊂ A c and Y = − Y − < D = { U ′− ( X, B ) < λξ } ∩ A n ⊂ A n with E [ ξY ] = 0 to obtain0 > E (cid:2) λξY + C − λξY − D (cid:3) = λ E [ ξY ] = 0 , leading to a contradiction. Hence U ′ + ( x B , B ) λ n ξ holds almost surely on A c while U ′ + ( X, B ) λ n ξ U ′− ( X, B ) a.s. on A n . Letting n → + ∞ , we obtain some λ > U ′ + ( x B , B ) λξ a.s. on A c ,U ′ + ( X, B ) λξ U ′− ( X, B ) a.s. on A. In conclusion (1)-(3), we know that for a finite optimal solution X , there exists λ ∈ [0 , + ∞ ] such that U ′ + ( x B , B ) λξ a.s. on the set { X = x B } ,U ′ + ( X, B ) λξ U ′− ( X, B ) a.s. on the set { X > x B } . Following (10), the proof is completed.
To prove Theorem 2, we need some further discussion on concavification (Lemma 1) and non-atomicmeasures. H n and G n are important tools in the proof of Theorem 2, and it is necessary to confirm theirmeasurability before we apply mathematical operations on them, which is stated in the following Lemma 3.Its proof is rather technical. Lemma 3.
Suppose that Assumption 3 holds. Then ˜ U , H n and G n are measurable functions on ( R × E, B ( R ) × E ) . roof of Lemma 3. (1) For ˜ U , thanks to Lemma 1, we have˜ U ( x, b ) = sup a x ca = c ( x − a ) U ( c, b ) + ( c − x ) U ( a, b ) c − a = sup a ′ ,c ′ > a ′ ,c ′ ) =(0 , a ′ U ( x + c ′ , b ) + c ′ U ( x − a ′ , b ) a ′ + c ′ = sup a ′ ,c ′∈ [0 , + ∞ ] ∩ Q ( a ′ ,c ′ ) =(0 , a ′ U ( x + c ′ , b ) + c ′ U ( x − a ′ , b ) a ′ + c ′ , which indicates that ˜ U is measurable.(2) Define A = { ( t, b ) : H n ( t, b ) < s } and A = (cid:26) ( t, b ) ∈ R × E : t > s and ˜ U ( x, b ) > U ( x, b ) + 1 n for every x ∈ [ s, t ] (cid:27) ∪ (( −∞ , s ) × E ) . We first show that A = A . It is obvious that A ⊂ A , and we only need to prove A ⊂ A . As t < s leads to H n ( t, b ) < s , it suffices to consider the case that t > s . If ˜ U ( x, b ) > U ( x, b ) + n holds for every x ∈ [ s, t ], then s > x b , and by definition of H n we know H n ( t, b ) s , and when the equality holds, we have x k ↑ s satisfying ˜ U ( x k , b ) U ( x k , b ) + n . Therefore, we derive a contradiction that ˜ U ( s, b ) > U ( s, b ) + n > lim x k ↑ s U ( x k , b ) + n > lim x k ↑ s ˜ U ( x k , b ) = ˜ U ( s, b ). As such, A ⊂ A , and hence A = A .Now we investigate the structure of A . Define A = (cid:26) ( t, b ) : t > s and ˜ U ( x, b ) > U ( x, b ) + 1 n for every x ∈ [ s, t ] (cid:27) . We are going to prove A = [ j > [ x ∈ Q ,x > s [ s, x ] × \ N > F j (cid:16) z ( N )1 ( x ) , z ( N )2 ( x ) , ..., z ( N ) N +1 ( x ) (cid:17) ! , (13)where F j ( z , z , ..., z m ) , T k m n b ∈ E : ˜ U ( z k , b ) > U ( z k , b ) + n + j o , and n z ( N ) i ( x ) o i N +1 denotes the N -uniform partition points of the interval [ s, x ]. In fact, if we denote the right side of (13) by A , then for( t, b ) ∈ A , we know that ˜ U ( x, b ) > U ( x, b ) + n holds for every x ∈ [ s, t ], and hence t > s > x b . Define θ = inf n x ∈ [ s, t ] : ˜ U ( x, b ) − U ( x, b ) o . We have { y n } ⊂ [ s, t ] satisfying y n → y and ˜ U ( y n , b ) − U ( y n , b ) → θ .As ˜ U ( · , b ) is continuous on ( x b , + ∞ ) and that U ( · , b ) is upper semicontinuous, we obtain θ = lim n → + ∞ (cid:16) ˜ U ( y n , b ) − U ( y n , b ) (cid:17) > ˜ U ( y, b ) − U ( y, b ) > θ, thus θ = ˜ U ( y, b ) − U ( y, b ) > n , and we have some j ∈ N + s.t. θ > n + j . Again using the continuity of ˜ U ( · , b )and upper semicontinuity of U ( · , b ), we know that ˜ U ( x, b ) > U ( x, b ) + n + j also holds on x ∈ [ s, t + δ ] forsome small δ >
0, that is, one can find t ′ ∈ Q such that t ′ > t and [ s, t ′ ] × { b } ⊂ A . Moreover, based on14efinition of F j , we know that b ∈ F j (cid:16) z ( N )1 ( t ′ ) , z ( N )2 ( t ′ ) , ..., z ( N ) N +1 ( t ′ ) (cid:17) for any N , thus( t, b ) ∈ [ s, t ′ ] × { b } ⊂ [ s, t ′ ] × \ N > F j (cid:16) z ( N )1 ( t ′ ) , z ( N )2 ( t ′ ) , ..., z ( N ) N +1 ( t ′ ) (cid:17) ⊂ A , and we know A ⊂ A . It suffices to show A ⊂ A .For ( t, b ) ∈ A , ( t, b ) ∈ [ s, x ] × T N > F j (cid:16) z ( N )1 ( x ) , z ( N )2 ( x ) , ..., z ( N ) N +1 ( x ) (cid:17) with some x ∈ Q and j ∈ N + , thatis, s t x and, ˜ U (cid:16) z ( N ) k ( x ) , b (cid:17) > U (cid:16) z ( N ) k ( x ) , b (cid:17) + n + j for any k, N satisfying 1 k N + 1. Notingthat ˜ U ( · , b ) is continuous and that U ( · , b ) is right continuous, we have ˜ U ( x, b ) > U ( x, b ) + n + j > U ( x, b ) + n on [ s, x ] ⊃ [ s, t ]. It follows that A ⊂ A , which leads to A = A .As for a given z , ˜ U ( z, b ) − U ( z, b ) is measurable in b , we know F j (cid:16) z ( N )1 ( x ) , z ( N )2 ( x ) , ..., z ( N ) N +1 ( x ) (cid:17) ∈ E . Therefore, A = A ∈ B ( R ) × E , and A = A = A ∪ (( −∞ , s ) × E ) is also a measurable set, which meansthat H n is a measurable function. Similarly, one can prove the measurability of G n .At last, for non-atomic measures, we need the following result in Sierpi´nski (1922): Lemma 4. If µ is a non-atomic measure on (Ω , F ) with µ (Ω) = c , then there exists a one-parameter familyof increasing measurable sets { A t } t c such that µ ( A t ) = t . In the light of Lemmas 1-4 above, we proceed to prove Theorem 2. The idea is to substitute X ( ω ) locallyby some ˆ H ( ω ) and ˆ G ( ω ) for some ω such that X ( ω ) does not lie on the concave part of U and ˆ H X ˆ G .We proceed to find a proper substitution of X with the formˆ X = ˆ H C + ˆ G D + X ( C ∪ D ) c , which increases the utility, while the budget value E [ ξX ] keeps unchanged. Figure 1 is provided to assist forunderstanding.Define H ( t, b ) = sup n x t : ˜ U ( x, b ) = U ( x, b ) o , G ( t, b ) = inf n x > t : ˜ U ( x, b ) = U ( x, b ) o . In the following proof, we assume that H ( t, b ) and G ( t, b ) are finite, and ˜ U ( H ( t, b ) , b ) = U ( H ( t, b ) , b ) > −∞ ,˜ U ( G ( t, b ) , b ) = U ( G ( t, b ) , b ) when ˜ U ( t, b ) > U ( t, b ). Then, based on Lemma 3, we know that H ( t, b ) and G ( t, b )are also measurable. Moreover, we define ˆ H b = inf { t ∈ R : H ( t, b ) < t } and we require E h ξ ˆ H − B i < + ∞ .These conditions are assumed to provide a concise proof. For the weaker case with only H n ( t, b ) and G n ( t, b )being finite and Assumption 4, the proof is similar; see also Remark 2. Proof of Theorem 2. (i)
Part I:
Let us first assume that the concavification problem is well-posed. It is15quivalent to study both problems with binding budget constraints, i.e., E [ ξX ] = x . We are going toprove (11) by contradiction. Suppose that α , sup X : E [ ξX ]= x U ( X,B ) > −∞ E [ ˜ U ( X, B )] > β , sup X : E [ ξX ]= x U ( X,B ) > −∞ E [ U ( X, B )] . (14)As such, we have some random variable X satisfying E [ ξX ] = x and E [ ˜ U ( X, B )] > β. Define S = n ( x, b ) ∈ R × E : ˜ U ( x, b ) > U ( x, b ) o , Q = { ω ∈ Ω : ( X ( ω ) , B ( ω )) ∈ S } . It follows that P [ Q ] >
0. Define two random variablesˆ H = H ( X, B ) , ˆ G = G ( X, B ) . On set Q , we have ˆ H < X < ˆ G . Hence ˜ U ( x, B ) is affine in x for ˆ H < x < ˆ G , that is, ˜ U ( x, B ) = a B x + c B ,where a B = ˜ U ( ˆ G,B ) − ˜ U ( ˆ H,B )ˆ G − ˆ H . Based on Lemmas 1 and 3, a B and c B are measurable on Q , and ˆ H , ˆ G , a B and c B are finite. In the following Steps 1-3, we desire to construct a new random variable ˆ X byslightly changing X on Q satisfying E [ ξ ˆ X ] = x and E [ U ( ˆ X, B )] > β , which will lead to a contradiction. Step 1:
Take a countable partition of R as R = [ n > P n , { P n } n > are bounded sets . This leads to a partition of Q as Q = [ n > Q n , Q n , Q ∩ n ω ∈ Ω : ( ˆ H, ˆ G, a B , c B , ξ ) ∈ P n o , and we proceed to make our substitution of X on every Q n that P [ Q n ] > Step 2:
For every n with P [ Q n ] >
0, we need to determine a partition Q n = C n ∪ D n and then wereplace X by ˆ H on the set C n and by ˆ G on the set D n . We need to make sure that the new variableˆ X satisfies E h ξ ˆ X Q n i = E [ ξX Q n ] (15)and E h ˜ U ( X, B ) Q n i E h ˜ U ( ˆ X, B ) Q n i = E h U ( ˆ X, B ) Q n i . (16)To this end, we denote p = E [ ξX Q n ]. Noting that ˆ H , ˆ G , a B , and ξ are bounded on Q n , we define F ( t ) = E (cid:2) ξG { ξ ta B }∩ Q n (cid:3) + E h ξ ˆ H { ξ>ta B }∩ Q n i , t > .
