aa r X i v : . [ q -f i n . P M ] A p r A NOTE ON THE QUANTILE FORMULATION
Zuo Quan Xu ∗ The Hong Kong Polytechnic University
07 April 2014
Many investment models in discrete or continuous-time settings boil down to max-imizing an objective of the quantile function of the decision variable. This quantileoptimization problem is known as the quantile formulation of the original investmentproblem. Under certain monotonicity assumptions, several schemes to solve such quan-tile optimization problems have been proposed in the literature. In this paper, we proposea change-of-variable and relaxation method to solve the quantile optimization problemswithout using the calculus of variations or making any monotonicity assumptions. Themethod is demonstrated through a portfolio choice problem under rank-dependent utilitytheory (RDUT). We show that this problem is equivalent to a classical Merton’s port-folio choice problem under expected utility theory with the same utility function but adifferent pricing kernel explicitly determined by the given pricing kernel and probabil-ity weighting function. With this result, the feasibility, well-posedness, attainability anduniqueness issues for the portfolio choice problem under RDUT are solved. It is alsoshown that solving functional optimization problems may reduce to solving probabilisticoptimization problems. The method is applicable to general models with law-invariantpreference measures including portfolio choice models under cumulative prospect theory(CPT) or RDUT, Yaari’s dual model, Lopes’ SP/A model, and optimal stopping modelsunder CPT or RDUT.
Key Words : Portfolio choice/selection, behavioral finance, law-invariant, quantileformulation, probability weighting/distortion function, change of variable, relaxationmethod, calculus of variations, CPT, RDUT, time consistency, atomic, atomless/non-atomic, functional optimization problem. ∗ Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong. Email: [email protected] . The author acknowledges financial supports from Hong Kong General Research Fund(No. 529711), Hong Kong Early Career Scheme (No. 533112), and The Hong Kong Polytechnic University. S -shaped utility function , areference point, and probability weighting/distortion functions. The last two are missing inEUT. In light of these theoretical developments, it is natural to consider investment problemsthat involve probability weighting functions. However, the probability weighting functionsmake these problems time-inconsistent so that these problems cannot be studied using onlyclassical dynamic programming or probabilistic approaches.Jin and Zhou (2008) initiated the study of portfolio choice problems under CPT with prob-ability weighting functions in continuous-time settings. They solved the problem by assumingthe monotonicity of a function related to the pricing kernel and probability weighting func-tion. However, this assumption is so restrictive that it excludes most probability weightingfunctions that are typically used, including that proposed by Tversky and Kahneman (1992),in the Black-Scholes market setting. Jin, Zhang, and Zhou (2011) considered the same portfo-lio choice problem under the scenario of a loss constraint with the same assumption. He andZhou (2011) investigated general models with law-invariant preference measures, includingthe classical Merton’s portfolio choice model under EUT, the mean-variance model, the goalreaching model, the Yaari’s dual model, the Lopes’ SP/A model, the behavioral model underCPT, and those explicitly involving VaR and CVaR in their objectives and/or constraints.Their work took a step forward and reduced the monotonicity assumption in Jin and Zhou(2008) to a piece-wise monotonicity assumption. The results cover the probability weightingfunctions proposed by Tversky and Kahneman (1992), Tversky and Fox (1995), and Prelec(1998). Xu and Zhou (2013) initiated the study of continuous-time optimal stopping problemunder CPT and solved the problem under the same assumption of piece-wise monotonicity asHe and Zhou (2011). By adopting the calculus of