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Optimal investment problem under behavioral setting: A Lagrange duality perspective

March 2021

Authors:

Xiuchun Bi

University of Science and Technology of China

Lvning Yuan

University of Science and Technology of China

Zhenyu Cui

Stevens Institute of Technology

Jiacheng Fan

Chinese University of Hong Kong

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In this paper, we consider the optimal investment problem with both probability distor- tion/weighting and general non-concave utility functions with possibly finite number of inflection points. Our model contains the model under cumulative prospect theory (CPT) as a special case, which has inverse S-shaped probability weighting and S-shaped utility function (i.e. one inflection point). Existing literature have shown the equivalent relationships (strong duality) between the concavified problem and the original one by either assuming the presence of probability weighting or the non-concavity of utility functions, but not both. In this paper, we combine both features and propose a step-wise relaxation method to handle general non-concave utility functions and probability distortion functions. The nec- essary and sufficient conditions on eliminating the duality gap for the Lagrange method based on the step-wise relaxation have been provided under this circumstance. We have applied this solution method to solve in closed-form several representative examples in mathematical behavioral finance including the CPT model, Value-at-Risk based risk management (VAR-RM) model with probability distortions, Yarri’s dual model and the goal reaching model. We obtain a closed-form optimal trading strategy for a special example of the CPT model, where a “distorted” Merton line has been shown exactly. The slope of the “distorted” Merton line is given by an inflation factor multiplied by the standard Merton ratio, and an interesting finding is that the inflation factor is solely dependent on the probability distortion rather than the non-concavity of the utility function.

A case ofˆUofˆ

…

An example for non-increasing functionˆUfunctionˆ functionˆU (x) and its general inverse function ( ˆ U ) −1 (x), where ( ˆ U ) −1 (·) is jumped at k and ( ˆ U ) −1 (k) can take any value in [a, b].

…

A multi-solution example for the situation in Lemma 3.1. Here all the solutions are optimal to Problem D while some may be optimal to problem C and Problem B.

…

Optimal strategy π * t respect to wealth level when parameters taking: r = 0.01, µ = 0.07, σ = 0.15, T = 1, t = 0.5, a 1 = 1.17, a 2 = 0.86, b = 0.33, k = 2, γ = 0.5, η = 1, x 0 = 0.2

…

Figures - uploaded by Zhenyu Cui

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Content uploaded by Zhenyu Cui

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A general approach for solving behavioral

optimal investment problems with non-concave

utility functions and probability weighting∗

Xiuchun Bi †Lvning Yuan ‡Zhenyu Cui §Jiacheng Fan ¶

Shuguang Zhang k

Abstract In this paper, we consider the optimal investment problem with both probability distor-

tion/weighting and general non-concave utility functions with possibly ﬁnite number of inﬂection points,

and propose a general approach for solving this problem. Existing literature has shown the equivalent

relationships (strong duality) between the concaviﬁed problem and the original one by either assuming

the presence of probability weighting or the non-concavity of utility functions, but not both. In this

paper, we combine both features and propose a step-wise relaxation Lagrange method to handle the

optimization problems under general non-concave utility functions and probability distortion functions.

The necessary and suﬃcient conditions on eliminating the duality gap for the Lagrange method have

been provided under this circumstance. We have applied this solution method to solve several rep-

resentative examples in mathematical behavioral ﬁnance: the CPT model which has inverse S-shaped

probability distortion and a S-shaped utility function (i.e. one inﬂection point), Value-at-Risk based risk

management (VAR-RM) model with probability distortion, Yarri’s dual model and the goal reaching

model. We obtain the closed-form optimal trading strategy for a special example of the CPT model,

where a “distorted” Merton line has been shown exactly. The slope of the “distorted” Merton line is

given by an inﬂation factor multiplied by the standard Merton ratio, and an interesting ﬁnding is that

the inﬂation factor is solely dependent on the probability distortion rather than the utility function.

Keywords: Non-concave Utility, Probability Distortion, Concaviﬁcation, Lagrange Du-

ality, Relaxation Method.

MSC(2020): Primary: 49J55, 91B16; Secondary: 49K45, 91G80.

∗Xiuchun Bi and L.Y. contributed equally to this work.

†School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang Guizhou,

550025, PR China. Email: xcbi@ustc.edu.cn

‡PhD candidate at Department of Statistics and Finance, School of Management, University of Science and

Technology of China, China. Email: yln1994@mail.ustc.edu.cn

§Corresponding author. School of Business, Stevens Institute of Technology, 1 Castle Point on Hudson, United

States. Email: zcui6@stevens.edu

¶Postdoctoral Research Fellow, Chinese University of Hong Kong, Hong Kong. Email: jfan6@stevens.edu

kCo-Corresponding author. Department of Statistics and Finance, School of Management, University of Science

and Technology of China, China. Email: sgzhang@ustc.edu.cn

1 Introduction

Optimal portfolio choice is a classic yet ever-growing ﬁeld, which concerns the fundamental ques-

tion on how to allocate ﬁnancial capital among a portfolio of assets and investment products.

Starting from Merton’s seminal work (Merton,1969), the expected utility maximization has be-

come the dominant model to characterize the individual’s investment decision making process

in continuous time. The classical expected utility theory (EUT) requires the following key as-

sumptions: the utility function U(·) is strictly concave (i.e. decision maker is risk-averse); the

decision maker is suﬃciently rational to objectively evaluate the true distribution of the traded

underlying assets. There exists an extensive amount of empirical literature showing that the

EUT may fail to explain a plethora of real market observations and two main approaches have

been introduced in the literature to relax the assumptions of EUT: introduction of non-concave

utility functions, and the incorporation of subjective probability distortion/weighting.

The relaxation of the strict concavity of the utility functions in the classical EUT framework

has been well documented in the literature and the nonconcavity may arise from miscellaneous

scenarios: the composition of a concave utility function with a nonconcave pay-oﬀ scheme will re-

sult in a overall non-concave utility function, which frequently arise from the ﬁnancial managers’

compensation problem, see for example Bensoussan et al. (2015); Bichuch and Sturm (2014);

Carpenter (2000); He and Kou (2018). A S-shaped utility function indicating the loss aversion

(based on a reference point) and the switching of risk propensity of individual investors natu-

rally arises from practice, see representative examples in Berkelaar et al. (2004); Kahneman and

Tversky (1979). Thirdly, objective functions can be assigned to be of special types which are

nonconcave in general, such as some risk measures, e.g. Value-at-Risk (Basak and Shapiro,2001),

or a probability of reaching a target as in the goal-reaching model, see Browne (1999,2000).

Another stream of literature that look at the modiﬁcation of the classical EUT framework

considers introducing probability distortion/weighting to incorporate into the decision process

the decision makers’ subjective judgments on the distribution of wealth dynamics. There have

been abundant amount of research considering the phenomena of probability distortion, see for

example, rank-dependent utility theory (RDUT) which consists of two components: the concave

utility of the outcome and the probability weighting/distortion function (Quiggin,1982,2012),

Yarri’s dual model (Yaari,1987), household portfolio under-diversiﬁcation (Polkovnichenko,2005)

and the demand for “lottery” stocks (Barberis and Huang,2008). Especially, the cumulative

prospect theory (CPT) model which incorporates an inverse S-shaped probability weighting and a

S-shaped utility function (Kahneman and Tversky,1979;Tversky and Kahneman,1992), modiﬁes

the classical EUT framework in terms of both probability distortions and nonconcave utilities.

To handle the expected utility maximization problem with constraints, a natural way is to apply

the Lagrange method to transform the original problem into a relaxed unconstrained version and

then refer to the ﬁrst order condition to characterize the optimal solutions which satisfy the given

constraints. However, the Lagrange method and conventional optimization tools such as convex

optimization and dynamic programming cannot be directly applied to the optimization problem

when both probability distortion and nonconcave utility functions are present. To the best of our

knowledge, Jin and Zhou (2008) ﬁrst formulated the mathematical behavioral portfolio selection

problem under the CPT model in continuous time and developed a splitting method to divide

the original problem into a concave part and a convex part such that two standard point-wise

optimization problems could be formulated. One could then apply the standard Lagrange method

to obtain the optimal wealth dynamics under the additional assumption of the monotonicity

for a function related to the pricing kernel and the distortion function. He and Zhou (2011a)

generalized this type of problems by including a series of behavioral ﬁnance models via the so

called “quantile formulation”1. A substantial amount of subsequent literature arise based on

this framework, see He and Zhou (2011b), Xia and Zhou (2016), He and Zhou (2016), He et al.

(2017), Xu (2016), etc. The idea of “quantile formulation” is central to the above development

since it translates the stochastic optimization problem to an equivalent functional optimization

problem with monotonicity constraint 2.

Instead of applying the Lagrange method directly, one could recover the concavity of the ob-

jective functions or constraints by applying concaviﬁcation3. By change of variables, Xu (2016),

under a RDUT model, transformed a quantile optimization problem from He and Zhou (2011b)

into a Merton-type problem and solve it by Lagrange method with the help of concaviﬁca-

tion. Thanks to the concaviﬁcation, the monotonicity condition can be removed. However, the

strict concavity assumption on the utility function still needs to be maintained so as to make

sure that the generated optimal solution is a valid quantile function. Recently, Van Bilsen and

Laeven (2020) considered a dynamic consumption problem where S-shaped utility and probabil-

ity weighting coexist, while the monotonicity conditions (c.f. Propositions 3 & 4 in Van Bilsen

and Laeven (2020)) are still required to obtain the analytical results. R¨uschendorf and Vanduﬀel

(2020) reformulated the optimal portfolio selection problem as an optimization problem on real

functions under monotonicity conditions and obtained a uniﬁed method to solve several problems

of interest such as the Merton problem, the rank-dependent utility theory (RDUT) problem, a

general form of the CPT problem as well as Yaari’s optimal investment problem. Boudt et al.