16s ˆ
H < X < G holds on Q , we have F (0) = E h ξ ˆ H Q n i < p , F (+ ∞ ) = E [ ξG Q n ] > p , and that F isnondecreasing and right-continuous. In the following, we construct C n and D n through two cases suchthat E [ ξ ( ˆ H C n + ˆ G D n )] = p. • If F ( σ ) = p holds for some σ >
0, we define C n = { ξ > σa B } ∩ Q n , D n = { ξ σa B } ∩ Q n , and then we have E h ξ ( ˆ H C n + ˆ G D n ) i = F ( σ ) = p . • If p / ∈ F ( R + ), we have σ > F ( σ − ) p < F ( σ ), and then 0 < F ( σ ) − F ( σ − ) = E [ ξ ( ˆ G − ˆ H ) { ξ = σa B }∩ Q n ] . Hence, the set Q ′ , { ξ = σa B }∩ Q n has a positive measure ε . As (Ω , F , P ) is non-atomic, its restriction on Q ′ is also non-atomic. Using Lemma 4, we obtain a family of increasingmeasurable sets { Q ′ t } t ε , satisfying P [ Q ′ t ] = t and Q ′ t ⊂ Q ′ . Define F ( t ) = E [ ξ ( ˆ G − ˆ H ) Q ′ t ]. It isverified that F is nondecreasing and continuous, and F (0) = 0 , F ( ε ) = F ( σ ) − F ( σ − ) > F ( σ ) − p. As such, one can find σ such that F ( σ ) = F ( σ ) − p . In this case we define C n = ( { ξ > σa B } ∩ Q n ) ∪ Q ′ σ , D n = ( { ξ σa B } ∩ Q n ) \ Q ′ σ , which leads to E h ξ ( ˆ H C n + ˆ G D n ) i = E h ξ ˆ H { ξ>σa B }∩ Q n i + E h ξ ˆ H Q ′ σ i + E h ξ ˆ G { ξ σa B }∩ Q n i − E h ξ ˆ G Q ′ σ i = F ( σ ) − F ( σ ) = p. Define ˆ X = ˆ H C n + ˆ G D n on Q n , and then (15) holds. Moreover, as ˆ H ˆ X ˆ G and ˆ H X ˆ G ,they are bounded on Q n . Hence ˜ U ( X, B ) = a B X + c B and ˜ U ( ˆ X, B ) = a B ˆ X + c B are also bounded on Q n . In both two cases we have E h(cid:16) ˜ U (cid:16) ˆ X, B (cid:17) − ˜ U ( X, B ) (cid:17) Q n i = E h(cid:16) a B (cid:16) ˆ X − X (cid:17)(cid:17) Q n i = E h a B ( ˆ G − X ) D n i − E h a B ( X − ˆ H ) C n i > E (cid:20) ξσ ( ˆ G − X ) D n (cid:21) − E (cid:20) ξσ ( X − ˆ H ) C n (cid:21) = 1 σ E h ξ (cid:16) ˆ X − X (cid:17) Q n i = 0 , and hence (16) holds. Step 3:
For every n with P [ Q n ] = 0, we set C n = Q n , D n = ∅ . Define C = [ n > C n , D = [ n > D n . C ∪ D = Q is a partition of Q . Defineˆ X = ˆ H C + ˆ G D + X Q c , which is consistent with our definition in Step 2 . Using (15), we have E [ ξX ] = E [ ξX Q c ] + X n > E [ ξX Q n ]= E h ξ ˆ X Q c i + X n > E h ξ ˆ X Q n i = E h ξ ˆ X + Q c i − E h ξ ˆ X − Q c i + X n > (cid:16) E h ξ ˆ X + Q n i − E h ξ ˆ X − Q n i(cid:17) . (17)As ˆ H < X on Q , based on the definition of ˆ H , for any t satisfying ˆ H < t < X , we have H ( t, B ) = ˆ H < t .Based on the definition of ˆ H b , we know ˆ H B t . Hence ˆ H B ˆ H and we have ˆ X > ˆ H > ˆ H B on Q .Hence E h ξ ˆ X − Q i E h ξ ˆ H − B Q i < + ∞ , and the series P n > E h ξ ˆ X − Q n i converges. As such, (17) indicates that the series P n > E h ξ ˆ X + Q n i alsoconverges. Then (17) leads to E [ ξ ˆ X ] = E [ ξX ] = x . (18)Moreover, based on Lemma 1, we know U ( ˆ X, B ) > −∞ , and then (3) indicates that the expectation E h U ( ˆ X, B ) i is well-defined. Using (16), we have (noting that X = ˆ X and ˜ U ( X, B ) = U ( X, B ) on Q c ) E h ˜ U ( X, B ) i = E h ˜ U ( X, B ) Q c i + X n > E h ˜ U ( X, B ) Q n i E h U (cid:16) ˆ X, B (cid:17) Q c i + X n > E h U (cid:16) ˆ X, B (cid:17) Q n i = E h U (cid:16) ˆ X, B (cid:17)i . Therefore, E h U (cid:16) ˆ X, B (cid:17)i > E h ˜ U ( X, B ) i > β , which contradicts to the definition (14) of β . Part II:
We are going to prove that the concavification problem is well-posed. For X satisfying E [ ξX ] x and ˜ U ( X, B ) > −∞ (which is equivalent to U ( X, B ) > −∞ ), we have that (3) holds with U replaced by ˜ U . Noting that ˜ U > U , if E h U ( X, B ) − i < + ∞ , we have immediately E h ˜ U ( X, B ) − i < + ∞ . It remains to consider the situation where E h U ( X, B ) + i < + ∞ . We discuss two cases: • If ˜ U ( X, B ) + = U ( X, B ) + a.s. , then E h ˜ U ( X, B ) + i < + ∞ . • If P h ˜ U ( X, B ) + > U ( X, B ) + i >
0, let us consider the two functions ˜ h = ˜ U ( · , b ) + and h = U ( · , b ) + for any given b ∈ E . Based on Lemmas 1 and 3, we have:181) For a, c ∈ R and a < c , if ˜ h > h on ( a, c ), then ˜ h is affine on ( a, c ).(2) For t ∈ R , define H ∗ ( t ) = sup { x t : ˜ h ( x ) = h ( x ) } , G ∗ ( t ) = inf { x > t : ˜ h ( x ) = h ( x ) } . If ˜ h ( t ) > h ( t ), then G ∗ ( t ) and H ∗ ( t ) ∈ R , and ˜ h ( G ∗ ( t )) = h ( G ∗ ( t )) ∈ R , ˜ h ( H ∗ ( t )) = h ( H ∗ ( t )) ∈ R , which also indicates H ( t ) H ∗ ( t ) < t < G ∗ ( t ) G ( t ), where H ( t ) and G ( t ) are defined in Lemma 1.Define S + = n ( x, b ) ∈ R × E : ˜ U ( x, b ) + > U ( x, b ) + o , Q + = (cid:8) ω ∈ Ω : ( X ( ω ) , B ( ω )) ∈ S + (cid:9) . Then P [ Q + ] >
0. Using the above two features, we can replicate our operations in
Part I toconstruct Q + n = C + n ∪ D + n and ˆ X = ˆ H ∗ C + + ˆ G ∗ D + + X Q + c withˆ H ∗ , H ∗ ( X, B ) , ˆ G ∗ , G ∗ ( X, B ) , C + , [ n > C + n , D + , [ n > D + n , Q + = [ n > Q + n . On each Q + n with P [ Q + n ] >
0, we have E [ ξ ˆ X Q + n ] = E [ ξX Q + n ]and E h ˜ U ( X, B ) + Q + n i E h ˜ U ( ˆ X, B ) + Q + n i = E h U ( ˆ X, B ) + Q + n i . (19)As discussed in Step 3 , we have E h ξ ˆ X i = E [ ξX ] x . Using (3), we have E h U ( ˆ X, B ) + i < + ∞ or E h U ( ˆ X, B ) − i < + ∞ . If E h U ( ˆ X, B ) + i < + ∞ , then, using (19), we have E h ˜ U ( X, B ) + i < + ∞ .If E h U ( ˆ X, B ) − i < + ∞ , then as ˜ U > U , we have E h ˜ U ( ˆ X, B ) − i < + ∞ .Noting that on C + ∪ D + = Q + , we have ˜ U ( X, B ) + > U ( X, B ) + >
0. Hence ˜ U ( X, B ) − = 0, andwe have E h ˜ U ( X, B ) − Q + i = 0 , E h ˜ U ( X, B ) − Q + c i = E (cid:20) ˜ U (cid:16) ˆ X, B (cid:17) − Q + c (cid:21) < + ∞ , which indicates that E h ˜ U ( X, B ) − i < + ∞ .Therefore, the concavification problem is well-posed.19ii) Suppose that X ∗ is a finite optimal solution of Problem (2). We have E h ˜ U ( X ∗ , B ) i > E [ U ( X ∗ , B )] = sup X : E [ ξX ]= x U ( X,B ) > −∞ E [ U ( X, B )] = sup X : E [ ξX ]= x U ( X,B ) > −∞ E h ˜ U ( X, B ) i . As such, X ∗ is also an optimal solution for sup X : E [ ξX ]= x U ( X,B ) > −∞ E h ˜ U ( X, B ) i , and E h ˜ U ( X ∗ , B ) i = E [ U ( X ∗ , B )] ∈ R , that is, ˜ U ( X ∗ , B ) = U ( X ∗ , B ) almost surely. Applying Theorem 1, we know that X ∗ ∈ X ˜ UB ( λξ ) forsome λ > • When λ < + ∞ , this means that U ( X ∗ , B ) − λξX ∗ = ˜ U ( X ∗ , B ) − λξX ∗ > ˜ U ( x, B ) − λξx > U ( x, B ) − λξx, ∀ x ∈ R . Therefore, X ∗ ∈ X UB ( λξ ). • When λ = + ∞ , we have X ∗ = x ˜ UB = x UB (based on Lemma 1). Hence, X ∗ ∈ X UB ( λξ ).Conversely, if there exists X ∗ ∈ X UB ( λξ ) satisfying E [ ξX ∗ ] = x and U ( X ∗ , B ) > −∞ , then: • If λ < + ∞ . For any other X satisfying the budget constraint and U ( X, B ) > −∞ , we have U ( X ∗ , B ) − λξX ∗ > U ( X, B ) − λξX > −∞ . Taking the expectation on both sides, we obtain E [ U ( X ∗ , B )] > E [ U ( X, B )], as such, E [ U ( X ∗ , B )] > sup X : E [ ξX ]= x U ( X,B ) > −∞ E [ U ( X, B )] > E [ U ( X ∗ , B )] . Thus, X ∗ is an optimal solution of Problem (2). • If λ = + ∞ , then X ∗ = x B . As U ( X, B ) > −∞ requires X > x B = X ∗ , we have E [ ξX ] > E [ ξX ∗ ] = x . This indicates that X ∗ is the unique random variable satisfying the constraint inProblem (2), and hence X ∗ is the optimal solution.Note that in this part we do not need the non-atomic condition. The proof is also valid for the “if”part in Theorem 1. Remark . For the weaker case with only H n and G n being finite, we should take some m ∈ N + in Part I with E [ ˜ U ( X, B )] > β + m . And then we make our operation on the set S = n ( x, b ) ∈ R × E : ˜ U ( x, b ) > U ( x, b ) + m o and replace X by H m ( X, B ) and G m ( X, B ). 20o close this section, we summarize that for the benchmark B and the multivariate U , a tractableapproach is proposed to determine (the existence and expression of) the finite optimal solution. However,in the abstract setting, we only know about a finite optimal solution X that X ∈ X B ( λξ ). Different fromthe classic case where the optimal solution is consequently determined, we have to prove the existence of ameasurable selection and find out its expression. The work will be done in Section 5. This section aims to study the feasibility of Problem (2). We define the feasible set I , { x : Problem (2) has a feasible solution for the initial value x } . In our model, there are two constraints on the return X : one is the budget constraint, i.e., E [ ξX ] = x , andthe other is a (state-dependent) lower bound constraint embedded in the domain of U , i.e., X > x B > −∞ almost surely. Using these constraints, the following Theorem 3 characterizes the structure of the feasibleset I . Theorem 3.
Suppose that Assumption 1 holds, and that the expectations E [ ξx B ] and E [ ξx B ] are well-defined.Exactly one of the following holds:(i) I = [ E [ ξx B ] , + ∞ ) ;(ii) I = (˜ x, + ∞ ) for some ˜ x > E [ ξx B ] , where ˜ x is allowed to take value in {±∞} .Moreover, if x = E [ ξx B ] , X = x B is the only possible feasible solution. Finally, if x > E [ ξx B ] , Problem (2) admits a bliss solution.Proof. To start with, it is clear that • If x < E [ ξx B ], then x / ∈ I ; • If x = E [ ξx B ], X = x B is the only possible feasible solution. • If x > E [ ξx B ], Problem (2) admits a bliss solution X > x B .The main difficulty appears in the case E [ ξx B ] < x < E [ ξx B ]. We first prove that the set I is connected,i.e., if x ∈ I , then [ x, + ∞ ) ⊂ I . For x ∈ I , suppose X is a feasible solution with respect to the initial value x . Setting X ′ = X + c { ξ
0. Noting that ∆ x U ( X, B ) , U ( X + , B ) − U ( X − , B ) is finite and21onnegative on A , we have A = { ∆ x U ( X, B ) < N } ∩ { ξ < N } with P [ A ] > N >
0. As U ( · , B ) is nondecreasing and right continuous, we rewrite A as A = [ n ∈ N + (cid:26) U ( X, B ) − U ( X − n , B ) < N (cid:27)! \ { ξ < N } . Therefore, there exists N > A , (cid:8) U ( X, B ) − U ( X − N , B ) < N (cid:9) ∩ { ξ < N } has positiveprobability. Define X ′ = X − N A . Based on the definition of A , we know X ′ > x B , and U ( X ′ , B ) >U ( X, B ) − N . As such, E [ U ( X ′ , B )] > E [ U ( X, B )] − N > −∞ . Further, E [ ξX ′ ] = E [ ξX ] − N E [ ξ A ] is afinite number x < x with x ∈ I , which also indicates [ x , + ∞ ) ⊂ I .As a result, we have either I = [ E [ ξx B ] , + ∞ ) or I = (˜ x, + ∞ ) for some ˜ x > E [ ξx B ], where ˜ x is allowedto take value in {±∞} .In the light of Theorem 3, the most complicated and unclear case of Problem (2) is E [ ξx B ] < x < E [ ξx B ].In particular, when x B x > x B is typically taken to be+ ∞ . When x B < + ∞ and E [ ξx B ] < + ∞ , it means that U ( · , B ) is a constant on [ x B , + ∞ ), and it is possibleto obtain a bliss solution for large x . Theorem 3 shows that I is a connected interval with right endpointbeing + ∞ , but we do not know its left endpoint, which depends highly on U and B . In the rest of the paper,we will focus mainly on the case that E [ ξx B ] < x < E [ ξx B ] with x ∈ I . Remark . Practically, x B > −∞ means that the manager cannot bear a loss exceeding − x B , and then thecondition x > E (cid:2) ξx B (cid:3) requires enough initial capital to face the potential risk. Moreover, if x is largeenough such that E [ ξx B ] < x < + ∞ , then the manager can reach the maximal utility without any risk.Hence the results indicate that people with high tolerance for loss are easy to find a satisfactory strategy,while people who are easy to be satisfactory or have enough initial capital will obtain a bliss solution. We are going to fully investigate the feasibility, finiteness, attainability and uniqueness of optimalsolution(s). The results are presented in Theorem 4 and summarized in Table 1 and Figure 2.