variations, Xia and Zhou (2012) achieveda breakthrough. They proposed and solved a portfolio choice problem under rank-dependentutility theory (RDUT) with no monotonicity assumptions. Their method also works for gen-eral models with law-invariant preference measures. However, they use techniques from thecalculus of variations and have extensive recourse to convex analysis, so their arguments arelengthy, technical, and difficult to follow.In this paper, without making any monotonicity assumptions, we propose a new and easy-to-follow method to study the portfolio choice problem under RDUT. A complete and compactargument replaces the lengthy calculus of variations argument in Xia and Zhou (2012). The A function is called S -shaped if it is convex on the left and concave on the right; and reverse S -shaped ifconcave on the left and convex on the right. . We investigate these issues bylinking the portfolio choice problem under RDUT to a classical Merton’s portfolio choiceproblem under EUT for which the issues have been completely solved in Jin, Xu and Zhou(2008).The remainder of this paper is organized as follows. In Section 2, we formulate a portfoliochoice problem under RDUT and define its quantile formulation. In Section 3, we introducea key step — making a change of variable — to formulate an equivalent quantile optimizationproblem, in which the probability weighting function is removed from the objective. Theproblem is then completely solved by a new relaxation method in Section 4. In Section 5,we demonstrate how to transform the portfolio choice problem under RDUT into an equiv-alent classical Merton’s portfolio choice problem under EUT. The feasibility, well-posedness,attainability and uniqueness issues for the portfolio choice problem under RDUT are alsoinvestigated in this section. We conclude the paper in Section 6.2 PROBLEM FORMULATION
Using martingale representation theory (see, e.g., Pliska (1986), Karatzas, Lehoczky, andShreve (1987), Cox and Huang (1989, 1991)), the dynamic portfolio choice problem under see, e.g., Jin, Xu and Zhou (2008) for the definitions of feasibility, well-posedness, attainability anduniqueness issues for a portfolio choice problem reduces to finding a random outcome X to sup X Z ∞ u ( x ) d (cid:0) − w (1 − F X ( x )) (cid:1) , (1) subject to E [ ρX ] = x , X > , where F X ( · ) is the probability distribution function of X ; w ( · ) is the probability weightingfunction which is differentiable and strictly increasing on [0 , with w (0) = 0 and w (1) = 1 ; u ( · ) is the utility function which is strictly increasing and second order differentiable on R + with u ′′ ( · ) < ; and ρ > is the pricing kernel, also called the stochastic discount factor orstate pricing density. We always have that E [ ρ ] < + ∞ .If w ( · ) is the identity function, i.e., w ( x ) = x for all x ∈ [0 , , then Z ∞ u ( x ) d (cid:0) − w (1 − F X ( x )) (cid:1) = Z ∞ u ( x ) d F X ( x ) = E [ u ( X )] , for any X > , and consequently, problem (1) reduces to a classical Merton’s portfolio choiceproblem under EUT: sup X E [ u ( X )] , subject to E [ ρX ] = x , X > . To tackle problem (1), in the literature (see, e.g., Jin and Zhou (2008), Jin, Zhang, andZhou (2011), He and Zhou (2011, 2012), Xia and Zhou (2012)), it is always assumed that
Assumption 1
The pricing kernel is atomless . Under this assumption, solving problem (1) then reduces to solving a quantile optimizationproblem sup G ( · ) ∈G x Z u ( G ( x )) w ′ (1 − x ) d x, (2)where the set G x is given by G x := (cid:26) G ( · ) ∈ G : Z G ( x ) F − ρ (1 − x ) d x = x (cid:27) , See, e.g., Xia and Zhou (2012). A random variable is called atomless or non-atomic if its cumulative distribution function is continuous,and called atomic otherwise. The quantile function Q ( · ) of a real-valued random variable is defined as the right-continuous inversefunction of its cumulative distribution function F ( · ) , that is Q ( x ) = sup { t ∈ R : F ( t ) x } , for all x ∈ (0 , ,with convention sup ∅ = −∞ . A real-valued random variable is atomless if and only if its quantile function