(2020) derived explicit solutions to the optimal payoﬀ problem for a general Yaari investor in a

variety of relevant cases. He and Jiang (2021) solved the Yaari problem under non-negativity

constraints in a slightly more general set-up.

Among all the literature we discussed above, without assumptions on the monotonicity condi-

tions, either the strict concavity of the utility functions or the non probability-distortion frame-

work need to be maintained in order to apply the Lagrange method. In this paper, our focus is to

propose a general approach for solving behavioral optimal investment problems with nonconcave

utility functions and probability weighting without the monotonicity assumption. The method

can handle more general settings and yield fully explicit solutions that are new to the literature.

1This idea of “quantile formulation” ﬁnds its pedigree in a more general “rearrangement formulation” for this

kind of optimization problems (Bernard et al.,2014a,2015b,2014b;Burgert and R¨uschendorf,2006;R¨uschendorf,

1983).

2To ensure the solutions generated by the Lagrange function is the true optimal solution to the original problem,

i.e. to be a valid quantile function, one needs to presume the monotonicity of a function related to the pricing

kernel and the distortion function.

3A recent paper Dai et al. (2019) presented solutions to non-concave utility maximization problems with

bounded portfolio constraints, which makes the concaviﬁcation principle fail. However, the probability distortion

is not considered there.

It is of interest to combine the general version of quantile formulation with the Lagrangian relax-

ation method, hence the key question we address in this paper is how far we can go by applying

the Lagrange method to solving a general quantile optimization problem when both non-concave

utilities and probability distortions4are present.

Due to the existence of non-concavity and probability weighting, a straightforward point-wise

optimization may not admit a valid quantile function which needs to be right continuous and

non-decreasing. In order to tackle this diﬃculty, we propose the solution procedures as follows.

First we set up a “relaxed” form of the original problem by replacing the utility function U

and the distortion related function ϕby their concave envelopes ˆ

Uand ˆϕrespectively, and then

solve this problem pointwisely by referring to the ﬁrst order condition of the corresponding

Lagrange function. The value of the concaviﬁed problem is essentially an upper bound of the

original value function since the main mathematical elements have been relaxed to their concave

envelopes. Furthermore, it is well known that the concave envelopes of non-concave functions

could be locally aﬃne (Reichlin,2013), as a result of the non-strict concavity, hence the optimal

solutions may not be unique. Based on meticulous analysis of the optimal solutions to the relaxed

point-wise optimization problem, we characterize the set containing all possible solutions which

are formally non-deceasing and right continuous on the domain of the objective function. We

then take a bottom-up approach by substituting those solutions back to the original problem by

removing the relaxation in a step-wise way 5to see if the optimizer derived from the point-wise

optimization problem satisﬁes the given constraint of the unconcaviﬁed problem. If that is the

case, then there must exist optimal quantiles to the original unconcaviﬁed optimization problem.

However, to complement the result of Xu (2016) that there is no gap between the two problems

when the utility is assumed to be strictly concave, our results show that the concaviﬁcation

may result in a strictly positive gap when we remove the concavity assumption on the utility

function. The existence of the gap is determined by the existence of the Lagrange multiplier. We

shall show in the following analysis that the existence of the Lagrange multiplier can be easily

justiﬁed by checking whether the initial value X0=x0belongs to some given intervals which

can be explicitly expressed in terms of the parameters of the utility function and the probability

distortion function.

To apply the solution method aforementioned to concrete behavioral ﬁnance examples, we have

considered a special case of the CPT model where the distortion functions are assumed to be the

same on both sides of the reference point and the S-shaped utility is designed to be bounded from

below. The optimal wealth process and corresponding trading strategies have been completely

solved in explicit closed-form when the initial wealth lies within the “solvable intervals” that

we mentioned above. According to the solutions, one is able to clearly observe how investors’

probability weighting and switching risk propensity inﬂuence their portfolio selections. We com-

pare the strategies to the original Merton problem where there exists neither non-concavity in

utility functions nor probability distortions. We ﬁnd that the existence of probability distor-

4Notice that the classical utility maximization problem can still be attributed to the quantile formulation

where the distortion function is w(x) = x. By the nature of the problem, to ﬁnd the optimal terminal wealth is

equivalent to ﬁnd the optimal distribution of the terminal portfolio pay-oﬀ.

5See problems A, B, C, D in the following context.

tions shall fundamentally change the trading pattern of the investors while the non-concavity

of the utility function would not. The resulting optimal strategies from distorted beliefs on the

portfolio dynamics tend to exhibit higher holdings of the risky assets along with the increase

of portfolio value as compared to cases without distortions. Both strategies converge to a con-

stant level where the non-distorted strategies converge to the standard Merton line while the

distorted strategies shall converge to a constant which can be expressed as a standard Merton

ratio multiplied by an inﬂation factor that is strictly larger than 1. Intuitively, this means that

the presence of probability distortion increases the proportion of wealth invested in the risky

stock, as compared to the original Merton model. One possible explanation for this phenomenon

can be found from the inverse S-shaped probability distortion functions: theoretically speaking,

when the portfolio value goes to inﬁnity, the investors are not supposed to be risk-averse anymore

since under this circumstance they would have inﬁnite ability to aﬀord all kinds of risks. Hence,

the over-weighting on the probability for those events which potentially generate extremely high

returns shall dominate those extremely negative events, thus an increase in the holdings of the

risky asset could be naturally expected. Similarly, to show the broad applicability of the proposed

method, we have applied the method to other examples including the utility maximization with

expected shortfall constraint (Armstrong and Brigo,2019), goal reaching model (Browne,1999)

and Yarri’s dual model (Yaari,1987), etc.

To summarize, the contributions of the paper are three-fold:

1. We propose a general approach to derive explicit closed-form solutions for optimal invest-

ment problems with both probability distortion and general non-concave utility function

with a ﬁnite number of inﬂection points.

2. We provide necessary and suﬃcient conditions on closing the duality gap for the Lagrange

method when combined with the step-wise relaxation method. The condition is dependent

on the initial wealth level, and we have explicitly identiﬁed the “solvable interval” for the

initial wealth.

3. We solved several representative examples in behavior ﬁnance using our uniﬁed approach.

In particular, we solved in explicit closed-form a special case of the CPT model in the Black-

Scholes model, and explicitly derived a “distorted Merton line”, which can be represented as

an inﬂation factor multiplying the standard Merton line. The inﬂation factor only depends

on the probability distortion, and the economic insight is that the existence of probability

distortions shall fundamentally change the trading pattern of the investors while the non-

concavity in the utility function would not.

The remainder of this paper is organized as follows. In Section 2, we introduce in details the

model formulation and key assumptions. In Section 3, the procedures on the concaviﬁcation and

the main solution method are provided. The optimal solutions for the general problem and the

equivalence conditions are also investigated and listed. Following from the results in Section 3, in

Section 4a series of concrete examples are presented, and closed-form solutions of those models

with speciﬁc parameter settings are obtained. The rigorous mathematical analysis of the solution

scheme to the general problem is delegated to Section 5. We conclude the paper in Section 6with

a discussion of future research directions. The proofs for the main theorems and propositions

including the discussion on the well-posedness can be found in the Appendix.

2 Problem Formulation

2.1 A Continuous-Time Market

Suppose that there is a frictionless market equipped with a complete ﬁltered probability space,

(Ω,F,P,{Ft}t≥0), where Pdenotes the physical/objective probability measure, and Ft=σ{Wτ:

0≤τ≤t}is the natural ﬁltration, under which W:= {Wt, t ≥0}is a standard Brownian

motion. Assume that there are two assets traded continuously, with one of the assets Bbeing

risk-free (e.g. bank account or risk-free bond), and the other asset being risky (e.g. stock or

mutual fund). Given the terminal time T > 0, the dynamics of these assets on the investment

horizon [0, T ] are governed by

dSt=µtStdt +σtStdWt,

dP B

t=rtPB

tdt, 0≤t≤T,

where rtis the risk-free interest rate, µt> rtis the growth rate of the risky asset, and σtis the

volatility. All coeﬃcients are assumed to be adapted to Ftwith

0|µt|+1

2|σt|2dt < ∞.(2.1)

Suppose that Xtis the total wealth of the agent at time t, and we require Xt≥0 for t∈[0, T ],

i.e. no bankruptcy constraint. Let πtbe the fraction/proportion of wealth invested in the risky

asset, and the remaining fraction 1 −πtwill be invested in the risk-free asset. Given the initial

wealth x0>0, the self-ﬁnancing wealth process evolves according to the following stochastic

diﬀerential equation (SDE):

dXt= [rt+πt(µt−rt)]Xtdt +σtπtXtdWt.(2.2)

In order to guarantee the existence and uniqueness of the solution to (2.2), it is natural to

assume that the portfolio process πton the interval [0,T] is admissible, with its precise deﬁnition

given as follows.

Deﬁnition 2.1 A portfolio policy πt:= π(t, x)on the interval [0,T] is admissible at x:= Xt>0,

which we denote as πt∈ A(x), if it satisﬁes

•πtis Ft-progressively measurable;

•EhRT

0π2

tdti<∞.

Remark 2.1 Note that the subsequent analysis is not dependent on the speciﬁc stochastic process

governing the wealth process X, except that we consider a complete market setting. Our analysis

holds for a general class of continuous-time or discrete-time stochastic processes.

2.2 Optimization Problems

Our goal is then to choose the optimal portfolio policy πt∈ A(x), for a given current time tand

wealth level x:= Xt>0, to maximize the following objective function:

J(t, x, πt) := Z∞

ω(P(U(Xt,x,πt

T)> y))dy =Z∞

U(y)d1−ω(1 −FXt,x,πt

T(y)),(2.3)

where FXis the cumulative distribution function of X,ω(·) is the probability distortion function

which is strictly increasing and continuously diﬀerentiable. It is deﬁned from [0,1] to [0,1] with

ω(0) = 0 and ω(1) = 1. U(·) is the utility function satisfying the following general assumption:

Assumption 2.1 1. U(·)is an upper semi-continuous and non-decreasing function mapping

from [0,+∞]to R.6

2. There exists a large positive real number M > 0, such that U=ˆ

Uon [M, ∞), where ˇ

Uis

the concave envelope of U, i.e. U(·)is ultimately a concave function on [M, ∞).7.