As we have discussed in Section 3, to find a finite optimal solution, we desire to determine λ ∈ [0 , + ∞ ]and X ∈ X B ( λξ ) satisfying E [ ξX ] = x . The key difficulty here is that we need to determine both a Lagrangemultiplier λ ∗ and a measurable selection of X B ( λ ∗ ξ ). If the desired λ ∗ does not exist, then Problem (2) iseither infinite or unattainable. In this section, we will investigate the issues of finiteness, attainability anduniqueness. Moreover, even if the optimal solution exists, we find that it may not be unique under somenovel conditions. For finiteness, we will also propose sufficient conditions to attain a finite optimal solutionfor Problem (2) in Section 6. 22irst, for any y ∈ [0 , + ∞ ] and b ∈ E , as the set X b ( y ) is non-empty, we define X b ( y ) , sup X b ( y ) , X b ( y ) , inf X b ( y ) . (20)These two quantities are maximum and minimum of the set X b ( y ). They are important in the measurableselection of the optimal solution. Some properties are listed in Lemma 5; its proof is given in Appendix A. Lemma 5.
Suppose that Assumption 1 holds. The functions V b (see (7) ), X b and X b satisfy:(i) for any y ∈ [0 , + ∞ ] , X b ( y ) > x b ; for any y ∈ (0 , + ∞ ] , X b ( y ) x b ;(ii) X b ( y ) ∈ X b ( y ) , X b ( y ) ∈ X b ( y ) , and X b ( y ) X b ( y ) holds for any y < y + ∞ ;(iii) both X b ( y ) and X b ( y ) are nonincreasing in y , and Borel-measurable in ( y, b ) ;(iv) if U ( · , b ) ∈ H , then lim y → X b ( y ) = lim y → X b ( y ) = x b , lim y → + ∞ X b ( y ) = lim y → + ∞ X b ( y ) = x b ;(v) if U ( · , b ) ∈ H , then for any y ∈ (0 , + ∞ ) , lim y → y − X b ( y ) = X b ( y ) , lim y → y + X b ( y ) = X b ( y ) . From the lemma we know that, when U ( · , b ) ∈ H , X b ( · ) is nonincreasing and right continuous on[0 , + ∞ ], while X b ( y ) is nonincreasing and left continuous. They both have countable discontinuous points.Moreover, X b ( · ) or X b ( · ) is discontinuous at y ∈ (0 , + ∞ ) if and only if its right limit does not equal to itsleft limit, i.e., X b ( y ) = X b ( y ), which means that X b ( y ) is not a singleton.For λ ∈ [0 , + ∞ ], we attempt to take X ∗ = X B ( λξ ) as a candidate of the measurable selection, and wedefine g ( λ ) = E [ ξX B ( λξ )] . (21)Here we introduce another standing assumption of the following sections: Assumption 6 (Well-definedness of g ) . The expectation in (21) is well-defined for any λ ∈ [0 , + ∞ ], and g ( λ ) > −∞ for any λ ∈ [0 , + ∞ ).Proposition 2 provides a sufficient condition of Assumption 6. The proof is given in Appendix A. Proposition 2. If E [ ξx B ] > −∞ , then Assumption 6 holds. Different from the literature, there are two novel features of g in our discussion: finiteness and continuity.For the finiteness, there are three possible cases in terms of g (Figure 2):Case 1: g ( λ ) < + ∞ for any λ > λ > g ( λ ) < + ∞ on ( λ , + ∞ ) , g ( λ ) = + ∞ on (0 , λ ) . (22)Case 3: g ( λ ) = + ∞ for any λ >
0. 23 (i) Case 1, continuous bliss non-unique non-uniqueinfeasible (ii) Case 1, discontinuous non-unique infeasibleunattainable (iii) Case 2, continuous, g ( λ ) < + ∞ = g ( λ − ) unattainable non-uniquenon-uniqueinfeasible (iv) Case 2, discontinuous, g ( λ ) < + ∞ = g ( λ − ) infeasible (v) Case 2, continuous, g ( λ ) = + ∞ non-uniqueinfeasible (vi) Case 2, discontinuous, g ( λ ) = + ∞ Figure 2: Some examples for g . Case 3 is not included because g is not finite. Especially, in Case 1, g (0) canbe finite or infinite and there is no bliss region for the latter. In Case 2, the term E [ ξX B ( λ ξ )] can be finiteor infinite, and there is no unattainable region for the latter. Meanwhile, in all three cases, the term g (+ ∞ )may equal to or less than 0, and then there will be no infeasible region.24n the literature, Case 1 is always assumed to ensure the existence of the optimal Lagrange multiplier λ ∗ (cf. Karatzas et al. (1987); Kramkov and Schachermayer (1999)). From the perspective of Theorem 5, inthe classic univariate problem (1), we can write g ( λ ) = E [ ξ ( u ′ ) − ( λξ )], which is a continuous function if ξ isnon-atomic. Therefore, for any x ∈ I , one can adopt the intermediate value theorem and find a Lagrangemultiplier λ ∗ > g ( λ ) = x and solve the problem (cf. Jin et al. (2008)).On the infiniteness issue of g , Case 2 and Case 3 are investigated in detail in this paper. In Theorem 4 wegive a complete investigation on all of three cases. For Case 3, Theorem 4(1) shows that either Problem (2)admits at most one feasible solution, or it is infinite. Case 2 is more complicated, as in this case the optimalLagrange multiplier may exist only for some x ∈ I , though we are only using X B ( λξ ) as a representationof X B ( λξ ), Theorem 4 shows that there may be no measurable selection as a finite optimal solution.On the continuity issue of g , g may be discontinuous in our model. In fact, based on Lemma 5, we knowthat X b ( y ) is nonincreasing and right continuous with respect to y . Under Assumption 6, the monotoneconvergence theorem implies that g is nonincreasing and right continuous, g ( λ ) = lim λ → λ + g ( λ ), which canbe finite or infinite. However, g is discontinuous at some λ if the set of random variables X B ( λξ ) is not asingleton, that is, X B ( λξ ) < X B ( λξ ) happens for a positive probability. In this case, to find an optimalsolution for x ∈ ( g ( λ ) , g ( λ − )), our basic technique is to “gradually” change X ∗ from X B ( λξ ) to X B ( λξ ).As such, the value E [ ξX ∗ ] will vary from g ( λ ) to g ( λ − ), and we will obtain an optimal solution.Significantly, we can construct infinitely many optimal solutions in this case. These results are concludedin Theorem 4. Noting that g ( λ ) = lim λ → λ + g ( λ ), which may be finite or infinite. In the following context, E (cid:2) ξX B ( λ ξ ) (cid:3) is also allowed to be + ∞ .In the following theorem, for simplicity we will use the expression “(unique)” to mean “unique underAssumption 5”. Theorem 4.
Suppose that (Ω , F , P ) is non-atomic and Assumptions 1-2 and 6 hold.(1) If Case 3 holds, then Problem (2) is infinite for any x ∈ I \{ E [ ξx B ] } .(2) if Case 1 or Case 2 holds, then we have λ ∈ [0 , + ∞ ) in (22) . Suppose that Assumption 3 holds and x ∈ I . We have:(a) if there exists a real number λ ∗ > λ such that g ( λ ∗ ) = x , then X ∗ = X B (cid:0) λ ∗ ξ (cid:1) is the (unique)optimal solution;(b) if g ( λ ∗ ) < x g ( λ ∗ − ) for some discontinuous point λ ∗ > λ , it has a positive probability that theset of optimal solutions X B (cid:0) λ ∗ ξ (cid:1) is not a singleton. Moreover, if x = g ( λ ∗ − ) , the optimal solutionis (unique) and is X ∗ = X B ( λ ∗ ξ ) ; if x < g ( λ ∗ − ) , there are infinitely many optimal solutions;(c1) (for Case 1, where λ = 0 ) if x > g ( λ ) = g (0) , then we have a bliss solution;(c2) (for Case 2, where λ > ) if x = E [ ξX B ( λ ξ )] , the optimal solution is (unique) and equals X B ( λ ξ ) ;if g ( λ ) < x < E [ ξX B ( λ ξ )] , there are infinitely many optimal solutions. if x > θ , E [ ξX B ( λ ξ )] , e have a sequence { ˆ X k } k > converging almost surely to X B ( λ ξ ) with E h ξ ˆ X k i = x and sup E [ ξX ] x U ( X,B ) > −∞ E [ U ( X, B )] = lim k → + ∞ E h U (cid:16) ˆ X k , B (cid:17)i E (cid:2) U (cid:0) X B ( λ ξ ) , B (cid:1)(cid:3) + λ ( x − θ ) . (23) Under some special cases, (23) holds as an equality. If Assumption 4 holds, Problem (2) has nofinite optimal solutions.
Theorem 4 gives tractable methods to find out optimal solutions of Problem (2) using g and answers thequestion that Theorem 2 leaves, while in most parts of Theorem 4 we do not use Assumption 4 which Theorem2 requires. Specifically, Theorem 2 shows that a necessary and sufficient condition for X to be an optimalsolution is that X ∈ X B ( λξ ) for some λ satisfying E [ ξX ] = x , but different from the univariate case wherethe term X can be defined as a function. Here we need an extra work of measurable selection, and Theorem 4proves the existence of such a selection by construction. (a) and (b) deal with the case x ∈ ( g (+ ∞ ) , g ( λ )],while (c1) and (c2) aim at the situation where x > g ( λ ), and the case that x g (+ ∞ ) has been alreadystudied in Theorem 3.The parts (c1)(c2) of Theorem 4 (when x > g ( λ )) are first investigated in this paper. Most literatureonly concentrates on Case 1. In our models, when a benchmark B is involved, the issue of measurableselection arises. Here, our g considers X B as a candidate. We have shown the existence of a measurableselection satisfying the budget constraint. Based on Theorems 3-4, we entirely figure out the (non-)existenceand (non-)uniqueness of an optimal solution for all scenarios. Moreover, under some sufficient conditions, wecan pass the limit in (23) into the expectation and derivesup E [ ξX ] x U ( X,B ) > −∞ E [ U ( X, B )] = E (cid:2) U (cid:0) X B ( λ ξ ) , B (cid:1)(cid:3) , which indicates that the optimal value will not increase when the initial value x begins to exceed the number α (cf. (14)); see Proposition 5 for details.The result of non-unique solutions is because of the generality of both U and B . On the one hand, if U is strictly concave, then X b ( y ) contains always one element and non-unique optimal solutions describedin (b) will not occur. On the other hand, if B is a constant, then X B ( λξ ) is not a singleton if and only if λξ lies in the discontinuous point set of X B . Noting that X B is decreasing, the set is countable. As such,the probability that the random set X B (cid:0) λ ∗ ξ (cid:1) is not a singleton equals to zero when the pricing kernel ξ isnon-atomic, which means that non-unique optimal solutions will not happen either. Therefore, it is only aspecial case of Problem (2). Remark . As B can be any measurable variable, the optimal solution X B ( λ ∗ ξ ) may not be a function of ξ .Moreover, as the function X b ( y ) performs no evident monotonicity in b , X B ( λ ∗ ξ ) may not be a decreasingfunction of ξ even if B is a function of ξ . This result is very different from the classic one obtained by quantileformulation (cf. Jin and Zhou (2008); He and Zhou (2011); Xu (2016)).26ase 1 Case 2 Case 3 x < E [ ξx B ] = g (+ ∞ ) infeasible (Thm 3) x = E [ ξx B ] = g (+ ∞ ) at most one feasible solution x B (Thm 3) x ∈ g ([ λ , + ∞ )) unique (Thm 4(2)(a)) infinite(Thm 4(1)) x ∈ ( g ( λ ∗ ) , g ( λ ∗ − )) non-unique (Thm 4(2)(b)) x = g ( λ ∗ − ) unique (Thm 4(2)(b)) x > g ( λ ) bliss(Thm 4(2)(c1)) x > E [ ξX B ( λ ξ )]: unattainable (Thm 4(2)(c2)) x = E [ ξX B ( λ ξ )]: unique (Thm 4(2)(c2)) x < E [ ξX B ( λ ξ )]: non-unique (Thm 4(2)(c2))Table 1: Solutions of Problem (2) when Assumptions 3-5 hold and inf I = E [ ξx B ]. Note: Denote by λ ∗ ∈ ( λ , + ∞ ) a discontinuous point of g .For the finiteness and attainability, as our results in Theorem 4 (2)(a)(b)(c1) (and (c2) when x θ )are tractable, Problem (2) is all attainable in these situations, and one can verify the finiteness directly. Forthe remaining case (c2) with x > θ , Theorem 4 indicates that θ is the largest initial value for which we canfind an optimal solution through g . For this initial value, we have an optimal solution X B ( λ ξ ), and we canverify the finiteness of Problem (2). We have two cases: • If Problem (2) is finite for α , then (23) shows that for the initial value x , Problem (2) is also finite.Moreover, Theorem 4 asserts that there is no optimal solution, i.e. the problem is unattainable for x . • If Problem (2) is infinite for α , then as x > θ and that U ( x, b ) is nondecreasing in x , we can simplyturn up the infinite optimal solution X B ( λ ξ ) for Problem (2) with initial value θ to obtain an infiniteoptimal solution with initial value x .In this light, we can use the function g and Theorem 4 to completely figure out the existence and expressionsof optimal solutions, finiteness and attainability of Problem (2), while Theorem 2 gives support on uniquenessof the optimal solution. The proof of Theorem 4 requires the following lemma:
Lemma 6.