isstrictly increasing. G denotes the set of all quantile functions: G := (cid:8) G ( · ) : (0 , R + , increasing and right-continuous with left limits (RCLL) (cid:9) , and F − ρ ( · ) ∈ G denotes the quantile function of the pricing kernel ρ . By Assumption 1, ρ isatomless, so F − ρ ( · ) is strictly increasing.Problem (1) and problem (2) are linked as follows. The optimal solution X ∗ to problem(1) and the optimal solution G ∗ ( · ) to problem (2) satisfy X ∗ = G ∗ (1 − F ρ ( ρ )) . (3)For this reason, problem (2) is called the quantile formulation of problem (1).Before Xia and Zhou (2012), problem (2) was partially solved under certain monotonicityassumptions in the literature. Xia and Zhou (2012) used the calculus of variations to tackle itwithout making those monotonicity assumptions, but their arguments are lengthy and com-plex. Moreover, they did not study the feasibility, well-posedness, attainability or uniquenessissues for problem (1).In this paper, we propose a simple change-of-variable and relaxation method to tackleproblem (2) without making any assumptions. We also solve the feasibility, well-posedness,attainability and uniqueness issues for problem (1) by linking the problem to a classicalMerton’s portfolio choice problem under EUT. Remark 1
In the literature, G x is often replaced by G x := (cid:26) G ( · ) ∈ G : Z G ( x ) F − ρ (1 − x ) d x x (cid:27) . However, there is no difference between considering problem (2) for G x or G x because theoptimal solution to problem (2) in G x , if it exists, must belong to G x . Remark 2
Here we assume that the pricing kernel is atomless as according to convention.However, if one studies economic equilibrium models with law-invariant preference measures(see, e.g., Xia and Zhou (2012)), the pricing kernel will be a part of the solution, so onecannot make a priori any assumption on it. The quantile formulation problem with an atomicpricing kernel is solved in Xu (2014). CHANGE OF VARIABLE
To tackle problem (2), our first main idea in this paper is to make a change of variable toremove the probability weighting function from the objective.Let ν : [0 , [0 , be the inverse mapping of x − w (1 − x ) , that is ν ( x ) := 1 − w − (1 − x ) , x ∈ [0 , . ν ( · ) is also a probability weighting function that is differentiable and strictly increasingon [0 , . It follows that Z u ( G ( x )) w ′ (1 − x ) d x = Z u ( G ( x )) d (1 − w (1 − x ))= Z u ( G ( x )) d ( ν − ( x )) = Z u ( G ( ν ( x ))) d x = Z u ( Q ( x )) d x, where Q ( x ) = G ( ν ( x )) , x ∈ (0 , . Note that G x = (cid:26) G ( · ) ∈ G : Z G ( x ) F − ρ (1 − x ) d x = x (cid:27) = (cid:26) G ( · ) ∈ G : Z G ( ν ( x )) F − ρ (1 − ν ( x )) ν ′ ( x ) d x = x (cid:27) . Therefore, we conclude that G ( · ) ∈ G x if and only if Q ( · ) ∈ Q , where Q := (cid:26) Q ( · ) : (0 , R + , increasing and RCLL with Z Q ( x ) ϕ ′ ( x ) d x = x (cid:27) = (cid:26) Q ( · ) ∈ G : Z Q ( x ) ϕ ′ ( x ) d x = x (cid:27) , and(4) ϕ ( x ) := − Z x F − ρ (1 − ν ( y )) ν ′ ( y ) d y = − Z ν ( x ) F − ρ (1 − y ) d y = − Z − ν ( x )0 F − ρ ( y ) d y = − Z w − (1 − x )0 F − ρ ( y ) d y, x ∈ [0 , . Note that ϕ ( · ) is a differentiable and strictly increasing function on [0 , with ϕ (0) = − E [ ρ ] and ϕ (1) = 0 .By making this change of variable, problem (2) has now been transformed into an equiv-alent problem: sup Q ( · ) ∈Q Z u ( Q ( x )) d x, (5)in which the probability weighting function does not appear in the objective. From now on,we focus on this problem.We point out here that although the objective of problem (5) does not involve the prob-ability weighting function, the constraint set Q does. So problem (5) is different from thespecial scenario of problem (2), in which w ( · ) is replaced by the identity function. We willstudy their relationship in Section 5. 