Remark 2.2 The restriction on the utility function U(·)is satisﬁed by almost all utility func-

tions considered in previous literature. In particular, the celebrated S-shape utility satisﬁes the

assumption with one kinked point, which is also the inﬂection point where the convex and concave

parts of the utility function meet.

For notation convenience, in the following, we shall denote Xt,x,πt

Tby XT. Let

v(t, x) := max

πt∈A(x)Z∞

U(y)d(1 −ω(1 −FXT(y))) (2.4)

be the value function. Our optimization problem can be rewritten in the following form,

v(t, x) = max

π∈A(x)Z∞

U(y)d(1 −ω(1 −FXT(y)))

subject to dXt= [rt+πt(µt−rt)]Xtdt +σtπtXtdWt, t ∈[0, T ],

X0=x0, Xt≥0,

(2.5)

where Xt≥0, t ∈[0, T ] is called the bankruptcy constraint. In the following, we shall denote

the initial wealth level as x0unless speciﬁed otherwise. In general, it is challenging to solve this

optimization problem due to the following two diﬃculties:

1. the utility function is not necessarily strictly concave, and it may not even be continuous,

which allows the possibility for the utility function to exhibit jumps.

2. there is presence of probability distortion, which destroys the time-consistency of the opti-

mization problem.

6The utility function may not necessarily be restricted to start from 0 and it can be any ﬁxed ﬁnite constant.

One could also deﬁne the utility function on the whole real line while deﬁning U(x) = −∞ if x < 0. Basically, in

this paper, we only consider the case that the wealth is non-negative.

7Intuitively, this means that the utility function is “eventually concave”. This is consistent with standard

utility theory, and means that the investor will not be risk-seeking when his wealth reaches a very large amount.

This technical assumption is needed to guarantee the well-posedness of the solution to the optimization problem.

The asymptotic elasticity condition AE(U) = lim supx→+∞xU0(x)/U (x)<1 is required instead in some related

literature (e.g. Bichuch and Sturm (2014)). Note that although this condition on asymptotic elasticity can imply

cases of eventually strictly concave, but it can not include the following special case : Uis eventually concave

with limx→+∞U(x)/x =k > 0.

To solve the dynamic stochastic optimization problem presented in (2.5), we ﬁrst set up to

solve the corresponding quantile formulation of the optimization problem. As is shown in He and

Zhou (2011b), we can transform (2.5) into a functional optimization problem with the auxiliary

assumption on the pricing kernel, which is deﬁned as

ρt= exp −Zt

(rs+θ2

s/2)dt −Zt

θsdWs,

with θt=µt−rt

σtdenoting the market price of risk.

Assumption 2.2 The pricing kernel is atomless.

Through the quantile formulation we can reduce problem (2.5) to the following functional

optimization problem:

sup

G(·)∈G Z1

U(G(x))ω0(1 −x)dx

subject to Z1

G(x)F−1

ρ(1 −x)dx =x0,

(2.6)

where F−1

ρ(·) denotes the quantile function of the pricing kernel ρ:= ρT, which is strictly in-

creasing due to Assumption 2.2. Here G(·) is the quantile function of the terminal wealth XT,

and Gis the set of all quantile functions formally deﬁned as

G:={G(·) : (0,1) 7−→ R, increasing and right-continuous with left limits (RCLL)}.

The optimal solution X∗

Tto problem (2.5) and the optimal solution G∗(·) to the transformed

problem (2.6) satisfy the following link:

X∗

T=G∗(1 −Fρ(ρT)).(2.7)

Furthermore, applying change of variables similarly as in Xu (2016), we have

Q(x) := G(1 −ω−1(1 −x)),(2.8)

ϕ(x) := −Zω−1(1−x)

F−1

ρ(y)dy, x ∈[0,1].(2.9)

Then (2.6) can be transformed into:

sup

Q(·)∈G Z1

U(Q(x))dx,

subject to Z1

Q(x)ϕ0(x)dx =x0.

(2.10)

Here Q(x) is also a quantile function, which is RCLL on (0,1). ϕ(x) is diﬀerentiable and strictly

increasing on [0,1] with ϕ(0) = −E[ρ] and ϕ(1) = 0. Observe that the probability weighting

function has now been removed from the objective function in the above formulation, and only

appears in the constraint. Due to this equivalence, without loss of generality, the next section

investigates the functional optimization problem (2.10) and proposes a solution methodology

based on the concave envelopes of Uand ϕ.

3 Solution Method

3.1 Concave envelopes of two functions

Let ˆ

Uand ˆϕ(·) be the concave envelopes of Uand ϕrespectively. It is obvious that these two

concave envelope functions satisfy the following analytical properties:

1. They are continuous, non-decreasing and concave.

2. ˆ

U≥Uand ˆϕ≥ϕ.

3. ˆ

U(resp. ˆϕ) is locally aﬃne on some open set of {ˆ

U=U}(resp. {ˆϕ=ϕ}).

3.2 Solution scheme

In this section, we discuss the main method to solve the functional optimization problem (2.10).

First, we shall assume that the problem (2.10) is well-posed, where well-posedness means that

the problem admits a ﬁnite maximum. The precise conditions for well-posedness are presented

in Appendix A.

Next, we propose the solution scheme through four steps of relaxation by using the Lagrangian

dual method. We will show that these four problems share a common solution set under some

technical conditions:

•Problem A:

VA(x0) = sup

Q∈G

JA(Q) = sup

Q∈G Z1

U(Q(x))dx, subject to Z1

Q(x)ϕ0(x)dx =x0.(3.1)

•Problem B:

VB(λ, x0) = sup

Q∈G

JB(Q, λ) = sup

Q∈G Z1

(U(Q(x)) −λQ(x)ϕ0(x))dx +λx0,(3.2)

•Problem C:

VC(λ, x0) = sup

Q∈G

JC(Q, λ) = sup

Q∈G Z1

(ˆ

U(Q(x)) −λϕ0(x)Q(x))dx +λx0.(3.3)

•Problem D:

VD(λ, x0) = sup

Q∈G

JD(Q, λ) = sup

Q∈G Z1

(ˆ

U(Q(x)) −λˆϕ0(x)Q(x))dx +λx0.(3.4)

In the above formulations, λ > 0 is a ﬁxed constant standing for the Lagrange multiplier.

Then we shall characterize the solutions to each problem above under the given assumptions,

and provide veriﬁcation results to link the solutions of the individual optimization problem to

each other. We take a bottom-up approach and start from the optimization problem D, and

then show that by restricting8the solution set by imposing certain constraints, we can recover

the solution set to the optimization problem C. Similarly, we can move froward and restrict the

solution set of problem C to obtain the solution set of problem B, until we arrive at the solution

set of problem A. Then the remaining task is to verify that the obtained solution set for problem

8Note that the solution set of problem C is a subset of the solution set of problem D due to relaxation.

A is not empty. We name our method the “relaxation approach” as we ﬁrst relax the optimization

problems and then trace back the optimal solution set by consecutive restrictions of the solution

sets from previously solved optimization problem.

3.3 General Characterization of Optimal Solutions

In this part, we aim to ﬁnd the connections between the optimal solutions to problems A-D and

give the characterizations of them.

Problem D in (3.4) can be solved by the point-wise optimization method. For a ﬁxed x, we

consider the following problem,

Q∗(x) = arg max

Q{ˆ

U(Q(x)) −λˆϕ0(x)Q(x)}.

Consequently, the optimal solutions possess the form

Q∗(x)=(ˆ

U0)−1(λˆϕ0(x)),(3.5)

where ( ˆ

U0)−1(·) denotes the general inverse of ˆ

U0(·).

Remark 3.3 We should treat the solution form (3.5)as a composite function with respect to x.

For some special λand x < y,(ˆ

U0)−1(λˆϕ0(x)) and (ˆ

U0)−1(λˆϕ0(y)) can be totally diﬀerent, even

if ˆϕ0(x) = ˆϕ0(y).

Now, we give more details for Q∗.

Proposition 3.1 The general inverse of ˆ

U0(·)may admit jump points. In particular, if ˆ

Uis

aﬃne in some intervals, (ˆ

U0)−1must admit at least one jump point (see Figure 1and Figure 2).

Proof.

We deﬁne

(ˆ

U0)−1

−(x) = inf{y≥0|ˆ

U0(y)< x},(3.6)

(ˆ

U0)−1

+(x) = sup{y≥0|ˆ

U0(y)> x}.(3.7)

Then for any general inverse ( ˆ

U0)−1,

(ˆ

U0)−1

+(x)≤(ˆ

U0)−1(x)≤(ˆ

U0)−1

−(x).

The concavity of ˆ

Uindicates that ( ˆ

U0)−1

−(x),(ˆ

U0)−1

+(x) are left-continuous and right-continuous

functions, respectively.

Then, we can deduce that: if there exists k > 0, b > a ≥0, with

0≤(ˆ

U0)−1

+(k)≤a<b≤(ˆ

U0)−1

−(k)

or equivalently,

U0(a+) = ˆ

U0(b−) = k,

or equivalently,

Uis aﬃne on interval (a, b). with slope k,

then ( ˆ

U)−1has a jump point at k.

In fact, if there exists a non-empty set (a, b)⊆ {x|ˆ

U(x)> U(x)}, then ˆ

Uis aﬃne on [a, b] from

the deﬁnition of the concave envelope. Hence the results follow naturally. This completes the

proof. 

Figure 1: A case of ˆ

U(·)

Figure 2: An example for non-increasing function ˆ

U0(x) and its general inverse function

(ˆ

U0)−1(x), where ( ˆ

U0)−1(·) is jumped at kand ( ˆ

U0)−1(k) can take any value in [a, b].