For any finite random variable Y in a non-atomic probability space (Ω , F , P ) with E [ Y + ] = + ∞ and any a > , there exists an event A ∈ F such that E [ Y A ] = a .Proof of Lemma 6. It suffices to find A ∈ F such that E [ Y + A ] = a . Hence we only need to consider thecase that Y >
0. Define F ( t ) = E [ Y { Y
0, as E [ ξX B ( λξ )] = + ∞ , we know that E [ ξ ( X B ( λξ ) − X ′ )] = + ∞ . Using Lemma 6, wehave A ∈ F satisfying E [ ξ ( X B ( λξ ) − X ′ ) A ] = x − x . Define X , X B ( λξ ) A + X ′ A c , then E [ ξX ] = E [ ξX B ( λξ ) A ] + x − E [ ξX ′ A ] = x , and E [ U ( X, B )] = E [ U ( X B ( λξ ) , B ) A ] + E [ U ( X ′ , B ) A c ] > E [ U ( X ′ , B ) + λξ ( X B ( λξ ) − X ′ ) A ]= E [ U ( X ′ , B )] + λ ( x − x ) . Letting λ → + ∞ , it follows that Problem (2) is infinite.(2) (a) For X ∗ = X B (cid:0) λ ∗ ξ (cid:1) ∈ X B (cid:0) λ ∗ ξ (cid:1) and any solution X satisfying E [ ξX ] = x = E [ ξX ∗ ], we have X ∗ > −∞ a.s. . Then, based on the definition of X b ( y ) in (8), we know U ( X ∗ , B ) > −∞ a.s. , andwe have U ( X ∗ , B ) − λ ∗ ξX ∗ > U ( X, B ) − λ ∗ ξX. (24)As such, E (cid:2) U ( X ∗ , B ) (cid:3) > E (cid:2) U ( X, B ) (cid:3) , (25)and the optimality of X ∗ then follows. As x ∈ I , we can take X in (25) as a feasible solution, andthen we know that the left side is not −∞ . When Assumption 5 holds, (25) takes “=” if and only if(24) takes “=” almost surely, we have X ∈ X B ( λ ∗ ξ ) happens almost surely. Based on the definitionof X b , we know X > X ∗ . As E [ ξX ] = E [ ξX ∗ ], we have X = X ∗ almost surely, and the uniquenessof X ∗ follows.(b) For x = g ( λ ∗ − ), as λ ∗ > λ , based on Lemma 5, we have g ( λ ∗ − ) = E [ ξX B ( λ ∗ ξ )]. Therefore, X ∗ = X B ( λ ∗ ξ ) is an optimal solution. Similar to (1), we know that X ∗ is the unique optimalsolution.As λ ∗ is a discontinuous point, we know g ( λ ∗ − ) > g ( λ ∗ ), i.e. the set { X B ( λ ∗ ξ ) = X B ( λ ∗ ξ ) } has a positive measure. We show that there in fact exists infinitely many optimal solutions when x < g ( λ ∗ − ). As the probability space is non-atomic, it admits a standard normal distribution W .For A ∈ B ( R ), define X A = X B ( λ ∗ ξ ) + (cid:2) X B ( λ ∗ ξ ) − X B ( λ ∗ ξ ) (cid:3) { W ∈ A } ∈ X B ( λ ∗ ξ ) . X = ξX B ( λ ∗ ξ ) , X = ξ (cid:2) X B ( λ ∗ ξ ) − X B ( λ ∗ ξ ) (cid:3) . Then E X = g ( λ ∗ ) < x , E X = g ( λ ∗ − ) − g ( λ ∗ )( , ρ ) , and E (cid:2) ξX A ( T ) (cid:3) = E X + E [ X { W ∈ A } ] = g ( λ ∗ ) + E [ X { W ∈ A } ] . Define s ( A ) , E [ X { W ∈ A } ]. We need to choose A such that s ( A ) = x − g ( λ ∗ ) , ρ ∈ (0 , ρ ).Choose k large enough such that ρ − ρ k < ρ , and denote further a , ρ − ρ , a n , ρ − ρ − ρ n + k , n > . Then for any i, j , a < a < a < ... < ρ , a j < a i + a < ρ + ρ . (26)Let g ( t ) = s (cid:0) ( −∞ , t ) (cid:1) . Using the monotone convergence theorem, we know g ( −∞ ) = 0 and g (+ ∞ ) = s ( R ) = E X = ρ , and that g is increasing and continuous. Therefore, for any n , thereexist δ n , ε n and ζ n such that δ n < ε n < ζ n , g ( δ n ) = a n , g ( ε n ) = a n + a , g ( ζ n ) = ρ + ρ ρ + a . Using (26), we know that for any i , j , δ < δ < ... < δ n , ε i > δ j . Define A n , ( −∞ , δ n ) ∪ [ ε n , ζ n ). We have s ( A n ) = g ( δ n ) + g ( ζ n ) − g ( ε n ) = ρ . Consequently, X A n ∈ X B ( λ ∗ ξ ), E (cid:2) ξX A n (cid:3) = x . We obtain that X A n is an optimal solution. Assuch, it remains to prove that for any i < j , P [ X A i ( T ) = X A j ( T )] > , (27)or equivalently, P [ X { W ∈ A i } = X { W ∈ A j } ] >
0. As δ i < δ j < ε i , let A = ( −∞ , δ j ), then E (cid:2) X { W ∈ A i } { W ∈ A } (cid:3) = g ( δ i ) = a i < a j = g ( δ j ) = E (cid:2) X { W ∈ A j } { W ∈ A } (cid:3) . g (0) = E [ ξx B ].(c2) For g ( λ ) < x E [ ξX B ( λ ξ )], we know g ( λ ) = E [ ξX B ( λ ξ )] < + ∞ , and again we obtain P [ A ] = ε > A = { X B ( λ ξ ) = X B ( λ ξ ) } . Here we also consider two cases: • If E [ ξX B ( λ ξ )] < ∞ , using the same methods in (2), we construct infinitely many optimalsolutions when x < E [ ξX B ( λ ξ )] and one (unique) optimal solution when x = E [ ξX B ( λ ξ )]. • If E [ ξX B ( λ ξ )] = + ∞ , then we use Lemma 6 to construct an optimal solution. As E [ ξX B ( λ ξ )] = g ( λ ) < x < + ∞ , we know that E [ ξ ( X B ( λ ξ ) − X B ( λ ξ ))] = + ∞ . Then Lemma 6 shows thatwe can find A ∈ F such that E [ ξ ( X B ( λ ξ ) − X B ( λ ξ )) A ] > x − g ( λ ) . Using the same method as in (2), we have A ⊂ A with E [ ξ ( X B ( λ ξ ) − X B ( λ ξ )) A ] = x − g ( λ ) . Let X A = X B ( λ ξ ) A + X B ( λ ξ ) A c ∈ X B ( λ ξ ) , we have E [ ξX A ] = E [ ξX B ( λ ξ )] + E [ ξ ( X B ( λ ξ ) − X B ( λ ξ )) A ] = g ( λ ) + x − g ( λ ) = x . Based on Theorem 2, we have found an optimal solution X A . Similarly as in (2), one canconstruct infinitely many optimal solutions.For x > E [ ξX B ( λ ξ )] , θ . We know θ > g ( λ ) > −∞ . We first prove that there is no finite optimalsolution. Suppose X ∗ is a finite optimal solution. Using Theorem 2, we know that X ∗ ∈ X B ( λξ )for some λ >
0. We consider two cases: • If λ > λ , then, based on Lemma 5, we know X ∗ X B ( λ ξ ), and hence x = E [ ξX ∗ ] E [ ξX B ( λ ξ )], which leads to a contradiction. • If λ < λ , then as X ∗ > X B ( λ ′ ξ ) for some 0 λ < λ ′ < λ , we know g ( λ ′ ) < + ∞ , which leadsto a contradiction.For (23), take a sequence { λ k } k > ⊂ (0 , λ ) with λ k ↑ λ . For each k , E (cid:2) ξ (cid:0) X B ( λ k ξ ) − X B ( λ ξ ) (cid:1)(cid:3) = g ( λ k ) − θ = + ∞ . A k such that E (cid:2) ξ (cid:0) X B ( λ k ξ ) − X B ( λ ξ ) (cid:1) A k (cid:3) = x − θ. Define ˆ X k = X B ( λ k ξ ) A k + X B ( λ ξ ) A ck . Then E [ ξ ˆ X k ] = x . Hence ˆ X k > −∞ a.s. . As ˆ X k alwayslocates in some X b ( y ), we know that U ( ˆ X k , B ) > −∞ a.s. . Suppose that X is a random variablesatisfying E [ ξX ] = x and U ( X, B ) > −∞ . Based on the definition of ˆ X k , we have ( U ( X, B ) − λ k ξX ) A k (cid:16) U (cid:16) ˆ X k , B (cid:17) − λ k ξ ˆ X k (cid:17) A k , ( U ( X, B ) − λ ξX ) A ck (cid:16) U (cid:16) ˆ X k , B (cid:17) − λ ξ ˆ X k (cid:17) A ck . Taking an expectation on both sides and adding the two inequalities, we get E [ U ( X, B )] − λ x − ( λ k − λ ) E [ ξX A k ] E h U (cid:16) ˆ X k , B (cid:17)i − λ x − ( λ k − λ ) E h ξ ˆ X k A k i . (28)Using Lemma 5, we know that X B ( λ k ξ ) converges to X B ( λ ξ ) almost surely. Hence X B ( λ k ξ ) alsoconverges to X B ( λ ξ ) in probability. For ε >
0, we have P h(cid:12)(cid:12)(cid:12) ˆ X k − X B ( λ ξ ) (cid:12)(cid:12)(cid:12) > ε i P (cid:2)(cid:12)(cid:12) X B ( λ ξ ) − X B ( λ ξ ) (cid:12)(cid:12) > ε (cid:3) . As such, ˆ X k also converges to X B ( λ ξ ) in probability, and we can find one of its subsequence con-verging to X B ( λ ξ ) almost surely, which is still denoted by { ˆ X k } k > for simplicity. As E (cid:2) ξX B ( λ ξ ) (cid:3) is finite, we know E h ξ ˆ X k A k i > E (cid:2) ξX B ( λ ξ ) A k (cid:3) > − E (cid:2) ξX B ( λ ξ ) − (cid:3) > −∞ , and E h ξ ˆ X k A k i = x − E h ξ ˆ X k A ck i = x − E (cid:2) ξX B ( λ ξ ) A ck (cid:3) x + E (cid:2) ξX B ( λ ξ ) − (cid:3) < + ∞ . Letting k → + ∞ on both sides of (28), we know E [ U ( X, B )] lim inf k → + ∞ E h U (cid:16) ˆ X k , B (cid:17)i . (29)Hence sup E [ ξX ] x U ( X,B ) > −∞ E [ U ( X, B )] lim inf k → + ∞ E h U (cid:16) ˆ X k , B (cid:17)i . E h ξ ˆ X k i = x and U ( ˆ X k , B ) > −∞ , we have E h U (cid:16) ˆ X k , B (cid:17)i sup E [ ξX ] x U ( X,B ) > −∞ E [ U ( X, B )] . As such, we obtain sup E [ ξX ] x U ( X,B ) > −∞ E [ U ( X, B )] = lim k → + ∞ E h U (cid:16) ˆ X k , B (cid:17)i . Moreover, as U (cid:16) ˆ X k , B (cid:17) U (cid:0) X B ( λ ξ ) , B (cid:1) + λ ξ (cid:16) ˆ X k − X B ( λ ξ ) (cid:17) , Y k , using Fatou’s lemma,we derive E (cid:20) lim inf k → + ∞ (cid:16) Y k − U (cid:16) ˆ X k , B (cid:17)(cid:17)(cid:21) lim inf k → + ∞ E h Y k − U (cid:16) ˆ X k , B (cid:17)i . (30)As ˆ X k > X B ( λ ξ ) and ˆ X k → X B ( λ ξ ), using the right continuity of U , we know U (cid:16) ˆ X k , B (cid:17) → U (cid:0) X B ( λ ξ ) , B (cid:1) . Moreover, as E [ Y k ] = E (cid:2) U (cid:0) X B ( λ ξ ) , B (cid:1)(cid:3) + λ ( x − θ ) , lim k → + ∞ Y k = U (cid:0) X B ( λ ξ ) , B (cid:1) , using (30), we obtainlim k → + ∞ E h U (cid:16) ˆ X k , B (cid:17)i E (cid:2) U (cid:0) X B ( λ ξ ) , B (cid:1)(cid:3) + λ ( x − θ ) . Finally, we find that (23) holds as an equality in the following Example 2.