6his change in the formulation of problem (2) is mathematically simple, but reveals theessence of the problem. In problem (5), the function ϕ ( · ) , rather than the probability weight-ing function and the quantile function of the pricing kernel, plays a key role; whereas, inproblem (2), the probability weighting function and the quantile function of the pricing ker-nel play separate roles in the objective and the constraint. Because the probability weightingfunction does not appear in the objective of problem (5), we can solve it by a new relax-ation approach. Moreover, this also suggests that it may be possible to link problem (5) to aproblem under EUT. This will be investigated after solving it.We also point out here that the new formulation explains why the function ϕ ′ ( · ) plays suchan important role in many existing models, such as those introduced by Jin and Zhou (2008),He and Zhou (2011), and Xia and Zhou (2012). In those works, the mysterious function ϕ ′ ( · ) is derived after lengthy analysis, and an explanation of why it should appear and play thekey role is never provided.In tackling problem (2), some studies assume ϕ ( · ) to satisfy various properties which arenot generally true in practice, and under these assumptions, the problem is partially solved.Here are some examples. Example 1
In Jin and Zhou (2008), the function F − ρ ( · ) w ′ ( · ) is assumed to be increasing in As-sumption 4.1. This is equivalent to ϕ ′ ( · ) being decreasing, i.e., ϕ ( · ) is a concave function. Infact, we have − w (1 − ν ( x )) = x, x ∈ [0 , , so ν ′ ( x ) = 1 w ′ (1 − ν ( x )) , x ∈ [0 , . And consequently, by (4) , ϕ ′ ( x ) = F − ρ (1 − ν ( x )) ν ′ ( x ) = F − ρ (1 − ν ( x )) w ′ (1 − ν ( x )) , x ∈ [0 , . (6) The equivalence follows immediately as ν ( · ) is increasing. Example 2
In He and Zhou (2011), the function w ′ (1 −· ) F − ρ (1 −· ) is assumed to be first strictlyincreasing and then strictly decreasing in Assumption 3.5 and many of the following results.By (6) , this is equivalent to ϕ ′ ( · ) being first strictly decreasing and then strictly increasing,i.e., ϕ ( · ) is a strictly reverse S -shaped function. Example 3
In He and Zhou (2012), the function w ′ (1 −· ) F − ρ (1 −· ) is assumed to be nondecreasingin Theorem 2, which is equivalent to ϕ ′ ( · ) being decreasing, i.e., ϕ ( · ) is a concave function.In Proposition 4-7, Theorem 4-6, and Corollary 1, the same function w ′ (1 −· ) F − ρ (1 −· ) is assumed tobe first strictly decreasing and then strictly increasing. This is equivalent to ϕ ′ ( · ) being firststrictly increasing and then strictly decreasing, i.e., ϕ ( · ) is a strictly S -shaped function. A NEW RELAXATION APPROACH
Our second main idea in this paper is to introduce a simple relaxation method to tackleproblem (5).The objective of problem (5) is concave with respect to the decision quantiles, so we canapply the Lagrange multiplier method. Problem (5) is equivalent to problem sup Q ( · ) ∈G J ( Q ( · )) = Z (cid:16) u ( Q ( x )) − λQ ( x ) ϕ ′ ( x ) (cid:17) d x, (7)for some Lagrange multiplier λ > in the sense that they admit the same optimal solution.A naive approach to tackling the foregoing problem (7) is to point-wise maximize itsLagrangian (the integrand in (7)) to get a point-wise solution Q ( x ) := arg max n y : u ( y ) − λyϕ ′ ( x ) o = ( u ′ ) − ( λϕ ′ ( x )) , x ∈ (0 , . However, this point-wise solution may not be a quantile function in G . In fact, Q ( · ) is aquantile function if and only if it is increasing, that is equivalent to ϕ ( · ) being concave. Thisis exactly what has been assumed in Jin and Zhou (2008) so as to solve the problem.The novel idea in this paper is to replace ϕ ( · ) by some function δ ( · ) in the Lagrangian ofproblem (7) so that:(i) The new cost function gives an upper bound to that in (7);(ii) The new problem can be solved by point-wise maximizing the new Lagrangian; and(iii) There is no gap between the new and old cost functions in the point-wise solution.This approach allows us to solve the problem completely without making any assumptions onthe function ϕ ( · ) .We first need to find a relaxed cost function. To this end, let δ ( · ) be an absolutelycontinuous function such that Z (cid:16) u ( Q ( x )) − λQ ( x ) ϕ ′ ( x ) (cid:17) d x Z (cid:16) u ( Q ( x )) − λQ ( x ) δ ′ ( x ) (cid:17) d x, (8)for every Q ( · ) ∈ G . Setting δ (0) = ϕ (0) and