Figure 3: A multi-solution example for the situation in Lemma 3.1. Here all the solutions are optimal

to problem D while some may be optimal to problem C and problem B.

Note that the uniqueness of the optimal solution to problem D is highly related to the structure

of ˆ

U, ˆϕand λ. The jump points of ( ˆ

U0)−1play an important role in such solutions. Now, we

denote Kas the set containing all the jump points of ( ˆ

U0)−1, which are countable. We have the

following result.

Lemma 3.1 The optimal solution Q∗(x)deﬁne in (3.5)is not unique if and only if

there exists a k∈ K and at least one non-empty interval (α, β)⊆[0,1] satisfying λˆϕ0(x) = k, ∀x∈(α, β).

Proof. (ˆ

U0)−1is well-deﬁned on R+\ K. If such a pair does not exist, which indicates

∀k∈ K,{x|λˆϕ0(x) = k}is a singleton (denoted as {y0}), then by the right continuity of Q∗(x),

we must have Q∗(y0)=(ˆ

U0)−1

−(k). Consequently, Q∗is unique.

If just one pair of k, (α, β) exists, then for any non-decreasing and right-continuous function

h: [α, β)→[( ˆ

U0)−1

+(k),(ˆ

U0)−1

−(k)], the function

Q∗(x) = ((ˆ

U0)−1

−(λˆϕ0(x)), x /∈[α, β)

h(x)x∈[α, β)

is optimal. So Q∗must not be unique.

Moreover, if there exist many pairs ki,(αi, βi), for each interval [αi, βi), then we can similarly

construct a function hiand

Q∗(x) = ((ˆ

U0)−1

−(λˆϕ0(x)), x /∈ ∪i[αi, βi)

hi(x)x∈[αi, βi)

is optimal to the problem D. This completes the proof. 

Let YD,YC,YBbe respectively the optimal solution set for problems D,C,B. By studying the

connections between problems D,C,B, we have the following result.

Theorem 3.1

(1) For any Q∈ YD,Qis optimal to the problem C if and only if

∀(α, β)⊆ {x: ˆϕ(x)> ϕ(x)}, Q(·)is a constant function on (α, β).(3.8)

(2) For any Q∈ YC,Qis optimal to the problem B if and only if

U(Q(x)) = U(Q(x)),∀x∈[0,1].(3.9)

Moreover, there exists a common solution for problems D,C,and B, which is given by

VD(λ, x0) = VC(λ, x0) = VB(λ, x0)

Proof. See Appendix B.1.

Remark 3.4 Figure 3represents some visualization of solutions to problems D,C,B in the in-

terval (α, β). In this special example, all functions are optimal to problem D, while Q1(x)

satisﬁes the second condition in Theorem 3.1 and Q2(x)satisﬁes the ﬁrst condition. Here

c1, c2are the corresponding parameters. In particular, if (α, β)⊂ {x: ˆϕ(x)> ϕ(x)}and

(a, b)⊂ {x:ˆ

U(x)> U(x)}, then the optimal solutions can only take either aor b.

This theorem gives us the procedure on how to select the optimal solutions to problem B from

the set YD. Next, we will consider whether there exists a λso that the corresponding Q(x)∈ YB

is optimal to the problem A. By the weak duality, we must have:

VA(x0)≤inf

λ>0VB(λ, x0).

The most important task is thus to check that the strong duality

VA(x0) = inf

λ>0VB(λ, x0) (3.10)

holds, so that we can ﬁnd the optimal solution to the problem A by selecting a subset from the

solution set YB.

Generally speaking, if we can ﬁnd a λ∗and the corresponding Q(x)∈ YBsatisfying the

constraint: Z1

Q(x)ϕ0(x)dx =x0,

then the strong duality must hold and Q(x) must be an optimal solution to problem A. If not,

then the strong duality does not hold, and we cannot get the optimal solution to problem A from

the solution set YB.

In order for the readers to better understand our relaxation method and how it works, in

subsequent sections we will solve four representative behavior ﬁnance models in the literature,

obtaining closed-form or semi-closed-form solutions using our general method. In these examples,

both non-concave utility and probability distortion are present. The more general case for uand

ϕwith ﬁnitely many inﬂection points will be delegated to Section 5.

Figure 4: Shape of ϕunder diﬀerent probability weighting models

4 Solutions for concrete models

Previous literature on the mathematical behavior portfolio choice considers tackling either the

ﬁrst or the second challenge, i.e. the non-concave utility function or the probability distortion.

However, to the best of authors’ knowledge, there is no uniﬁed approach to handle the optimal

investment problem when both challenges are present.9Our method can solve all these combina-

tions in a uniﬁed way. Furthermore, previous literature considers the S-shape utility and the

inverse-S-shaped probability weighting as representative examples, while our proposed method

goes beyond that and can handle even more general utility functions and probability distortion

functions with a ﬁnite number of inﬂection points.

4.1 Example 1: A Special CPT model

In this subsection, we consider a S-shape utility function with an inverse S-shaped distortion

function. To illustrate the main ideas of our procedure, for convenience, we only consider the

case when the function ϕ(·) is also of S-shape10.

Following our procedure, we rewrite the original optimization problem as problem A:

VA(x0) = sup

Q∈G

JA(Q) = sup

Q∈G Z1

U(Q(y))dy,

subject to Z1

Q(y)ϕ0(y)dy =x0.

9The cumulative prospect theory (CPT) is one example in which both challenges appear. In the literature, the

optimal investment problem under the CPT model has been only solved for a special class of power-type S-shaped

utility functions and also for a special class of probability distortion functions, see for example sections 5.1 and

5.2 in He and Zhou (2011a).

10When the pricing kernel of the market follows log-normal distribution, almost all the inverse S-shape distortion

in He and Zhou (2016); Tversky and Fox (1995); Tversky and Kahneman (1992) will lead to a S-shape function

for ϕ, see Figure 4.

We introduce the Lagrange multiplier and transform problem A into its unconstrained form:

problem B

VB(λ, x0) = sup

Q∈G

JB(Q) = sup

Q∈G Z1

(U(Q(y)) −λϕ0(y)Q(y))dy +λx0,(4.1)

where λ > 0 is a ﬁxed constant corresponding to the Lagrange multiplier. After replacing the

two functions U(·) and ϕ(·) by their concave envelopes step by step, we obtain

problem C

VC(λ, x0) = sup

Q∈G

JC(Q) = sup

Q∈G Z1

(ˆ

U(Q(y)) −λϕ0(y)Q(y))dy +λx0.(4.2)

problem D:

VD(λ, x0) = sup

Q∈G

JD(Q) = sup

Q∈G Z1

(ˆ

U(Q(y)) −λˆϕ0(y)Q(y))dy +λx0.(4.3)

We ﬁrst solve problem D, and then go back to solve problem A through investigating the

solution sets of problem C and problem B in order. Let ξ:= inf{x > 0 : U(x) = ˆ

U(x)},

c:= inf{x > 0 : ϕ(x) = ˆϕ(x)}, then

{y≥0 : ˆ

U > U }= (0, ξ ),

{y∈[0,1] : ˆϕ>ϕ}= (0, c).(4.4)

Let λ0:= U0(ξ)

ϕ0(c). We obtain the following results on translating from problem D to problem

•If λ6=λ0, then problem D can be solved by point-wise optimization. By Theorem 3.1, it

is easy to check that,

Q∗

1(λ, y) = ((U0)−1(λˆϕ(y)),if λˆϕ0(y)≤U0(ξ),

0,otherwise (4.5)

is the unique solution to problem D and is also the optimal solution to problems C and

•If λ=λ0, then problem D has solutions given by:

Q∗(y) = ((U0)−1(λ0ˆϕ0(y)),if λ0ˆϕ0(y)< U0(ξ),

any non-decreasing function on [0, ξ],if λ0ˆϕ0(y) = U0(ξ).

By using Theorem 3.1, we have that

Q∗

2(y) = ((U0)−1(λ0ˆϕ0(y)),if λ0ˆϕ0(y)< U0(ξ),

0,otherwise, (4.6)

Q∗

3(y) = ((U0)−1(λ0ˆϕ0(y)),if λ0ˆϕ0(y)< U0(ξ),

ξ, if λ0ˆϕ0(y) = U0(ξ),(4.7)

are all the common solution to problems D, C, and B. In fact

Q∗

2(y) = lim

λ→λ+

Q∗

1(λ, y),

Q∗

3(y) = lim

λ→λ−

Q∗

1(λ, y).

Deﬁne

f(λ) = Z1

Q∗

1(λ, y)ϕ0(y)dy, λ 6=λ0,

xL=Z1

Q∗

2(y)ϕ0(y)dy =f(λ+),

xU=Z1

Q∗

3(y)ϕ0(y)dy =f(λ−),

f(λ0) = xLor xU.

and we have the following result.

Theorem 4.2 Assume that f(λ)<∞for any λ∈R+, then

(1) if x0/∈[xL, xU], then Q∗

1(λ∗, y)is the unique solution to problem A and VA(x0) =

VB(λ∗, x0),where λ∗is the unique solution to f(λ) = x0;

(2) if x0=xL, then Q∗

2(y)is the unique solution to problem A and VA(xL) = VB(λ0, xL);

(3) if x0=xU, then Q∗

3(y)is the unique solution to problem A and VA(xU) = VB(λ0, xU);

(4) if x0∈(xL, xU), then the strong duality does not hold, and our method fails to ﬁnd the

optimal solution.

Proof. The proof follows straightforwardly from the strictly decreasing property of f(λ). 

We next give an explicit numerical example. Let the utility function be

U(x) = ((x−η)γ,if x≥η,

−k(η−x)γ,if 0 ≤x < η, (4.8)

where ηis the reference point of the agent, 0 < γ < 1 is the risk aversion factor and k > 1 (see

Figure 5), then Uis a S-shape utility function.