Example 2 (Sharpness of (23)) . Suppose that ξ is uniformly distributed on [1 , u ( x ) = x + 1 , x > , x, x , − ∞ , x < . Then we can compute X ( y ) = , y > , , y < , + ∞ , y < . and g ( λ ) = , λ > , − λ λ , λ , + ∞ , λ < . . Hence λ = 1, X ( λ ξ ) = 1 a.s. , θ , E (cid:2) ξX ( λ ξ ) (cid:3) = and E (cid:2) u (cid:0) X ( λ ξ ) (cid:1)(cid:3) = 2. For x > θ , using (43), weknow sup X : E [ ξX ] x u ( X ) > −∞ E [ u ( X )] E (cid:2) u (cid:0) X ( λ ξ ) (cid:1)(cid:3) + λ ( x − θ ) = x + 12 . (31)32or ε >
0, define X ε = [1+ ε, ( ξ ) + (cid:18) x − ε + 2 ε (cid:19) [1 , ε ] ( ξ ) > . Then E [ ξX ε ] = 4 − (1 + ε ) ε + 2 ε (cid:18) x − ε + 2 ε (cid:19) = x , and E [ u ( X ε )] = 2(1 − ε ) + (2 + 2 x − ε + 2 ε ) ε = 2 + 2 x −
32 + ε .
Letting ε →
0, we know that (31) takes “=”.
We have formulated solutions of Problem (2) completely, with various assumptions on Case 1 (existenceof the Lagrange multiplier), well-posedness and finiteness of Problem (2). In this section, we propose theirsufficient conditions in Proposition 3.
Proposition 3.
Suppose that E [ ξx B ] > −∞ , ξ ∈ L (Ω) , and Assumption 1 and both of the followings hold:(1) There exist u ( b ) > , u ( b ) > , K ( b ) > and δ ∈ (0 , such that for any x > K ( b ) , U ( x, b ) u ( b ) + u ( b ) x δ with ξ − δ u ( B ) ∈ L − δ (Ω) and ξ − δ u ( B ) ∈ L − δ (Ω) , and ξK ( B ) ∈ L (Ω) ;(2) There exist θ ( b ) > and γ ( b ) = U ( x b + θ ( b ) , b ) such that ξ − δ γ ( B ) ∈ L − δ (Ω) , ξθ ( B ) ∈ L (Ω) .Then Case 1 holds, and Problem (2) is well-posed and finite for any x ∈ R .Proof of Proposition 3. Denote γ ( b ) = U ( x b + θ ( b ) , b ) > −∞ . We prove X b ( y ) (cid:0) K ( b ) y (cid:1) − δ + K ( b ) , (32)where K ( b ) = 2( u ( b ) + u ( b ) + | γ ( b ) | ) , K ( b ) = K ( b ) + 2 θ ( b ) + 2 | x b | .If x , X b ( y ) > (cid:0) K ( b ) y (cid:1) − δ + K ( b ), then, based on the definition of X b ( y ), we know that for any x ′ > x b , U ( x ′ , b ) − yx ′ U ( x, b ) − yx. Letting x ′ = x , we have x y h U ( x, b ) − U (cid:16) x , b (cid:17)i . (33)As x > K ( b ), we know x > x b + θ ( b ) and U ( x , b ) > γ ( b ). In addition, as x > K ( b ) > K ( b ), we also have33 ( x, b ) u ( b ) + u ( b ) x δ , and then (33) leads to x y (cid:2) u ( b ) + u ( b ) x δ − γ ( b ) (cid:3) , or x δ (cid:0) x − δ − u ( b ) y (cid:1) y ( u ( b ) − γ ( b )) . (34)However, as x > (cid:0) K ( b ) y (cid:1) − δ , we have x δ >
1, and x − δ − u ( b ) y > u ( b ) + 2 | γ ( b ) | y . As such, (34) leads to y ( u ( b ) + | γ ( b ) | ) < y ( u ( b ) − γ ( b )), which is a contradiction. Thus (32) holds.Now, for λ >
0, using (32), we have X B ( λξ ) (cid:0) K ( B ) λξ (cid:1) − δ + K ( B ). As such, there exists K > X B ( λξ ) K (cid:0) λ − − δ ξ − − δ K ( B ) − δ + θ ( B ) + | x B | + K ( B ) (cid:1) , (35)and it follows that for some K > ξX B ( λξ ) + K (cid:20) ξ (cid:0) | x B | + θ ( B ) + K ( B ) (cid:1) + λ − − δ ξ − δ − δ (cid:18) u ( B ) − δ + u ( B ) − δ + | γ ( B ) | − δ (cid:19)(cid:21) . Based on our conditions in Proposition 3, we know that E [ ξX B ( λξ ) + ] < + ∞ , ∀ λ > . (36)Combining (36) with (51), we know g ( λ ) = E [ ξX B ( λξ )] ∈ R , that is, Case 1 holds.For the well-posedness and finiteness of Problem (2), suppose that we have some X satisfying E [ ξX ] x , and U ( X, B ) > −∞ . We first derive from the definition of X b ( y ) that U ( X, B ) − λξX U ( X B ( λξ ) , B ) − λξX B ( λξ ) . (37)Hence U ( X, B ) + U ( X B ( λξ ) , B ) + + ( λξX − λξX B ( λξ )) + U ( X B ( λξ ) , B ) + + | λξX | + | λξX B ( λξ ) | . (38)Noting that E [ | ξX | ] = E (cid:2) ξX + (cid:3) + E (cid:2) ξX − (cid:3) = E [ ξX ] + 2 E (cid:2) ξX − (cid:3) x + 2 E (cid:2) ξx − B (cid:3) < + ∞ , E [ | ξX B ( λξ ) | ] g ( λ ) + 2 E (cid:2) ξx − B (cid:3) < + ∞ . Taking an expectation on both sides of (38), we obtain E [ U ( X, B ) + ] E h U ( X B ( λξ ) , B ) + i + C ( λ ) , (39)where C ( λ ) is a finite number depending on λ . We proceed to prove E [ U ( X B ( λξ ) , B ) + ] < + ∞ . Using (35),we know U ( X B ( λξ ) , B ) + u ( B ) K δ (cid:0) λ − − δ ξ − − δ K ( B ) − δ + θ ( B )+ | x B | + K ( B ) (cid:1) δ + u ( B ) . For simplicity, we take λ = 1 and derive for some K > U ( X B ( ξ ) , B ) + K (cid:20) u ( B ) + u ( B ) θ ( B ) δ + u ( B ) K ( B ) δ + u ( B ) | x B | δ + ξ − δ − δ (cid:18) u ( B ) − δ + u ( B ) u ( B ) δ − δ + u ( B ) | γ ( B ) | δ − δ (cid:19)(cid:21) + u ( B ) . (40)As E [ ξ − δ − δ u ( B ) − δ ] < + ∞ and E [ ξ ] < + ∞ , using H¨older’s inequality, we obtain E [ u ( B )] = E (cid:20)(cid:18) ξ − δ − δ u ( B ) − δ (cid:19) − δ ξ δ (cid:21) < + ∞ . Similarly, we can write u ( B ) = (cid:18) ξ − δ − δ u ( B ) − δ (cid:19) − δ ξ δ ,u ( B ) θ ( B ) δ = (cid:18) ξ − δ − δ u ( B ) − δ (cid:19) − δ (cid:18) ξθ ( B ) (cid:19) δ ,u ( B ) K ( B ) δ = (cid:18) ξ − δ − δ u ( B ) − δ (cid:19) − δ (cid:18) ξK ( B ) (cid:19) δ ,u ( B ) | x B | δ = (cid:18) ξ − δ − δ u ( B ) − δ (cid:19) − δ (cid:18) ξ | x B | (cid:19) δ ,ξ − δ − δ u ( B ) u ( B ) δ − δ = (cid:18) ξ − δ − δ u ( B ) − δ (cid:19) − δ (cid:18) ξ − δ − δ u ( B ) − δ (cid:19) δ ,ξ − δ − δ u ( B ) | γ ( B ) | δ − δ = (cid:18) ξ − δ − δ u ( B ) − δ (cid:19) − δ (cid:18) ξ − δ − δ | γ ( B ) | − δ (cid:19) δ . As such, the expectation of each term at left is finite, and (40) leads to E [ U ( X B ( ξ ) , B ) + ] < + ∞ . Using (39),we know E [ U ( X, B ) + ] < + ∞ , and alsosup E [ ξX ] x U ( X,B ) > −∞ E [ U ( X, B )] E h U ( X B ( ξ ) , B ) + i + C (1) < + ∞ . That is, Problem (2) is well-posed and finite. 35oting that the benchmark variable b is involved in every parameter above, we have to give integrabilityconditions on each of them. The conditions we proposed in Proposition 3 seem complicated, but it is indeedeasy to satisfy; see the following examples. Example 3. If B is a constant, then u ( B ) , u ( B ) , K ( B ) , θ ( B ) and γ ( B ) are all constants, and hence weonly need E [ ξ ] < + ∞ and E [ ξ − δ − δ ] < + ∞ , which hold if ξ is lognormal. Example 4.
Fix b ∈ R . For a S-shaped utility U ( x, b ) = ( x − b ) p , x > b, − k ( b − x ) p , x b, if we take u ( b ) = 1, u ( b ) = 0, K ( b ) = 0, δ = p , θ ( b ) = b , γ ( b ) = 0, then Proposition 3 requires that E [ ξ ] < + ∞ , E [ ξ − p − p ] < + ∞ and E [ ξB ] < + ∞ . If we use θ ( b ) = 0, γ ( b ) = − kb p , then E [ ξB ] < + ∞ can bereplaced by E [ ξ − p − p B p − p ] < + ∞ .Based on Remark 1 and (37)-(39) in the proof of Proposition 3, we can also verify the finiteness ofProblem (2) using g and X B . We have: Proposition 4.
Suppose that x ∈ R and that there exists λ ∈ [0 , + ∞ ) such that g ( λ ) and E [ U ( X B ( λξ ) , B )] are well-defined and finite. If Assumption 2 holds or E [ ξx B ] > −∞ , then Problem (2) is well-posed and finite. In the light of Proposition 4, we can also study the finiteness of Problem (2) using g and J ( λ ) , E [ U ( X B ( λξ ) , B )] , λ ∈ (0 , + ∞ ) . (41) Assumption 7 (Well-definedness of J ) . The expectation in (41) is well-defined for any λ ∈ (0 , + ∞ ) and J ( λ ) > −∞ for any λ ∈ (0 , + ∞ ).Under assumption 7, similarly as λ in (22), we define λ ∈ [0 , + ∞ ] from J ( λ ) < + ∞ for λ ∈ ( λ , + ∞ ) , J ( λ ) = + ∞ for λ ∈ (0 , λ ) . We have the following results on the optimal value of Problem (2):
Proposition 5.