δ (1) = ϕ (1) and applying Fubini’s theorem, theinequality (8) is equivalent to Z (cid:16) ϕ ( x ) − δ ( x ) (cid:17) d Q ( x ) , (9)for every Q ( · ) ∈ G , which is clearly equivalent to δ ( · ) dominating ϕ ( · ) on [0 , .8n this case, we have(10) Z (cid:16) u ( Q ( x )) − λQ ( x ) ϕ ′ ( x ) (cid:17) d x Z (cid:16) u ( Q ( x )) − λQ ( x ) δ ′ ( x ) (cid:17) d x Z (cid:16) u ( Q ( x )) − λQ ( x ) δ ′ ( x ) (cid:17) d x, where the last inequality is obtained by point-wise maximizing the new Lagrangian: Q ( x ) := arg max n y : u ( y ) − λyδ ′ ( x ) o = ( u ′ ) − ( λδ ′ ( x )) , x ∈ [0 , . (11)To make Q ( · ) a quantile function, we require δ ( · ) to be concave.To make Q ( · ) an optimal solution to problem (7), it is sufficient, by (10), to have Z (cid:16) u ( Q ( x )) − λQ ( x ) ϕ ′ ( x ) (cid:17) d x = Z (cid:16) u ( Q ( x )) − λQ ( x ) δ ′ ( x ) (cid:17) d x, (12)or equivalently, Z ( u ′ ) − ( λδ ′ ( x )) (cid:16) ϕ ′ ( x ) − δ ′ ( x ) (cid:17) d x = 0 . Applying Fubini’s theorem and using δ (0) = ϕ (0) and δ (1) = ϕ (1) , the above identity isequivalent to(13) Z ( u ′ ) − ( λδ ′ ( x )) (cid:16) ϕ ′ ( x ) − δ ′ ( x ) (cid:17) d x = Z (cid:16) δ ( x ) − ϕ ( x ) (cid:17) d (cid:16) ( u ′ ) − ( λδ ′ ( x )) (cid:17) = λ Z (cid:16) δ ( x ) − ϕ ( x ) (cid:17) u ′′ (cid:16) ( u ′ ) − ( λδ ′ ( x )) (cid:17) d δ ′ ( x ) = 0 . Since δ ( · ) dominates ϕ ( · ) on [0 , , u ′′ ( · ) < , and δ ( · ) is concave, by the last identity, δ ′ ( · ) must be constant on any sub interval of { x ∈ [0 ,
1] : δ ( x ) > ϕ ( x ) } .Putting all of the requirements on δ ( · ) obtained thus far together, we see that δ ( · ) should(i) dominate ϕ ( · ) on [0 , with δ (0) = ϕ (0) and δ (1) = ϕ (1) ;(ii) be concave on [0 , ; and(iii) be affine on { x ∈ [0 ,
1] : δ ( x ) > ϕ ( x ) } .Therefore, we conclude that δ ( · ) must be the concave envelope of ϕ ( · ) on [0 , : δ ( x ) = sup a x b ( b − x ) ϕ ( a ) + ( x − a ) ϕ ( b ) b − a , x ∈ [0 , . (14)On the other hand, if δ ( · ) is the concave envelope of ϕ ( · ) on [0 , , then (9) and (13) holdtrue. This further implies, by (10) and (12), that Q ( · ) defined in (11) is an optimal solutionto problem (7).Putting all of the results obtained thus far together and noting that u ( · ) is strictly concave,we conclude that 9 heorem 1 Problem (7) admits a unique optimal solution ( u ′ ) − ( λδ ′ ( x )) , x ∈ (0 , , where δ ( · ) defined in (14) is the concave envelope of ϕ ( · ) on [0 , .Problem (5) admits an optimal solution if and only if Z ( u ′ ) − ( λδ ′ ( x )) ϕ ′ ( x ) d x = x admits a solution λ > , in which case ( u ′ ) − ( λδ ′ ( x )) , x ∈ (0 , , is the unique optimal solution to problem (5) . Proof . The foregoing argument shows that ( u ′ ) − ( λδ ′ ( x )) , x ∈ (0 , , is an optimal solution to problem (7). Since u ( · ) is strictly concave, the optimal solution isunique.Suppose problem (5) admits an optimal solution. Then the solution must be an optimalsolution to problem (7) for some λ > , so it must be of the form ( u ′ ) − ( λδ ′ ( x )) , x ∈ (0 , . This should be a feasible solution to problem (5), so Z ( u ′ ) − ( λδ ′ ( x )) ϕ ′ ( x ) d x = x . On the other hand, suppose that Z ( u ′ ) − ( λδ ′ ( x )) ϕ ′ ( x ) d x = x holds true for some λ > . Note that Z Q ( x ) ϕ ′ ( x ) d x = x for all Q ( · ) ∈ Q , so sup Q ( · ) ∈Q Z u ( Q ( x )) d x = sup Q ( · ) ∈Q Z (cid:16) u ( Q ( x )) − λQ ( x ) ϕ ′ ( x ) (cid:17) d x + λx sup Q ( · ) ∈G Z (cid:16) u ( Q ( x )) − λQ ( x ) ϕ ′ ( x ) (cid:17) d x + λx , Q ⊆ G . The optimization problem on the right-hand sideis nothing but problem (7), so the unique solution is ( u ′ ) − ( λδ ′ ( x )) , x ∈ (0 , . This solution belongs to Q as R ( u ′ ) − ( λδ ′ ( x )) ϕ ′ ( x ) d x = x , so it is a feasible solution tothe problem on the left-hand side, and consequently, it is an optimal solution to problem (5).Since u ( · ) is strictly concave, the optimal solution to problem (5) is unique. The proof iscomplete. (cid:3) By Theorem 1, the optimal solution to problem (2) is given by G ∗ ( x ) = ( u ′ ) − ( λδ ′ ( ν − ( x ))) = ( u ′ ) − ( λδ ′ (1 − w (1 − x ))) , x ∈ (0 , , which is the same as the last identity on page 14 in Xia and Zhou (2012). That is, ourapproach yields the same result as in Xia and Zhou (2012). It is clear that our change-of-variable and relaxation approach is much simpler and neater than the calculus of variationsapproach in Xia and Zhou (2012), which has extensive recourse to convex analysis. If ϕ ( · ) is assumed to take special shape, such as reverse S -shaped function in He and Zhou (2011), S -shaped function in He and Zhou (2012), then we can get explicit expression for δ ( · ) , andconsequently, G ∗ ( · ) reduces to the results obtained in those works.The feasibility, well-posedness, attainability and uniqueness issues for problem (1) are veryimportant and hard to answer. To avoid these issues, various assumptions are used in theliterature to ensure the existence and uniqueness of solutions (see, e.g., Jin and Zhou (2008),Jin, Zhang, and Zhou (2011), He and Zhou (2011, 2012)). In the following section, withTheorem 1, we will link problem (1) to a classical Merton’s portfolio choice problem underEUT, for which the feasibility, well-posedness, attainability and uniqueness issues are studiedin Jin, Xu, and Zhou (2008). This connection also develops a new way to solve problem (1),which avoids dealing with the quantile formulation problem (2).5 A LINK BETWEEN MODELS UNDER RDUT AND EUT
By Theorem 1, it is clear that a quantile function is an optimal solution to problem (5) if andonly if it is an optimal solution to problem sup Q ( · ) ∈ e Q Z u ( Q ( x )) d x, (15)where e Q := (cid:26) Q ( · ) ∈ G : Z Q ( x ) δ ′ ( x ) d x = x (cid:27) . δ ′ ( · ) is decreasing, function F − e ρ ( x ) := δ ′ (1 − x ) , x ∈ (0 , , belongs to G and can be regarded as the quantile function of some positive random variable e ρ . It is possible to choose e ρ to be comonotonic with ρ , which is henceforth assumed. Then e Q = (cid:26) Q ( · ) ∈ G : Z Q ( x ) δ ′ ( x ) d x = x (cid:27) = (cid:26) Q ( · ) ∈ G : Z Q ( x ) F − e ρ (1 − x ) d x = x (cid:27) . Now, we see that problem (15) can be regarded as a special case of problem (2), in whichthe probability weighting function w ( · ) is replaced by the identity function and the pricingkernel ρ is replaced by e ρ .We point out here that the new pricing kernel e ρ may be atomic, which does not satisfyAssumption 1. In fact, e ρ is atomless if and only if its quantile function F − e ρ ( · ) is strictlyincreasing. This is equivalent to δ ( · ) being strictly concave as F − e ρ ( · ) = δ ′ (1 − · ) , and alsoequivalent to ϕ ( · ) being strictly concave as δ ( · ) is the concave envelope of ϕ ( · ) .Recalling the relationship between problem (1) and problem (2), it is natural to linkproblem (15) to a portfolio choice problem sup X Z ∞ u ( x ) d F X ( x ) , subject to E [ e ρX ] = x , X > . Note that Z ∞ u ( x ) d F X ( x ) = E [ u ( X )] , for any X > , so the above problem is the same as problem sup X E [ u ( X )] , (16) subject to E [ e ρX ] = x , X > . This is a classical Merton’s portfolio choice problem under EUT.Under the assumption that ρ is atomless, we have linked problem (2) to problem (1).However, we cannot directly link problem (15) to problem (16) as before, because the newpricing kernel e ρ in problem (16) may not be atomless.The following result from Xu (2014), where no assumption on e ρ is required, links problem(15) to problem (16). Two random variables X and Y are said to be comonotonic if ( X ( ω ′ ) − X ( ω ))( Y ( ω ′ ) − Y ( ω )) > almostsurely under P ⊗ P . In fact, e ρ = δ ′ (1 − F ρ ( ρ )) in the current setting. Xu (2014) proved that e ρ can be chosen to be comonotonicwith ρ even if ρ is not atomless. heorem 2 If e X ∗ is an optimal solution to problem (16) , then its quantile function is anoptimal solution to problem (15) .On the other hand, if e Q ∗ ( · ) is an optimal solution to problem (15) , then e X ∗ := e Q ∗ (1 − U ) is an optimal solution to problem (16) , where U is any random variable uniformly distributedon the unit interval (0 , and comonotonic with e ρ . With this result, we can link problem (16) to problem (1).