For convenience, we consider a particular probability distortion function from He and Zhou

(2016) and Jin and Zhou (2008) (see Figure 7):

w(x) = (k1Φ(Φ−1(x) + a1),if x≤b,

1−k2Φ(a2−Φ−1(x)),if x > b, (4.9)

where a1≥0, a2≥0, b ∈[0,1], and

k1=exp{(a1+a2)Φ−1(b) + a2

1/2}

exp{(a1+a2)Φ−1(b) + a2

1/2}Φ(Φ−1(b) + a1) + exp{a2

2/2}Φ(a2−Φ−1(b)),

k2=exp{a2

2/2}

exp{(a1+a2)Φ−1(b) + a2

1/2}Φ(Φ−1(b) + a1) + exp{a2

2/2}Φ(a2−Φ−1(b)).

Now we assume that the market parameters r(·), µ(·), σ(·) are all deterministic. Let

˜µ=−ZT

(r(t) + θ2

t/2)dt, ˜σ=sZT

θ2

tdt, (4.10)

At=−ZT

(rs+θ2

s/2)ds, Bt=sZT

θ2

sds, (4.11)

Figure 5: Uand its concave envelope ˆ

U, with

parameters: η= 1, k = 2, γ = 0.5

Figure 6: ϕand its concave envelope ˆϕwith

parameters: r= 0.01, µ = 0.07, σ = 0.15, T =

1, a1= 1.17, a2= 0.86, b = 0.33

then,

ϕ(x) = −exp{˜µ+ ˜σ2/2}Φ(Φ−1(w−1(1 −x)) −˜σ)

=(−exp{˜µ+ ˜σ2/2}Φ(a2−˜σ−Φ−1(x

k2)) if x < 1−w(b),

−exp{˜µ+ ˜σ2/2}Φ(Φ−1(1−x

k1)−a1−˜σ) if x≥1−w(b).

Here we assume a2>˜σto guarantee that ϕ(x) is a S-shaped function, see an illustration in

Figure 6.

Using the notations ξ, c in (4.4), At, Btin (4.11), we obtain

xL=−ηϕ(c) + k1γ

λ01

1−γ

exp (a1˜σ+a2

1/2−˜µ)γ

1−γ

+γ2(a1+ ˜σ)2

2(1 −γ)2ΦΦ−11−c

k1+γ(a1+ ˜σ)

1−γ,

xU=xL+ (ϕ(c)−ϕ(0))ξ.

(4.12)

Theorem 4.3 Assume f(λ)<+∞for any λ∈R+and a2>˜σ.

(1) If x0≤xL, then f(λ) = x0admits a unique solution λ∗, and the optimal terminal wealth

is given by

X∗

T= (η+c2ρ−c3)1{ρ≤c1},(4.13)

Figure 7: An example of the distortion function in (4.9) when parameters take: a1= 1.17, a2= 0.86, b =

0.33

where

c1= exp (˜σ(ln k1−a2

1/2 + ln U0(ξ)

λ∗) + a1˜µ

a1+ ˜σ),

c2=k1γ

λ∗1

1−γ

exp −1

1−γa2

1/2−a1˜µ

˜σ,

c3=1

1−γ1 + a1

˜σ.

(4.14)

The optimal wealth process is

X∗

t=ηexp{B2

t/2 + At}Φln c1/ρt−At

−Bt

+c2exp{(1 −c3)At+ (1 −c3)2B2

t/2}ρ−c3

tΦln c1/ρt−At

−(1 −c3)Bt,

and the optimal portfolio is

π∗

t=µt−rt

σ2

tX∗(t)η

exp{B2

t/2 + At}φ(ln c1/ρt−At

−Bt)

+c2c3exp (1 −c3)At+ (1 −c3)2B2

t/2ρ−c3

tΦln c1/ρt−At

−(1 −c3)Bt

+c2

exp{(1 −c3)At+ (1 −c3)2B2

t/2}ρ−c3

tφln c1/ρt−At

−(1 −c3)Bt.

Moreover,

X∗

t→+∞, π∗

t→µt−rt

σ2

t(1 −γ)1 + a1

˜σ,as ρt→0,

X∗

t→0, π∗

t→+∞,as ρt→+∞.

(4.15)

(2) If x0≥xU,then f(λ) = x0admits a unique root λ∗, and the optimal wealth is given by

X∗

T=η+c2ρ−c31{ρ≤c1}+c2c−c3

11{ρ>c1}(4.16)

where

c1= exp ˜σΦ−11−c

k1+ ˜µ−a1˜σ,

c2=k1γ

λ∗1

1−γ

exp −1

1−γ−a1˜µ

˜σ+a2

1/2,

c3=1

1−γ1 + a1

˜σ.

The optimal wealth process

X∗

t=ηexp{At+B2

t/2}+c2exp{(1 −c3)At+ (1 −c3)2B2

t/2}ρ−c3

tΦln(c1/ρt)−At

−Bt(1 −c3)

+c2c−c3

1exp{B2

t/2 + At}1−Φln(c1/ρt)−At

−Bt,

and the optimal portfolio

π∗

t=µt−rt

σ2

tX∗(t)c2c3exp{(1 −c3)At+ (1 −c3)2B2

t/2}ρ−c3Φln c1/ρt−At

−(1 −c3)Bt

+c2

exp{(1 −c3)At+ (1 −c3)2B2

t/2}ρ−c3φln c1/ρt−At

−(1 −c3)Bt

−c2

c−c3

1exp{B2

t/2 + At}φln c1/ρt−At

−Bt.

Moreover,

X∗

t→+∞, π∗

t→µt−rt

σ2

t(1 −γ)1 + a1

˜σ,as ρt→0,

X∗

t→(η+c2c−c3

1) exp{At+B2

t/2}, π∗(t)→0,as ρt→+∞.

(4.17)

Proof. See Appendix C.1.

One can observe that the analytical solutions given by (4.13) and (4.16) contain a lot of

interesting economic insights. First, based on the quantitative structure of the solutions, the

value of the optimal wealth, if not trivial (i.e. X∗

T6= 0), would always be obtained from the right

side of the reference point, which indicates that the investor shall drive the optimal expected

utility from the concave part of her utility function. This ﬁnding can be intuitively explained

since the “true” utility function we are using for the computation is the concaviﬁed version of the

original one, thus the optimization was conducted by nature on a concave function. Second, we

can clearly observe from the limiting forms of the optimal trading strategies (4.15) and (4.17),

that a new type of “Merton Line” has appeared in the limiting expression when the optimal wealth

evolves to inﬁnity. When there exists probability distortion, the “Merton Line” is signiﬁcantly

ampliﬁed by a positive inﬂation factor as compared to the original form of the classical Merton

solution. This shows that the probability distortion shall fundamentally change the trading

structure of the investment process and consistently increases the holding of risky assets when

the accumulated wealth goes to inﬁnity. Simultaneously, when the wealth of investors lies at

a relatively low level, the investor shall signiﬁcantly decrease the exposure to risky asset which

can be observed directly from the Figures 8and 9. This coincides with the intuition behind the

construction of the probability distortion functions, i.e. the low probability events have been

distorted to be overestimated by the positive prospect holding of the investors.

Figure 8: Optimal strategy π∗

trespect to wealth

level when parameters taking: r= 0.01, µ =

0.07, σ = 0.15, T = 1, t = 0.5, a1= 1.17, a2=

0.86, b = 0.33, k = 2, γ = 0.5, η = 1, x0= 0.2

Figure 9: Optimal strategy π∗

trespect to wealth

level when parameters taking: r= 0.01, µ =

0.07, σ = 0.15, T = 1, t = 0.5, a1= 1.17, a2=

0.86, b = 0.33, k = 2, γ = 0.5, η = 1, x0= 1.5

If we take a closer look at Figures 8and 9, then we can ﬁnd that the optimal proportion being

invested in the risky asset seems to experience no change for the non-distorted case if we vary

the initial endowment x0, which can be observed in a clearer way in Figure 10. We can see from

Figure 10 on the bottom right that all the investment ratios corresponding to diﬀerent values of

x0will coincide with each other once the values of X∗

tare the same for the non-distorted situation.

We can mathematically show that under the setting of our CPT framework, the distortion shall

ruin the time consistency of the individual investor’s investment strategies from Theorem 4.3

since diﬀerent x0values result in diﬀerent c1and c2. If there exists no distortion, then using the

notations in (4.10) and (4.11), we get the optimal terminal wealth

X∗

T= (η+Acpρ−p)1{ρ≤c},

where p=1

1−γ, A =γ

u0(ξ)p,and cis the unique solution that solves: E[ρX∗

T] = x0. It is clear

that diﬀerent x0will lead to diﬀerent values of c. Since log(ρ/ρt)∼N(At, B2

t) and is adapted to

Ft, the optimal wealth process can be computed as

X∗

t=ηc1Φ(dt(c/ρt)) + c2(c/ρt)pΦ(dt(c/ρt) + pBt)

with c1exp{At+B2

t/2}, c2=Aexp{(1 −p)At+ (1 −p)2B2

t/2}, dt(x) = log x−At

Bt−Bt, and the

optimal value invested in stock is

π∗

tX∗

t=µt−rt

σ2

tηc1

φ(dt(c/ρt)) + c2

(c/ρt)pφ(d(c/ρt) + pBt) + c2p(c/ρt)pΦ(dt(c/ρt) + pBt).

As a result, X∗(t) and π∗(t) are two functions with respect to c/ρt, and X∗

tis strictly increasing

with respect to c/ρt, then the same X∗(t) shall result the same c/ρt, which will also lead to the

same π∗

Figure 10: Comparison of optimal wealth and investment strategies without probability distortion

under diﬀerent levels of initial wealth

4.2 Example 2: VaR-RM model with distortion function and non-

concave utility

In this example, we consider an extension of the Value-at-Risk based risk management (VaR-RM)

model proposed in Basak and Shapiro (2001) and solve it using our new method. The model1is

sup

X≥0Z+∞

U(x)d(1 −w(1 −FX(x))),

s.t. E[ρX] = x0,

Hg(X)≤ −x1.