Suppose that Assumptions 2, 3, 6, 7 hold and x ∈ I \{ E [ ξx B ] } , then(i) If λ = + ∞ , then Problem (2) is infinite;(ii) If λ < + ∞ , but λ = + ∞ , then Problem (2) is infinite;(iii) If λ < + ∞ , and λ < + ∞ , then Problem (2) is finite, and we have λ λ . Moreover, if λ < λ ,then the limit in (23) can be passed into the expectation and we have sup E [ ξX ] x U ( X,B ) > −∞ E [ U ( X, B )] = E (cid:2) U (cid:0) X B ( λ ξ ) , B (cid:1)(cid:3) . (42)36n this light, we can first compute g and J to verify the finiteness of Problem (2), then we use Theorem4 to find a finite solution or an infinite solution. Proof of Proposition 5. (i) is the same as Theorem 4 (1).For (ii), if x ∈ ( g (+ ∞ ) , g ( λ ))(may be empty) and x = g ( λ ), using Theorem 4, we can find an optimalsolution X ∗ ∈ X B ( λξ ) for some λ ∈ [ λ , + ∞ ). Hence X ∗ > X B ( λξ ), and we have E [ U ( X ∗ , B )] > J ( λ ) =+ ∞ .If x > g ( λ ), then as U ( x, b ) is nondecreasing in x , we know that the optimal value of Problem (2) isnondecreasing in the initial value. Hence Problem (2) with an initial value x > g ( λ ) is also infinite.For (iii), as λ < + ∞ and λ < + ∞ , there exists λ ∈ (0 , + ∞ ) such that g ( λ ) < + ∞ and J ( λ ) < + ∞ .Then g ( λ ) and J ( λ ) are all finite. Using Proposition 4, we know that Problem (2) is finite.If λ > λ , then we define ˆ x = g (cid:0) λ + λ (cid:1) . Using Theorem 4, we know that Problem (2) with initialvalue ˆ x admits an optimal solution X ∗ = X B (cid:0) λ + λ ξ (cid:1) , and the optimal value equals J (cid:0) λ + λ (cid:1) = + ∞ ,which contradicts to Proposition 4. Thus, λ λ .Suppose that λ < λ . We desire to prove (42) in (c2) of Theorem 4. In fact, in this case we can takesome λ ′ ∈ ( λ , λ ). Noting that in (29), we have ˆ X k X B ( λ ′ ξ ) for k sufficiently large, using Fatou’s Lemma,we obtain lim sup k → + ∞ E h U (cid:16) ˆ X k , B (cid:17)i E (cid:20) lim sup k → + ∞ U (cid:16) ˆ X k , B (cid:17)(cid:21) = E (cid:2) U (cid:0) X B ( λ ξ ) , B (cid:1)(cid:3) . Hence (23) leads to sup E [ ξX ] x U ( X,B ) > −∞ E [ U ( X, B )] E (cid:2) U (cid:0) X B ( λ ξ ) , B (cid:1)(cid:3) . As the optimal value of Problem (2) is nondecreasing with the initial value, we havesup E [ ξX ] x U ( X,B ) > −∞ E [ U ( X, B )] > E (cid:2) U (cid:0) X B ( λ ξ ) , B (cid:1)(cid:3) . Therefore, (42) holds.
In this section, we connect our results to the univariate framework of general utilities, while we assumethat u satisfies good conditions (differentiable and strictly concave) in Problem (1). The univariate utilityis state-independent and the univariate framework depends only on the distribution of the wealth X , andhence the objective is distribution-invariant. Fix u ∈ H . For y >
0, define the conjugate set function X ofthe utility u as X ( y ) = arg sup x ∈ R ∪{±∞} [ u ( x ) − yx ] . X ( y ) = inf X ( y ) , X ( y ) = sup X ( y )as well as g ( λ ) = E [ ξX ( λξ )]. Similar as in (22), we define λ .In this section we assume that the univariate version of Assumptions 2-4 and 6 hold. Using the resultsin Sections 3-6, we propose a complete result for the univariate utility u ∈ H in Theorem 5 and the followingdiscussion. Theorem 5.
Suppose that (Ω , F , P ) is non-atomic and that the univariate version of Assumptions 3-6 hold.(1) X ∗ is an optimal solution of Problem (1) if and only if there exists λ ∗ > such that X ∗ ∈ X ( λ ∗ ξ ) and E [ ξX ∗ ] = x .(2) The optimal solution is unique if the set X ( y ) is singleton for any y > or ξ is non-atomic. Theset of optimal solutions X ( λ ∗ ξ ) may be non-unique if u is non-concave and ξ is atomic. Theorem 5 is a complete result for general (discontinuous and non-concave) utilities, which extends theunivariate results in literature; see Jin et al. (2008) and Reichlin (2013). Under our framework, Theorem 5becomes a direct corollary of the multivariate version Theorems 1-2. Thus, our framework also serves tooffer a proof for the univariate result. Apart from the uniqueness, for x ∈ R we also have (for simplicity, weassume ξ is integrable, then the feasible set I = [ x E [ ξ ] , + ∞ ) or ( x E [ ξ ] , + ∞ ), which depends on the type of x ):(i) When λ = + ∞ , the problem is infinite for every x ∈ ( x E [ ξ ] , + ∞ ). For x < x E [ ξ ], there is nofeasible solutions. While for x = x E [ ξ ], there is at most one feasible solution, which depends on thetype of x .(ii) When λ ∈ (0 , + ∞ ), the optimal solution(s) exists for all x ∈ (cid:0) x E [ ξ ] , E (cid:2) ξX ( λ ξ ) (cid:3)(cid:3) , and one candirectly verify the finiteness or infiniteness of the solution. For x > E (cid:2) ξX ( λ ξ ) (cid:3) , θ , we have twocases: • If the problem with initial value θ is finite, the problem for x is finite and unattainable, and we canfind a sequence { ˆ X k } converges to X ( λ ξ ) almost surely satisfying E h ξ ˆ X k i = x , u ( ˆ X k ) > −∞ and sup X : E [ ξX ] x u ( X ) > −∞ E [ u ( X )] = lim k → + ∞ E h u ( ˆ X k ) i E (cid:2) u (cid:0) X ( λ ξ ) (cid:1)(cid:3) + λ ( x − θ ) . (43)There are situations where the inequality becomes equity, see Example 2. • If the problem with initial value θ is infinite, then it is also infinite with initial value x , and anoptimal solution X ∗ can be obtained by simply turn up the optimal solution X ( λ ξ ) for initialvalue θ to satisfy E [ ξX ∗ ] = x . To prove Theorem 5, we should only regard B as a constant in Theorems 1-2, or just simply delete all the “ B ” appeared inthe proof. λ = 0, then for every x > x E [ ξ ] there exists an optimal solution, and the finiteness orinfiniteness can be verified directly.In Reichlin (2013), it is claimed that there exists ˜ x ∈ (0 , + ∞ ] such that for x ∈ (0 , ˜ x ), Problem (1)admits an optimal solution. While in Theorem 5, ˜ x can be indeed determined as ˜ x = E (cid:2) ξX ( λ ξ ) (cid:3) , and thecase when x > ˜ x is also supplemented. The result of a differentiable and strictly concave function u existsin Jin et al. (2008) and Karatzas and Shreve (1998); in this case, we have X = ( u ′ ) − and X ∗ = ( u ′ ) − ( λ ∗ ξ ).A key insight of Theorem 5 is that there may also exist non-unique solutions in the univariate framework.It may only happen if u is non-concave and ξ is atomic. In this case, the random set X ( λ ∗ ξ ) is not a singletonand g ( λ ) := E [ ξ X ( λξ )] is discontinuous at the Lagrange multiplier λ ∗ . The optimal solution is unique if ξ is non-atomic. However, in the multivariate framework, even if ξ is non-atomic, there may exist non-uniquesolutions. It is because of the state-dependent utility U and the stochastic benchmark B . They lead to anon-singleton random set X ( λ ∗ ξ ).Traditionally, the optimal solution for the affine utility is known to be X ∗ = + ∞ if the pricing kernel ξ is non-atomic. We finally propose Example 5, showing that for the affine utility, the optimal solution mayexist if ξ is atomic. Example 5 (Affine utility) . Fix k ∈ (0 , + ∞ ) and L ∈ R . Let u ( x ) = kx, x > L. For any x > L E [ ξ ], we find that the optimal solution exists if and only if ξ min , ess-inf ξ ∈ (0 , + ∞ ) and ξ has an atom at ξ min . Indeed, if the optimal solution exists, it is given by X ∗ ∈ X ( λ ∗ ξ ), where λ ∗ ∈ (0 , + ∞ )is a constant (to be determined) and X ( y ) = L, y > k ;[ L, + ∞ ] , y = k ;+ ∞ , y < k. To satisfy E [ ξX ∗ ] = x , we obtain that λ ∗ = k/ξ min and the optimal solution is X ∗ = L, ξ > ξ min ; x − L E (cid:2) ξ { ξ>ξ min } (cid:3) ξ min P [ ξ = ξ min ] , ξ = ξ min . Applications in finance
In this section, we formulate a constraint utility optimization problem with state-dependent benchmarks:sup X > E [ U ( X, B )]subject to E [ ξX ] x , P [ X > B ] > − α, (44)where U is the S-shaped utility function defined in Example 4, B is a performance benchmark (referencelevel) and B is a risk benchmark. ξ is a log-normal pricing kernel with the form ξ = exp( − ( r + 12 θ ) T − θ √ T W ) , where r is the risk-free rate of interest, θ is the risk premium, and W follows a standard normal distribution.In the literature of risk management with VaR constraint (cf. Nguyen and Stadje (2020)), B is alwaystaken as a constant L , which means the worst level of return under the given confidence level. However,when the market state is not bad, it is reasonable to require a higher level of return. If we raise L only underthe situation when the market performs not poorly, the risk of the return may not be increased. That is, weallow B (and also B ) to be stochastic.In the next subsection, we will propose a general solution to Problem (44). In consideration of thenumerical results, we will only consider three simple Plans as follows: (where L i and w are constants) Plan I : B = L , B = L . Plan II : B = L , B = L + ( L − L ) { W >w } . Note that B is state-dependent. Plan III : B = L + ( L − L ) { W >w } , B = L + ( L − L ) { W >w } . Note that both B and B arestate-dependent.Plan I is the most classic case. In Plan II, we adjust the level of return in the VaR constraint from L to L when the market is not bad, while in Plan III, the performance benchmark B will also be raised from L to L . For simplicity, we also assume B > B .To solve Problem (44), we first apply Lagrange method to convert it into a problem as Problem (2).Noting that to solve an optimization problem with risk management, we often use the quantile formulation,but this method requires that the value function being invariant when replacing X by a random variablewith same distribution. As B and B may be stochastic, the quantile formulation is not valid. For µ >
0, define the modified utility function U µ ( x, ( b , b )) = U ( x, b ) + µ { x > b } , x > , −∞ , x < . b = ( b , b ) and B = ( B , B ). We consider the converted problem without the risk constraint:sup X > E [ U µ ( X, B )]subject to E [ ξX ] x . (45)Under Plan I, II and III, we have the following result: Theorem 6.
For every x > , Problem (45) admits a unique finite optimal solution X µ = X µB ( λ ( µ ) ξ ) with λ ( µ ) as a function of µ . When b > b , the function X µb ( y ) is given as follows: ○ . if y ( b ) y µ ( b ) , then ○ . if y ( b ) > y µ ( b ) , then X µb ( y ) = b + (cid:0) py ) − p , y < y ( b ) ,b , y y < y µ ( b ) , , y > y µ ( b ) , X µb ( y ) = b + (cid:0) py ) − p , y < y µ ( b ) , , y > y µ ( b ) , where y ( b ) , y µ ( b ) and y µ ( b ) are defined by y ( b ) = p ( d ( k ) b − b ) − p ,y µ ( b ) = b ( µ + kb p − k ( b − b ) p ) ,y µ ( b ) = p ( d ( k + µb − p ) b ) − p , and d ( s ) denotes the solution of the equation sx p = p ( x + 1) .Proof of Theorem 6. The expression of X µb ( y ) can be derived after some routine but trivial computation. Weproceed to prove that Problem (45) admits a unique finite optimal solution. To apply Theorem 4, we needthe condition that Problem (45) is finite. Indeed, if we take u ( b ) = µ , u ( b ) = k , K ( b ) = 0, δ = p , θ ( b ) = b , γ ( b ) = µ { b > b } , then Proposition 3 indicates that Problem (45) is finite, and the Case 1 in Section 5 holds.Based on Theorem 4, finite optimal solution exists for every x > g (+ ∞ ) = 0 (noting that x b = 0).From the expression of X µb ( y ), we know that the set D µb := { y : X µb ( y ) = X µb ( y ) } = { y ( b ) , y µ ( b ) } , y ( b ) y µ ( b ) { y µ ( b ) } , y ( b ) > y µ ( b ) . (46)As such, in Plan I, II, III, as B only takes value in a finite set, and ξ is continuously distributed, we have forevery λ > P [ X µB ( λξ ) = X µB ( λξ )] P [ λξ = y ( B )] + P [ λξ = y µ ( B )] + P [ λξ = y µ ( B )] = 0 . Therefore, using Theorem 4(2), we know that g ( λ ) = E [ ξX B ( λξ )] is continuous, and Problem (45) admits aunique optimal solution X µ = X µB ( λ ( µ ) ξ ). 41ased on Theorem 6, we solve the equation P [ X µ > B ] = 1 − α to determine the Lagrange multiplier µ ∗ , and then X ∗ := X µ ∗ gives a finite optimal solution of Problem (44). Remark . In fact, the existence of µ ∗ is not a trivial issue, and for some initial value x there may be nosuch µ ∗ . But this is not the key point of this paper; e.g., see Wei (2018) for details. The following figure shows the numerical result of the relation between X ∗ and ξ in Plan I, II and IIIrespectively. The parameter is taken as r = 0 . θ = 0 . T = 10, p = 0 . k = 2 . x = 30, L = 60, L = 70, L = 40, L = 50, w = − α = 0 .