Theorem 3
Let e X ∗ be an optimal solution to problem (16) and e Q ∗ ( · ) be its quantile function.Then X ∗ := e Q ∗ (1 − w ( F ρ ( ρ ))) is an optimal solution to problem (1) .On the other hand, if X ∗ is an optimal solution to problem (1) , then there exists a uniquequantile function e Q ∗ ( · ) such that X ∗ = e Q ∗ (1 − w ( F ρ ( ρ ))) . Moreover, e Q ∗ (1 − U ) is an optimal solution to problem (16) , where U is any random variableuniformly distributed on the unit interval (0 , and comonotonic with e ρ . Proof . Suppose that e X ∗ is an optimal solution to problem (16) and e Q ∗ ( · ) is its quantilefunction. By Theorem 2, e Q ∗ ( · ) is an optimal solution to problem (15) and problem (5).Consequently, G ∗ ( x ) := e Q ∗ ( ν − ( x )) , x ∈ (0 , , is an optimal solution to problem (2). Hence, by (3), X ∗ = G ∗ (1 − F ρ ( ρ )) = e Q ∗ ( ν − (1 − F ρ ( ρ ))) = e Q ∗ (1 − w ( F ρ ( ρ ))) is an optimal solution to problem (1).On the other hand, if X ∗ is an optimal solution to problem (1). Then by (3), X ∗ = G ∗ (1 − F ρ ( ρ )) , where G ∗ ( · ) is an optimal solution to problem (2). Consequently, e Q ∗ ( x ) := G ∗ ( ν ( x )) , x ∈ (0 , , is an optimal solution to problem (5) and problem (15). By Theorem 2, e Q ∗ (1 − U ) is anoptimal solution to problem (16). The proof is complete. (cid:3) Remark 3
The optimal solution to problem (16) can be obtained by the Lagrange multipliermethod directly. Consequently, its quantile function can be obtained without solving problem (15) . Such approach to solving an investment problem under RDUT without using quantileoptimization technique has never appeared in the literature to the best of our knowledge.On the other hand, this result also tells us that a functional optimization problem (2) can besolved via solving a probabilistic optimization problem (16) . It is an important and challengingquestion whether we can apply this idea to other functional optimization problems.
Remark 4
The new pricing kernel e ρ does not depend on the utility function u ( · ) . Remark 5
Problem (1) is time-inconsistent, whereas problem (16) is time-consistent. Itwould be interesting to study their relationships as time changes. CONCLUDING REMARKS
In this paper, we consider a portfolio choice problem under RDUT. We propose a short, neat,and easy-to-follow method to solve the problem. The method consists of two key ideas. Thefirst is making a change of variable to reveal the key function that we need to consider inthe quantile formulation problem. The second is relaxing the Lagrangian so as to find anachievable upper bound. Our approach can also be adopted to deal with portfolio choiceand optimal stopping problems under CPT/RDUT as well as many other models with law-invariant preference measures.The second contribution of this paper is showing that solving a portfolio choice problemunder RDUT is equivalent to solving a classical Merton’s portfolio choice problem underEUT. The latter avoids studying the quantile optimization problem and can be solved bythe classical dynamic programming and probabilistic approaches. Theorem 2 obtained by Xu(2014) plays a key role in connecting these two problems as the new pricing kernel cannot beassumed to be atomless in general. 14he third contribution of this paper is solving the feasibility, well-posedness, attainabilityand uniqueness issues for the portfolio choice problem under RDUT.Last but not least, we show that solving functional optimization problems may reduce tosolving probabilistic optimization problems. This idea may be applicable to other functionaloptimization problems.
Acknowledgments.
The author is grateful to the editors and anonymous referees forcarefully reading the manuscript and making useful suggestions that have led to a muchimproved version of the paper. References[1]
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