(4.18)

Here U(·) is a non-concave utility function (e.g. S-shaped utility), H(·) is a given risk measure,

and

Hg(X) = −Z1

G(x)m(dx),

where G(·) is the quantile function of X,mis a ﬁxed probability measure on [0,1]. The basic

limitations of (x0, x1) is clearly analyzed in Wei (2018).

By using the quantile formulation X=G(1 −Fρ(ρ)) and change of variables, we have

Q(x) = G(1 −ω−1(1 −x)),

1To our best knowledge, there are two type of extensions to Basak and Shapiro (2001): one considers a weighted

VaR as the constraint(see Wei (2018) ); the other considers the non-concave utility with Var/ES constraint (see

Armstrong and Brigo (2019)). Here we use an example combining both non-concave utility and weighted VaR

constraint with probability distortion.

and we can transform the above problem into the form of problem A:

sup

G(·)∈G Z1

U(Q(x))dy,

s.t. Z1

Q(x)ϕ0(x)dx =x0,

Q(x)m1(dx)≥x1,

(4.19)

where

ϕ(x) = −Zw−1(1−x)

F−1

ρ(y)dy,

and

m1(dx) = mdx

w0(w−1(1 −x)).

To tackle this problem, we need to ﬁrst use the Lagrange multiplier method to transform it into

the non-constraint form, i.e. problem B:

sup

QZ1

(U(Q(x)) −λ1ϕ0(x)Q(x))dx +λ2Z1

Q(x)m1(dx) + λ1x0−λ2x1.(4.20)

Let

ϕλ(x) := ϕ(x) + λm1([x, 1]),

where λ=λ2/λ1, then we can transform the problem into

sup

QZ1

(U(Q(x)) −λ1ϕ0

λ(x)Q(x))dx +λ1(x0−λx1).(4.21)

Now, we turn to solving problem (4.21) by using the concaviﬁcation procedure. We shall ﬁrst

solve the following, i.e. problem D

sup

QZ1

(ˆ

U(Q(x)) −λ1ˆϕ0

λ(x)Q(x))dx +λ1(x0−λx1).(4.22)

It is easy to analyze the optimal solution of this problem and we can then get the optimal solution

of (4.21) by referring to Theorem 3.1.

However, for the general form of U(·), m(·), it is hard for us to ﬁnd the explicit conditions

under which the Lagrange duality gap will diminish. Thus, we consider a simple case: Uis a

S-shaped utility (i.e. ﬁrst strictly convex then strictly concave), w(x) = x, and the risk measure

is the expected shortfall (Armstrong and Brigo,2019;Wei,2018). Then the problem is

sup

G(·)∈G Z1

U(G(x))dy,

s.t. Z1

G(x)F−1

ρ(1 −x)dx =x0,

αZα

G(x)dx ≥x1,

(4.23)

where x1, x0>0.

We denote the concave envelope of Uas ˆ

Uand deﬁne ξ= inf{x > 0 : ˆ

U(x) = U(x)},then

{x≥0 : ˆ

U > U }= (0, ξ ),

and

U0(ξ)ξ=U(ξ)−U(0).

We then calculate the function

ϕλ(x) = −Z1

F−1

ρ(1 −y)dy +λ1−x

α1{x<α}.(4.24)

It is easy to see that ϕλ(x) is locally concave on (0, α) and (α, 1), but not globally concave on

(0,1). Thus there must exist 0 < y1≤α≤y2<1 (here if λ= 0, then y1=α=y2) such that

ϕ0

λ(y1) = ϕ0

λ(y2) = ϕλ(y2)−ϕλ(y1)

y2−y1

=: k, (4.25)

where kis a function w.r.t λ. Hence we have

ˆϕ0

λ(y) = 









F−1

ρ(1 −y)−λ

α, y < y1,

k, y ∈[y1, y2],

F−1

ρ(1 −y), y > y2.

By using Theorem 3.1, the optimal solution to problem (4.20) is

Gλ,λ1(y) = (0,if λ1ˆϕ0

λ(y)> U0(ξ),

(U0)−1(λ1ˆϕ0

λ(y)),if λ1ˆϕ0

λ(y)≤U0(ξ).(4.26)

Note that the expected shortfall (ES) constraint implies

λ1k≤U0(ξ).

Set

g(c) = R1

cF−1

ρ(1 −y)dy

α−c.(4.27)

It is not hard to verify that g(c) is decreasing ﬁrst and then increasing on the interval [0, α), thus

g(c) must admit a unique minimizer. Let

c∗= arg min

c∈[0,α)g(c),

R=g(c∗),

(4.28)

then we have the following result.

Lemma 4.2

(1) The problem (4.23)admits no optimal solution when x0< αRx1.

(2) If we treat λ, y2, k as the function of y1, then λ(·), y2(·)are continuously decreasing, while

k(·)is continuously increasing with respect to y1. Moreover, the reasonable range of y1is

[c∗, α]and the ranges of λ, y2, k are [0, αF −1

ρ(1 −c∗)],[α, 1],[0, F −1

ρ(1 −α)], respectively.

Proof. See Appendix C.2.

Let

f(λ, λ1) := 1

αZα

Gλ,λ1(y)dy,

g(λ, λ1) := Z1

Gλ,λ1(y)F−1

ρ(1 −y)dy.

For ﬁxed x1, λ, let λ1=f1(λ, x1) be the solution of f(λ, λ1) = x1. Set ∆(x1) = g(0, f1(0, x1)).

Lemma 4.3 Assume 0< λ1k < U0(ξ).

(1) Both fand gare continuously increasing w.r.t λwhile continuously decreasing w.r.t λ1.

(2) f1(·, x1)is continuously increasing w.r.t. λ.

(3) g(λ, f1(λ, x1)) is a continuous function w.r.t. λ.

Proof. See Appendix C.2.

The main tool to check whether our approach works or not, is to actually solve the following

equation system:

f(λ, λ1) = x1,

g(λ, λ1) = x0.(4.29)

Using the continuity property in Lemma 4.3, one can deduce the following results.

Theorem 4.4 Suppose g(λ, λ1)has range R+for any λ1>0.

(1) If x0>∆(x1), then the ES constraint is redundant. Problem (4.23)must admit a unique

optimal solution.

(2) If x1≥ξ(1 −c∗

α), then for any x0∈[αRx1,∆(x1)], the equation system (4.29)admits at

least one solution (λ∗, λ∗

1)and the corresponding Gλ∗,λ∗

1(y)solves problem (4.23).

(3) If 0< x1< ξ(1 −c∗

α), then λ≤¯

λ, where λ=¯

λis the unique solution to (4.25)when taking

y1=α1−x1

ξ. Let

L(x1) = min

λ∈[0,¯

λ]g(λ, f1(λ, x1)),

–if x0=αRx1, then problem (4.23)admits a unique feasible solution which is also

optimal.

–if x0∈[L(x1),∆(x1)], then the equation system (4.29)admits at least one optimal

solution (λ∗, λ∗

1)and the corresponding Gλ∗,λ∗

1(y)solves problem (4.23).

–if x0∈(αRx1, L(x1)), then the equation system (4.29)admits no solution. The strong

duality does not hold, thus our method fails to solve problem (4.23).

Proof. See Appendix C.2.



4.3 Example 3: Yaari’s dual model

We consider the Yaari’s dual model in Yaari (1987), which was also solved by quantile formulation

method in Boudt et al. (2020); He and Jiang (2021); He and Zhou (2011b). The optimization

problem is formulated as

max

XTZ∞

ω(P(XT> x))dx (4.30)

subject to E[ρTXT] = x0,XT≥0 and XTis FTadapted. The structure of the problem under

quantile formulation is

V(x0) = sup

Q∈G Z1

Q(y)dy,

s.t. Z1

Q(y)ϕ0(y)dy =x0.

(4.31)

Then, we will follow our procedure to tackle this problem. Firstly, by using the Lagrange

multiplier methods, we consider the non-constraint problem:

V(λ, x0) := sup

Q∈G

J(Q) = Z1

(Q(y)−λϕ0(y)Q(y))dy +λx0.(4.32)

Secondly, replacing ϕby its concave envelope ˆϕ, the problem becomes

sup

Q∈G

J1(Q) = Z1

(Q(y)−λϕ0(y)Q(y))dy +λx0.(4.33)

Thus, the optimal solution to this problem is:

Q∗(y) = 









0,if λˆϕ0(y)>1,

any non-decreasing and right continuous function,if λˆϕ0(y) = 1,

+∞,if λˆϕ0(y)<1.

Now, we ﬁrst consider the special case when there exists 0 <c<1, such that

ˆϕ(y)> ϕ(y),∀y∈(c, 1); ˆϕ(y) = ϕ(y) otherwise. (4.34)

In fact, these conditions are the Assumptions 6,7 in He and Zhou (2011b). Let λ0=1−c

−ϕ(c), and

recalling Theorem 3.1, we can get the optimal solution to problem (4.32):

(1) if λ6=λ0, then

Q∗(y) = (0, λ ˆϕ0(y)>1,

+∞, λ ˆϕ0(y)≤1.

(2) if λ=λ0, then

Q∗(y) = (0, y < c,

c1, y ∈[c, 1),(4.35)

where c1is an arbitrary non-negative number.

Thus

V(λ, x0) = (+∞,if λ < λ0,

λx0,if λ≥λ0.

The weak duality implies:

V(x0)≤inf

λ>0V(λ, x0) = λ0x0.

By setting c1=x0

−ϕ(c), the optimal solution (4.35) satisﬁes the constraint in the prime problem,

which indicates that the strong duality holds, or equivalently,

V(x0) = inf

λ>0V(λ, x0).

Hence

Q∗(y) = x0

−ϕ(c)1{y≥c}(4.36)

is the optimal solution to problem (4.31).

The next theorem will discuss the existence conditions of the optimal solution to (4.31).

Theorem 4.5 Let k= limy↑1ˆϕ0(y), and c= inf{y∈(0,1) : ˆϕ0(y) = k}.