05. The top axis shows the cumulative probability of ξ from leftto right. X * Plan IPlan IIPlan III P Figure 3: relationship between X ∗ and ξ From the figure we see that, if we adjust the return level higher in the VaR when the market is not bad,that is, we change from Plan I to Plan II, then the return X ∗ performs better when ξ ∈ [0 . , .
22] with astable increment from L (= 40) to L (= 50), and the probability is about 10.8%. While for ξ < .
85, thereturn of Plan I is higher than that of Plan II. That is, when the market performs well, Plan I gives higherreturn than Plan II, and the gap will increase rapidly from 0 as ξ decreases. Moreover, for a small probability42 < B ) increases the return when the market is nottoo bad by reducing the return when the market is quite good, which suits bearish managers or people withhigh risk aversion.In Plan III, the manager has a higher anticipation than Plan II when the market performs not bad.The result shows that his return becomes higher than Plan II when ξ < .
53 with probability about 54 . ξ decreases (which means that the market state being better). Whilefor ξ ∈ [0 . , . B ) increases the return when the market is relatively good byreducing the return when the market is relatively bad, which suits bullish managers or people with low riskaversion.As Plan I cares least on the risk among three plans, its return for ξ < . Remark . In Plan II and III, we have P [ X ∗ > L ] = 0 .
95, which is the first stage of the risk value B . Forthe second stage, we have P [ X ∗ > L ] = 0 .
84, and events of reaching the second stage is equivalent to theevent that ξ < .
22. As a possible extension, we can consider risk variable with multiple stages such as: B = f ( ξ ) = L , ξ < ξL , ξ < ξ < ξ ...L n ξ < ξ n , this may become an alternative for the modal with multiple risk constraints as P [ X > L i ] > − α i , i = 1 , , ..., n. Remark . Apart from Plan I, II, III, if we take B − B = d ( k ) (cid:0) ξp (cid:1) p − , then y ( B ) = ξ , and g ( λ ) may bediscontinuous at λ = 1. In this case, Theorem 4 shows that there may exist infinitely many optimal solutions. We propose a framework of multivariate utility optimization with general benchmarks. We give adetailed and complete discussion on the feasibility, finiteness and attainability in our framework (2). Wefind that: (i) the optimal solutions may consist of a random set and there may be infinitely many; (ii) theLagrange multiplier may not exist because the function g in (21) is discontinuous or infinite at some λ ∗ ; (iii)the measurability issue may arise when applying the concavification to a multivariate utility function; (iv)the optimal solution may not be a decreasing function or even not a function of ξ ; (v) the measurabilityissue may arise when selecting a candidate from the non-unique optimal solutions. In this paper, we fully43ddress these technical issues, especially for measurability, and we do not assume a prior that the optimalLagrange multiplier exists. In addition to (i) and (v), it is of interest to study how to find a suitable criteriaand further select the best one from non-unique optimal solutions in future research.We stress that the framework is not suitable for probability distortion. When the reference level isdeterministic in cumulative prospect theory, Problem (1) can be further modelled with a probability distor-tion on the distribution of the wealth X , which can be solved by the quantile formulation approach. Asthe benchmark B may be stochastic, the quantile formulation approach does not work in this framework.Moreover, the distortion simply on the distribution of X is inappropriate (a distortion on B should also bespecified). If one desires to define a distortion on the joint distribution of ( X, B ), it will lead to a ratherheavy model and the cumulative distribution function for B may be not well-defined on the general space E .Therefore, the distortion is not considered in the framework. Acknowledgements.
The authors acknowledge support from the National Natural Science Foundationof China (Grant Nos. 11871036, 11471183). The authors are grateful to the members of the group of Math-ematical Finance and Actuarial Science at the Department of Mathematical Sciences, Tsinghua Universityfor their feedback and useful conversations.
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A Additional proofs
Proof of Lemma 1.
We point out that a nondecreasing and upper semi-continuous function is right contin-uous. (1) is trivial by the definition of the concavification and properties of concave functions. We prove(2)-(6).(2) As ˜ h ( x ) = h ( x ) = −∞ on ( −∞ , x ), we know a > x . Then for any [ a ′ , b ′ ] ⊂ ( a, b ), as h ( x ) is uppersemicontinuous and ˜ h ( x ) is continuous, ˜ h ( x ) − h ( x ) admits a minimum value ε > a ′ , b ′ ]. If ˜ h is not affineon [ a ′ , b ′ ], then on [ a ′ , b ′ ] we can find one of its linear interpolation ˆ h = ˜ g such that 0 ˜ h ( x ) − ˆ h ( x ) ε .We extend the domain of ˆ h to R by just letting ˆ h = ˜ h on R \ [ a ′ , b ′ ]. Therefore, ˆ h is concave and satisfiesˆ h ( x ) > h ( x ), while ˆ h ( x ) < ˜ h ( x ) at some points of [ a ′ , b ′ ], which contradicts to the fact that ˜ h is theconcavification of h . Thus ˜ h is affine on any [ a ′ , b ′ ] ⊂ ( a, b ), and then affine on ( a, b ).(3) Step 1:
We prove the second assertion by contradiction.If there exists δ > x such that ˜ h ( x ) > h ( x ) on ( x, δ ), then ˜ h is affine on ( x, δ ), and we can assume˜ h ( x ) = αx + β . As x > −∞ , we have lim x → x + ˜ h ( x ) − h ( x ) = + ∞ , that is, one can find ε ∈ ( x, δ ) such that˜ h ( x ) − h ( x ) > x ∈ ( x, ε ). Defineˆ h ( x ) = ( α + 1 ε − x )( x − ε ) + αε + β, x ∈ [ x, ε ] , ˜ h ( x ) , x ∈ ( ε, + ∞ ) , − ∞ , x ∈ ( −∞ , x ) . Then ˆ h is a concave function, as we have just turned up the slope of ˜ h on [ x, ε ]. On ( x, + ∞ ) we haveˆ h ( x ) − ˜ h ( x ) = x − εε − x , x ∈ ( x, ε )0 , x ∈ ( ε, + ∞ ) ∈ ( − , , as such, h ( x ) < ˜ h ( x ) − < ˆ h ( x ) ˜ h ( x ) on ( x, ε ) and h ( x ) ˜ h ( x ) = ˆ h ( x ) on [ ε, + ∞ ). At the point x , as h is46ight continuous, we know h ( x ) ˆ h ( x ). Thus, the function min { ˆ h, ˜ h } is a convex function that is no smallerthan h , and is smaller than ˜ h at some points, which contradicts with the fact that ˜ h is the concavification of h . Step 2:
We prove the first assertion.When h ( x ) = −∞ , as ˜ h is nondecreasing, using the second assertion proved above, we know that ˜ h is right continuous at x and ˜ h ( x ) = −∞ . As such, it remains to study the case when h ( x ) > −∞ . If˜ h ( x ) > h ( x ), as h is right continuous and that ˜ h is nondecreasing, there exists δ > x and ε > h ( x ) > h ( x ) + ε holds on [ x, δ ]. Using the same methods as in (2), one can get a contradiction. Hence˜ h ( x ) = h ( x ).If ˜ h is not right continuous at x , then h ( x ) = ˜ h ( x ) < lim x → x + ˜ h ( x ), noting that h is right continuous, againwe have a δ > x and ε > h ( x ) > h ( x ) + ε holds on [ x, δ ], the proof then follows.(4) Case 1:
If there exists x > x such that ˜ h ( x ) = h ( x ) > −∞ . Suppose that ˜ h > h on ( x , + ∞ ),then ˜ h is affine on ( x , + ∞ ) with ˜ h ( x ) = αx + β , α > α = 0, then for x > x , we have h ( x ) < ˜ h ( x ) = ˜ h ( x ) = h ( x ), which contradicts to the fact that h isnondecreasing.If α >
0, noting that lim inf x → + ∞ h ( x ) x >
0, we have lim x → + ∞ h ( x ) x = 0. For x sufficiently large, we have h ( x ) < α x + β . In this case, we can replace the right tail of ˜ h by a linear function with lower slope, whichleads to a contradiction.Therefore, we have x ∈ ( x , + ∞ ) satisfying ˜ h ( x ) = h ( x ). In this light, if sup { x : ˜ h ( x ) = h ( x ) } = x ∗ < + ∞ , as that ˜ h is continuous on ( x, + ∞ ) and that h is nondecreasing, we have ˜ h ( x ∗ ) h ( x ∗ ) ˜ h ( x ∗ ).Hence ˜ h ( x ∗ ) = h ( x ∗ ). Then we have x ′ ∈ ( x ∗ , + ∞ ) such that ˜ h ( x ′ ) = h ( x ′ ), which is a contradiction, andwe know sup { x : ˜ h ( x ) = h ( x ) } = + ∞ . Therefore, G n ( t ) is finite. Case 2:
If for every x > x we have ˜ h ( x ) > h ( x ) or ˜ h ( x ) = h ( x ) = −∞ , then, based on (3), weknow x = −∞ . Hence ˜ h > h on R . As such, ˜ h ( x ) is affine on R . Assume ˜ h ( x ) = αx + β . Similar as in Case1 , we know α = 0, and ˜ h is a constant β on R . As ˜ h is the concavification of h , for every n ∈ N + and x ∈ R ,there exists x n > x such that h ( x n ) > β − n = ˜ h ( x n ) − n . Therefore, G n ( t ) is always finite.(5) For ˜ h ( t ) > h ( t ) + n , if x = −∞ , then as H n ( t ) > −∞ we know h ( H n ( t )) > −∞ . If x > −∞ ,then using (3) we know that there exists t ′ < t such that ˜ h ( t ′ ) = h ( t ) > −∞ . Then H n ( t ) > t ′ and h ( H n ( t )) > −∞ . The rest assertions can be derived directly from the continuity of ˜ h and the right continuityof h . (6) It only needs to prove for x > x . As ˜ h is concave, we have˜ h ( x ) > ( b − x )˜ h ( a ) + ( x − a )˜ h ( b ) b − a > ( b − x ) h ( a ) + ( x − a ) h ( b ) b − a . For ∀ n ∈ N + , if ˜ h ( x ) > h ( x ) + n , based on the fact that h ∈ H , we find a < x < b with ˜ h linear on [ a, b ],47 h ( x ) > h ( x ) on ( a, b ) and h ( a ) > ˜ h ( a ) − n , h ( b ) > ˜ h ( b ) − n . As such,˜ h ( x ) = ( b − x )˜ h ( a ) + ( x − a )˜ h ( b ) b − a ( b − x ) h ( a ) + ( x − a ) h ( b ) b − a + 1 n . If ˜ h ( x ) h ( x ) + n , then we take a = x < b and write˜ h ( x ) h ( x ) + 1 n = ( b − x ) h ( a ) + ( x − a ) h ( b ) b − a + 1 n . Thus ˜ h ( x ) = sup a x ba = b ( x − a ) h ( b ) + ( b − x ) h ( a ) b − a . To prove Lemma 5, we need:
Lemma 7.
Suppose that u : R → R ∪ {−∞} is a proper function with a concavification ˜ u : R → R ∪ {−∞} .Then for y ∈ (0 , + ∞ ) we have X u ( y ) ⊂ X ˜ u ( y ) , where the term is defined as in (8) with b omitted.Proof of Lemma 7. Denote by f ∗ ( y ) the convex conjugate of f , i.e. f ∗ ( y ) = sup x ∈ R { xy − f ( x ) } . Then we have ˜ u = − ( − u ) ∗∗ (cf. Rockafellar (1970)). Using the Fenchel-Moreau Theorem, we know V ( y ) , sup x ∈ R { ˜ u ( x ) − xy } = sup x ∈ R { x ( − y ) − ( − u ) ∗∗ ( x ) } = ( − u ) ∗∗∗ ( − y ) = ( − u ) ∗ ( − y ) = sup x ∈ R { u ( x ) − xy } . For x ∈ X u ( y ), we consider three cases:(i) x ∈ R . In this case, for any x ′ ∈ R we have˜ u ( x ) − xy > u ( x ) − xy = sup t ∈ R { u ( t ) − ty } = sup t ∈ R { ˜ u ( t ) − ty } . Hence x ∈ X ˜ u ( y ).(ii) x = + ∞ . Then there exists x n ↑ + ∞ with u ( x n ) − x n y → sup x ∈ R { u ( x ) − xy } = V ( y ). As V ( y ) > ˜ u ( x n ) − x n y > u ( x n ) − x n y , we know ˜ u ( x n ) − x n y → V ( y ). Hence + ∞ ∈ X ˜ u ( y ).(iii) x = −∞ . The proof is similar as in (ii). Proof of Lemma 5.