(1) If c= 1, then there is no optimal solution to problem (4.31).

(2) If c < 1and k > 0, then problem (4.31)admits an optimal solution. Moreover, if ˆϕ(y)>

ϕ(y),∀y∈(c, 1), the optimal solution is unique with the same form in (4.36).

Proof. See Appendix C.3 

4.4 Example 4: Goal Reaching Model

In this example we consider the goal reaching model put forward in Browne (1999), see also

Cvitanic et al. (2019). The original form of the optimization problem is

max

P(XT≥b),(4.37)

which is equivalent to

max

E[1{XT≥b}] (4.38)

subject to E[ρTXT] = x0,XT≥0 and XTis FTadapted, where bis a constant target. When

we consider probability distortion, the objective function can be transformed into the following

form:

Z+∞

U(y)d(1 −w(1 −FXT(y))) = Z+∞

d(1 −w(1 −FXT(y))) = w(1 −FXT(B)) = w(Eu(XT)),

where U(y) = 1{y≥B}, and B > 0 is the goal of the agent. Since w(·) is strictly increasing, the

problem is equivalent to the optimization problem when there is no distortion function at all.

Accordingly, the non-distorted problem has the quantile formulation form (also see He and

Zhou (2011b)):

V(x0) := sup

G∈G Z1

1{G(x)≥B}dx

s.t. Z1

G(x)F−1

ρ(1 −x)dx =x0,

(4.39)

where ρ > 0 is a continuous random variable with distribution Fρ.

Using the Lagrange multiplier methods with parameter λ > 0, we get the unconstrained

problem:

V(λ, x0) := sup

G∈G Z1

(1{G(x)≥B}−λG(x)F−1

ρ(1 −x))dx +λx0.(4.40)

The concave envelope of U(x) = 1{x≥B}is

U(x) = (x

B, x < B,

1, x ≥B.

Hence, we consider the problem :

sup

G∈G Z1

(ˆ

U(G(x)) −λG(x)F−1

ρ(1 −x))dx +λx0,

whose optimal solution can be easily obtained as

G∗(x) = (0, λF −1

ρ(1 −x)>1

B, λF −1

ρ(1 −x)≤1

And such a solution is also optimal to problem (4.40). Thus,

V(λ, x0) = Fρ1

λB +λ x0−BZ1

λB

ydFρ(y)!.

The ﬁrst order derivative is

λ(λ, x0) = x0−BZ1

λB

ydFρ(y) = x0−BE[ρ1{λ∗Bρ≤1}],

which is a strictly decreasing function w.r.t. λ. The equation

BE[ρ1{λ∗Bρ≤1}] = x0

admits a unique solution λ∗when 0 ≤x0≤BE[ρ]. If x0reaches the lower or upper bounds,

then we let λ∗be respectively +∞or 0+. Hence, if 0 < x0< BEρ, then

V(x0) = inf

λ>0V(λ, x0) = V(λ∗, x0),

or equivalently, the strong duality holds. Then

G∗(x) = B1{λ∗BF −1

ρ(1−x)≤1}

is the optimal solution to problem (4.39).

5 Solutions for the general model

Previously we demonstrate that our method can solve representative behavior ﬁnance portfolio

choice problems in the literature, under our proposed uniﬁed solution framework. In this section,

we shall characterize the solutions for the general case of Uand ϕ, i.e. when the utility function

exhibit a ﬁnite number of inﬂection points (reference points), and the probability distortion

function can also have multiple inﬂection points.

5.1 Notations

First we shall introduce some notations for convenience. For a monotone function f(x), let

f(x+) := lim

y↓xf(y)

f(x−) := lim

y↑xf(y)

be the right and left limit at point x.

Recalling Proposition 3.1 and Lemma 3.1, we shall pay more attention to the situation when

YDis not a singleton. Since Kis the jump points set of ( ˆ

U0)−1(·), for every ki∈ K, we can

ﬁnd the largest open interval (ai, bi) on which ˆ

Uis aﬃne with slope ki. We call this interval the

Aﬃne Interval of slope ki. In fact

ai= ( ˆ

U0)−1(ki+), bi= ( ˆ

U0)−1(ki−).

Conveniently, we call Kthe Slopes Set of ˆ

Uon the domain R+.

To avoid complicated computations in the construction of the optimal solution, we separate

Kinto three disjoint subsets in the following way: for every slope k∈ K, and its corresponding

aﬃne interval (a, b), let

K1:= k∈ K|(a, b)⊆ {x∈R+|ˆ

U(x)> U(x)},

K2:= k∈ K|(a, b)⊆ {x∈ R+|ˆ

U(x) = U(x)},

K3:= K \ (K1∪ K2).

For intuitive illustrations, please see Figure 11 and Figure 12.

Similarly, we can deﬁne Las the Slope Set of ˆϕon [0,1], the corresponding largest aﬃne

interval (αi, βi) for any li∈ L and separate Linto L1,L2and L3. Set

Λ := k

l|k∈ K, l ∈ L,

Λ0:= λ∈Λ|there exist n > 1 and diﬀerent ki∈ K and li∈ L with λ=ki

, i = 1,2,· · · , n.

Figure 11: An example for the situations ki∈ K1

and kj∈ K2.

Figure 12: An example for the situation k∈ K3.

The three sets K3,L3,Λ0will make the form of the optimal solutions complex, thus we shall

discuss them in the last subsection.

5.2 Optimal solutions under the case: K3=∅,L3=∅,Λ0=∅

In the case: K3=∅,L3=∅,Λ0=∅, we can easily get YBby using Theorem 3.1.

Proposition 5.2 Using the notations deﬁned in Section 5.1, we have that the optimal solutions

to problem B are given by

(1) If λ /∈Λ, then

Q∗

λ(x)=(ˆ

U0)−1(λˆϕ0(x)−).(5.1)

(2) If λ∈Λ, then we can ﬁnd the unique pair k∈ K, l ∈ L, such that λ=k

land their

corresponding aﬃne intervals (a, b),(α, β).

(i) if k∈ K1and l∈ L1, then

Q∗

λ(x) = ((ˆ

U0)−1(λˆϕ0(x)−), x /∈[α, β)

aor b, x ∈[α, β)(5.2)

(ii) if k∈ K1and l∈ L2, then

Q∗

λ(x) = ((ˆ

U0)−1(λˆϕ0(x)−), x /∈[α, β)

aI[α,c1)+bI[c1,β), x ∈[α, β)(5.3)

where c1is a constant in [α, β].

(iii) if k∈ K2and l∈ L1, then

Q∗

λ(x) = ((ˆ

U0)−1(λˆϕ0(x)−), x /∈[α, β)

c2, x ∈[α, β)(5.4)

where c2is a constant in [a, b].

(iv) if k∈ K2and l∈ L2, then

Q∗

λ(x) = ((ˆ

U0)−1(λˆϕ0(x)−), x /∈[α, β)

h(x), x ∈[α, β)(5.5)

where h: [α, β)→[a, b]is an arbitrary non-decreasing and right-continuous function.

Assumption 5.3 There exists a λ0<+∞such that

λ0= inf λ > 0 : Z1

(ˆ

U0)−1(λˆϕ0(x)+)ϕ0(x)dx < +∞.

This assumption is intimately related to the well-posedness of the original optimization problem.

One should notice that if this assumption is not satisﬁed, the original problem has to be ill-posed

while this does not mean the well-posedness would be guaranteed if this assumption is satisﬁed.

The detailed discussion can be found in Appendix A.

For any λ /∈Λ, Q∗

λin (5.1) is unique, thus we can deﬁne

f(λ) := Z1

Q∗

λ(x)ϕ0(x)dx.

As a result, f(λ) is strictly decreasing due to the fact that ( ˆ

U0)−1is non-increasing and can not

be a constant function. Such a result forces us to only consider the case λ≥λ0. Accordingly,

the optimal Lagrange multiplier λ∗, if it exists, must be a decreasing function respect to x0. In

other words, we may discuss the two cases: λ=λ0,λ>λ0, or x0≥f(λ0+), x0< f(λ0+), to get

the suﬃcient and necessary conditions under which there exist a unique λ∗and a corresponding

Q∗

λsatisfying f(λ∗) = x0, i.e.

Q∗

λ(x)ϕ0(x)dx =x0.(5.6)

The case corresponding to x0=f(λ0+) <+∞is easy to solve by letting ( ˆ

U0)−1(λˆϕ0(x−)+) be

the optimal solution. Thus, in the following we shall consider the other cases.

Remark 5.5 If f(λ0+) = +∞, then the case for x0>+∞is redundant, and one only needs to

consider x0< f(λ0+). Thus, we assume that f(λ0+) <+∞in the next two theorems.

Set

k0:= lim

x→+∞

U0(x),

l0:= lim

x→1−ˆϕ0(x),

M:= ( ˆ

U0)−1(k0+),

¯c:= ( ˆϕ0)−1(l0+).

Theorem 5.6 Assume that x0> f (λ0+).

(1) If k0= 0,¯

M < +∞, then λ0= 0 and f(λ0+) = ¯

ME[ρ]. Moreover, any quantile function

Qwhich satisﬁes the constraint (5.6)and Q≥¯

Mis an optimal solution to Problem A.

(2) If k0= 0,¯

M= +∞, then

(I) if l0= 0 and ¯c < 1, then λ0= +∞, the problem is ill-posed.

(II) if l0= 0 and ¯c= 1, then the strong duality does not hold.

(III) if l0>0, then λ0= 0 and f(0+) = +∞.

(3) If k0>0,l0= 0, then λ0= +∞, or equally, the initial problem is ill-posed.

(4) If k0>0,l0>0, then λ0=k0

l0. Thus,

(I) if ¯

M= +∞or if ¯

M < +∞and ¯c= 1, then f(λ0+) <+∞, the strong duality does

not hold.

(II) if ¯

M < +∞and ¯c < 1, then the strong duality holds. Moreover, if l0∈ L1, then the

optimal solution is unique.