As in this lemma, only (iii) is relevant to b . For simplicity, we will first prove (i)(ii)(iv)(v)omitting the notation of b , and then prove (iii). Also, as terms in the lemma may equal to ±∞ , we will proveby contradiction. 48i) Suppose x ∗ , X ( y ) < x for some y ∈ [0 , + ∞ ], then x > −∞ . As X (+ ∞ ) = { x } , we know y < + ∞ .Hence there exists a sequence of real numbers { x n } ↓ x ∗ with U ( x n ) − x n y → V ( y ). However, for x n closed enough to x ∗ we have x n < x , which means U ( x n ) = −∞ . Hence V ( y ) = −∞ , which contradicts toAssumption 1.Suppose x ∗∗ , X ( y ) > x for some y ∈ (0 , + ∞ ], then x < + ∞ . As x > x , we know y < + ∞ . Hencethere exists a sequence of real numbers { x n } ↑ x ∗∗ with U ( x n ) − x n y → V ( y ). However, for x n closed enoughto x ∗∗ we have x n > x , which means U ( x n ) = U ( x ). As such, U ( x n ) − x n y = U ( x ) − x n y → U ( x ) − x ∗∗ y .That is, V ( y ) = U ( x ) − x ∗∗ y < U ( x ) − xy , which contradicts to the definition (7) of V .(ii) Step 1:
We prove x ∗ , X ( y ) ∈ X ( y ).For y = + ∞ , we have X ( y ) = { x } . Hence X ( y ) = X ( y ) = x ∈ X ( y ).For y ∈ [0 , + ∞ ), we have a sequence of real numbers { x n } ↓ (or ↑ ) x ∗ with U ( x n ) − x n y → V ( y ). If x ∗ ∈ {±∞} , then by the definition (8) of X ( y ) we know x ∗ ∈ X ( y ). If x ∗ ∈ R , then it follows from the uppersemicontinuity of U that V ( y ) = lim n → + ∞ U ( x n ) − x ∗ y U ( x ∗ ) − x ∗ y. Based on the definition (7) of V , we know V ( y ) = U ( x ∗ ) − x ∗ y , i.e. x ∗ ∈ X ( y ). Similarly, we have X ( y ) ∈ X ( y ). Step 2:
We prove X ( y ) X ( y ) for 0 y < y + ∞ .If y = + ∞ , then X ( y ) = x X ( y ) (using (i) above).If y < + ∞ , suppose that we have 0 y < y such that x ∗∗ , X ( y ) > x ∗ , X ( y ). We have asequence of real numbers { x n } ↓ (or ↑ ) x ∗ with U ( x n ) − x n y → V ( y ), and a sequence of real numbers { x ′ n } ↓ (or ↑ ) x ∗∗ with U ( x ′ n ) − x ′ n y → V ( y ), and we can assume that U ( x n ) − x n y > V ( y ) − n > U ( x ′ n ) − x ′ n y − n ,U ( x ′ n ) − x ′ n y > V ( y ) − n > U ( x n ) − x n y − n . (47)As x ∗∗ > x ∗ > x , for n sufficiently large we have x ′ n > x and U ( x ′ n ) ∈ R . Then the first inequality in (47)indicates U ( x n ) ∈ R , and we can add two inequalities in (47) to derive( x n − x ′ n )( y − y ) > n . Letting n → + ∞ , as x n − x ′ n → x ∗ − x ∗∗ <
0, we obtain a contradiction.(iii) Based on (ii), for 0 y < y + ∞ , we have X ( y ) X ( y ) X ( y ) X ( y ). The proof of themeasurability will be given at the last of the proof.(iv) Because of the nonincreasing property just shown in (iii), both two limits exist (including thetendency to infinity). Step 1:
We prove l , lim y → X ( y ) = x . 49f l ∈ R , then there exist real numbers y n ↓ x n ↑ l such that X ( y n ) = x n ∈ X ( y n ). As such, forany x ∈ R , U ( l ) − x n y n > U ( x n ) − x n y n > U ( x ) − xy n . (48)Letting n → + ∞ , (48) leads to U ( l ) > U ( x ), and we have x l . But from (i) we know l x , thus l = x .If l = + ∞ , it follows from (i) that + ∞ x , and hence x b = + ∞ = l .If l = −∞ , then for any y ∈ (0 , + ∞ ) we have X ( y ) = −∞ . Using Lemma 7, we know −∞ ∈ X U ( y ) ⊂X ˜ U ( y ). As ˜ U is concave, we have ˜ U ′ ( −∞ ) y for every y >
0. Noting that ˜ U is nondecreasing, ˜ U ′ ( −∞ ) > U ′ ( −∞ ) = 0. This indicates that ˜ U is constant on R . Based on the assumption that U ∈ H , weknow that U is also constant on R . Therefore, x = x = −∞ = l .Similarly, we can prove lim y → X ( y ) = x . Step 2:
We prove L , lim y → + ∞ X ( y ) = x . Suppose that L > x , then
L > −∞ , U ( L − ) > −∞ and U ( L ) > −∞ .If L ∈ R , there exist real numbers y n ↑ + ∞ and x n ↓ L such that X ( y n ) = x n . For any x ∈ R , U ( x n ) − x n y n > U ( x ) − xy n . (49)For v >
0, we take x = x n − vy n > x for some large n , and (49) yields U ( x n ) > U ( x n − vy n ) + v. Letting n → + ∞ , we obtain U ( L ) > U ( L − ) + v, which is a contradiction as v is arbitrary. Thus L cannot be larger than x . As L > x we know L = x .If L = + ∞ , then for any y ∈ [0 , + ∞ ), we have X ( y ) = + ∞ ∈ X ˜ U ( y ). As such, ˜ U ′ (+ ∞ ) > y for every y < + ∞ , which contradicts to the fact that ˜ U is concave.In conclusion, L = x . Similarly, we can prove lim y → + ∞ X ( y ) = x .(v) Step 1:
We prove l , lim y → y + X ( y ) = X ( y ). Suppose l < X ( y ), then l < + ∞ .If l ∈ R , again we have real numbers y n ↓ y as well as x n = X ( y n ) ↑ l satisfying (48). When n tendsto infinity, the inequality shows that for any x ∈ R , U ( l ) − ly > U ( x ) − xy , which means l ∈ X ( y ). As such, l > X ( y ), which is a contradiction.If l = −∞ , then for any y ∈ ( y , + ∞ ) we have X ( y ) = −∞ . As we have discussed in (iv), this indicates˜ U ′ ( −∞ ) y for any y > y , i.e., ˜ U ′ ( −∞ ) y . Let us discuss several cases.50a) If l ′ , X ( y ) ∈ R , and for any x < l ′ , we have U ( x ) = ˜ U ( x ).In this case, for x < l ′ we have U ( x ) − xy = ˜ U ( x ) − xy > ˜ U ( l ′ ) − l ′ y > U ( l ′ ) − l ′ y = V ( y ) , which means x ∈ X ( y ). This contradicts to the definition of X ( y ) and the fact x < l ′ .(b) If l ′ ∈ R , and there exists x < l ′ such that U ( x ) < ˜ U ( x ).As U ∈ H , there exists a nonincreasing sequence { x ′ k } with x ′ k x and ˜ U ( x ′ k ) < U ( x ′ k ) + k . As such,for l ′ > x > x ′ k , using the fact ˜ U ′ ( −∞ ) y , we have U ( x ′ k ) − x ′ k y + 1 k > ˜ U ( x ′ k ) − x ′ k y > ˜ U ( l ′ ) − l ′ y > U ( l ′ ) − l ′ y = V ( y ) . Letting k → + ∞ , we know lim k → + ∞ x ′ k ∈ X ( y ), which is also a contradiction because lim k → + ∞ x ′ k < l ′ .(c) If l ′ = + ∞ .In this case we have x ′′ k increasing to + ∞ with U ( x ′′ k ) − x ′′ k y > V ( y ) − k . Fix x ∈ R , using thenonincreasing sequence { x ′ k } in (b), we derive for k sufficiently large that U ( x ′ k ) − x ′ k y + 1 k > ˜ U ( x ′ k ) − x ′ k y > ˜ U ( x ′′ k ) − x ′′ k y > U ( x ′′ k ) − x ′′ k y > V ( y ) − k . Letting k → + ∞ , we know lim k → + ∞ x ′ k ∈ X ( y ), which is also a contradiction because lim k → + ∞ x ′ k x < l ′ .Concluding (a)-(c), we have proved l ′ = −∞ = l . Step 2:
We prove L , lim y → y − X ( y ) = X ( y ). Suppose L > X ( y ), then L > −∞ .If L ∈ R , the proof is all the same as in Step 1 .If L = + ∞ , then for any y ∈ [0 , y ) we have X ( y ) = + ∞ . As we have discussed in (iv), this indicates˜ U ′ (+ ∞ ) > y for any y ∈ [0 , y ), i.e., ˜ U ′ (+ ∞ ) > y . Using the same method in Step 1 , we can prove L ′ , X ( y ) = + ∞ = L .(iii) At last, we prove the measurability of V b ( y ) , X b ( y ) and X b ( y ). Step 1:
We prove that V b ( y ) is measurable.Noting that U ( · , b ) is nondecreasing, for t ∈ R , V b ( y ) t is equivalent to U ( x, b ) − xy t for all x ∈ Q ,that is, ( y, b ) ∈ T x ∈ Q S x , where S x = { ( y, b ) ∈ [0 , + ∞ ) × E : U ( x, b ) − xy t } . For given x , as U ( x, b ) is ameasurable function of b , we know that U ( x, b ) − xy is a measurable function of ( y, b ) and S x ∈ B ([0 , + ∞ )) ×E .Therefore, we obtain ∩ x ∈ Q S x ∈ B ([0 , + ∞ )) × E , which leads to the measurability of V b ( y ) in ( y, b ). Step 2:
We prove that X b ( y ) is measurable.We proceed to show for ( y, b ) ∈ [0 , + ∞ ) × E and t ∈ R : X b ( y ) > t ⇐⇒ ∃ n ∈ N + s.t. U ( x, b ) − xy V b ( y ) − n holds for all x t, if V b ( y ) < + ∞ ; ∃ n ∈ N + s.t. U ( x, b ) − xy n holds for all x t, if V b ( y ) = + ∞ ; (50)51 ⇐ : This case is trivial, as we can directly verify that every real number x t does not belong to X b ( y )and −∞ / ∈ X b ( y ). • ⇒ : If V b ( y ) < + ∞ . Suppose ∀ n ∈ N + there exists x n t such that U ( x n , b ) − x n y > V b ( y ) − n .Then there exists a subsequence of { x n } converging to some x ∗ ∈ [ −∞ , t ]. Hence x ∗ ∈ X b ( y ), which isa contradiction.If V b ( y ) = + ∞ , the discussion is similar.Noting that in the definition of X b ( y ), y is allowed to be + ∞ . As such, X b ( y ) > t is equivalent to( y, b ) ∈ [ n ∈ N + \ x ∈ Q ∪{ t } x t e S nx [ [ n ∈ N + \ x ∈ Q ∪{ t } x t e T nx [ W, where e S nx , (cid:26) ( y, b ) ∈ [0 , + ∞ ) × E : U ( x, b ) − xy V b ( y ) − n (cid:27) ∩ { ( y, b ) ∈ [0 , + ∞ ) × E : V b ( y ) < + ∞} , e T nx , { ( y, b ) ∈ [0 , + ∞ ) × E : U ( x, b ) − xy n } ∩ { ( y, b ) ∈ [0 , + ∞ ) × E : V b ( y ) = + ∞} ,W = { + ∞} × { b ∈ E : x b > t } . Based on the measurability of V b ( y ), we know that U ( x, b ) − xy − V b ( y ) is measurable in ( y, b ) for given x .Hence e S nx ∈ B ([0 , + ∞ )) × E ⊂ B ([0 , + ∞ ]) × E . Similarly, we have e T nx ∈ B ([0 , + ∞ ]) × E . We consider W .As x b > t is equivalent to ∃ n ∈ N + s.t. U ( x, b ) = −∞ , ∀ x ∈ Q ∩ ( −∞ , t + 1 n ] . Then x b > t ⇐⇒ b ∈ [ n ∈ N + \ x ∈ Q x t + 1 n { b ∈ E : U ( x, b ) = −∞} . As U ( x, · ) is measurable on ( E, E ), we know { b ∈ E : x b > t } ∈ E . Hence W ∈ B ([0 , + ∞ ]) × E . Therefore,we know { ( y, b ) ∈ [0 , + ∞ ] × E : X b ( y ) > t } ∈ B ([0 , + ∞ ]) × E , and X b ( y ) is measurable.Using the same method, one can prove the measurability of X b ( y ). Proof of Proposition 2. As X B ( λξ ) > x B and E [ ξx B ] > −∞ , we know E (cid:2) ξX B ( λξ ) − (cid:3) E [ ξx − B ] < + ∞ . (51)Hence the expectation in (21) is well-defined, and is larger than −∞ for any λ ∈ [0 , + ∞∞