Proof. See Appendix D.1.

Let

Λ1=k

l> λ0|k∈ K1, l ∈ L1,

Λ2=k

l> λ0|k∈ K2, l ∈ L2.

(5.7)

Theorem 5.7 Assume that x0< f (λ0+).

(1) If x0∈Sλ∈Λ1(f(λ+), f (λ−)), then the strong duality (3.10)does not hold.

(2) If x0/∈Sλ∈Λ1(f(λ+), f(λ−)), then the strong duality (3.10)holds. There exists a λ∗and a

Q∗

λ∗solving (5.6), and such a function is the optimal solution to Problem A. Moreover,

if x0/∈Sλ∈Λ1∪Λ2(f(λ+), f (λ−)), then the optimal solution is unique.

Proof. See Appendix D.2.

Remark 5.6 For the case of x0∈Sλ∈Λ1(f(λ+), f (λ−)), the strong duality fails and the La-

grange method can not be directly applied to this case. To fully resolve this case is a future

research direction. A possible approach11 is to utilize the rationalization approach of Bernard

et al. (2015a) and combine it with results in Section 5 of Reichlin (2013).

Corollary 5.1 Assume that x0< f (λ0+).

If the utility function Uis concave, then Sλ∈Λ1(f(λ+), f (λ−)) is empty and the optimal solu-

tions to problem A are in the forms given in Proposition 5.2.

Moreover, if Uis strictly concave, then Sλ∈Λ2(f(λ+), f (λ−)) is empty and the optimal solution

to problem A is unique with the form (5.1), which is consistent with Xu (2016).

Corollary 5.2 Assume that x0< f (λ0+).

If ϕis concave, then Sλ∈Λ1(f(λ+), f (λ−)) is empty, thus, problem A admits optimal solution

in the forms given in Proposition 5.2.

Moreover if there is no probability distortion, then Sλ∈Λ1(f(λ+), f(λ−)) is empty too. Thus,

the problem A must admit an unique solution in the form (5.1)which is consistent with Reichlin

(2013).

11We thank Stephan Sturm for this suggestion based on private communication.

Remark 5.7 According to Xu (2016), a new pricing kernel ˜ρwith F−1

˜ρ(1 −x) = ˆϕ0(x)is given

with which the problem can be transformed into the following form:

sup

E[u(X)]

s.t. E[˜ρX] = x0.

(5.8)

From Reichlin (2013), only when ˜ρis atom-less12, this problem is equal to its concaviﬁed form

for all x0. In this paper, ˜ρmay contain an atom, but we only need to limit the value of x0out of

Sλ∈Λ1(f(λ+), f (λ−)) rather than Sλ∈Λ(f(λ+), f (λ−)).

Remark 5.8 For Yaari’s dual model, U(x) = x, then U0(x) = 1 and the general inverse (non-

increasing one) is given by

(U0)−1(x) = 









+∞, x < 1,

∈[0,+∞), x = 1,

0, x > 1.

•If l0= 0, then problem A is ill-posed.

•If l0>0and ¯c= 1, then the strong duality does not hold. In fact, Theorem 4.5 tells us

that we cannot attain the optimum of problem A. If l0>0and ¯c < 1, then for any c < 1

with ϕ(c) = l0, the form (4.36)is the optimal solution to problem A.

Remark 5.9 For the goal reaching case, ˆ

U0(x) = 1

B,if x∈(0, B);ˆ

U0(x) = 0,if x > B.

Similarly, we can get

(ˆ

U0)−1(x) = B1{x< 1

B}.

(ˆ

U0)−1(0) = +∞,we get f(0−) = +∞,f(0+) = BE[ρ].Thus if x0≥f(0+),then the optimal

multiplier is λ∗= 0, and any quantile Q∗(x)≥Bsatisfying R1

0Q∗(x)ϕ0(x)dx = 0 is an optimal

solution to problem A. If x0≤f(0+), then problem A admits a unique solution.

5.3 Solutions when at least one of K3,L3,Λ0is not empty

We discuss this case in the following three sub-categories.

5.3.1 K3=∅,L3=∅,Λ06=∅

For the case K3=∅,L3=∅,Λ06=∅, there exists at least one λ∈Λ0, and two sequences ki, li

with λ=ki

liand corresponding aﬃne intervals (ai, bi),(αi, βi), i = 1,2,· · · , n.

Now, for this special λ, using the results in (5.2), we can easily get

Q∗

λ+(x) = ((ˆ

U0)−1(λˆϕ0(x)−), x /∈S[αi, βi)

ai, x ∈[αi, βi)

Q∗

λ−(x) = ((ˆ

U0)−1(λˆϕ0(x)−), x /∈S[αi, βi)

bi, x ∈[αi, βi)

12We thank Stephan Sturm for pointing out that the general case of countable number of atoms can be handled

following similar steps as in Lemma 6.3 in Bichuch and Sturm (2014).

and

f(λ+) = Z1

Q∗

λ+(x)ϕ(x)dx,

f(λ−) = Z1

Q∗

λ−(x)ϕ(x)dx.

The strictly decreasing property of f(λ) implies that we can only consider x0∈[f(λ+), f(λ−)].

Let Iibe the set deﬁned through:

1. If ki∈ K1, li∈ L1,Ii={0,(bi−ai)(ϕ(βi)−ϕ(αi))}.

2. Otherwise, Ii= [0,(bi−ai)(ϕ(βi)−ϕ(αi))].

Theorem 5.8 If x0∈[f(λ+), f (λ−)], then the strong duality (3.10)holds if and only if the

equation

nx=x0−f(λ+)

with the N-dimension variable x, admits at least one solution with x∗∈ ⊗n

i=1Ii.

Proof. The proof follows from Theorem 5.7 and its proof is given in Appendix D.2.

Now, if we get a solution x∗= (x∗

1,· · · , x∗

n), then λ∗=λand we can construct the optimal

solution Q∗(y) to problem A in each interval [αi, βi) with Rβi

αi(Q∗(x)−ai)ϕ0(x)dx =x∗

iin the

forms of Proposition 5.2.

5.3.2 K36=∅,L36=∅,Λ0=∅

Next, we turn to the situation K36=∅,L36=∅,Λ0=∅. Similarly, we only need to consider the

case: there exists one λ, with λ=k

l, and the corresponding (a, b),(α, β) can be written as:

[a, b) =

[

i=1

[ai, bi),

[α, β) =

[

i=1

[αi, βi),

where

ai=bi−1, a1=a, bn=n, ˆ

U(ai) = U(ai), i = 2,3,· · · , n,

αi=βi−1, α1=α, βm=β, ˆϕ(αi) = ϕ(αi), i = 2,3,· · · , m,

and the sub-intervals (ai, bi) belongs to either {ˆ

U=U}or {ˆ

U > U }, and the sub-interval (αj, βj)

belongs to either {ˆϕ=ϕ}or {ˆϕ>ϕ}.

Set

H:= {x∈[a, b] : ˆ

U(x) = U(x)},

and a sequence of non-deceasing and right-continuous functions {hi}, i = 1,2,· · · , m:

hi: [αi, βi)→ H

satisfying

hi(βi−)≤hi+1(αi+1),

hiis constant if (αi, βi)⊆ { ˆϕ>ϕ}.

We denote

xi:= Zβi

αi

(hi(x)−a)ϕ0(x)dx.

Now, we obtain the following result.

Theorem 5.9 If λ, H,{hi}are deﬁned above, then the optimal solution to problem B can be

written as:

Q∗

λ(x) = Q∗

λ+(x) +

i=1

(hi(x)−a)1[αi,βi).

If x0∈[f(λ+), f (λ−)], then the strong duality (3.10)holds if and only if there exists such a

function sequence {hi}and x= (x1, x2,· · · , xm)deﬁned above solving:

mx=x0−f(λ+).

Proof. We utilize Theorem 3.1 and the proofs in Theorem 5.7 to obtain the results. 

5.3.3 K36=∅,L36=∅,Λ06=∅

Finally, if K36=∅,L36=∅,Λ06=∅, then for λ∈Λ0and x0∈[f(λ+), f (λ−)], we can combine

the solution construction of the former two cases and Theorem 5.8, Theorem 5.9 to solve this

problem.

6 Conclusion

In this paper, we propose to combine the relaxation method with the Lagrange method, and

provide a general solution method to the optimal investment problem with the presence of both

non-concave utility functions and nonlinear probability distortion. We obtain explicit closed-

form solutions in several representative behavior ﬁnance models. In particular, we obtain a

“distorted Merton line” in a special case of the CPT model. An interesting ﬁnding is that only

the probability distortion, but not the non-concave utility function, increases the proportion of

wealth invested in the risky stock, as compared to the standard Merton model.

Note that we only require that the market is complete, i.e. there exists a unique state-price

density or pricing kernel, and do not restrict the underlying asset price dynamics. Thus the

method is general enough and can be extended to other stochastic processes beyond the Black-

Scholes model. We leave the investigations of behavior portfolio choice for alternative stochastic

models as future research directions. In addition, the extension to possibly incomplete markets

is of great interest and left as a future research direction.

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Online Supplementary Appendix

A Well-posedness

We study the well-posedness problem in the following three cases:

1. There exists one C∈R, such that U(x)< C, ∀x∈[0,∞). This type of utilities include the

utility function arising from the goal reaching problem, exponential utility, etc.

2. limx→∞ U(x)

x=k > 0. This type of utilities include the linear utility function, or the

asymptotic linear utility function.

3. limx→∞ U(x)

x= 0. This type of utilities include the CRRA utilities, etc.

Case 1: In this case, U(X)< C means E[ˆ

U(X)] < C, so the problem is well-posed.

Case 2: In this case, we give a crucial lemma which is a simple extension of the corresponding

result in He and Zhou (2011b).

Lemma A.1 Problem (2.10)is well-posed if lim inf y→1ϕ0(y)>0and ill-posed if lim supy→1ϕ0(y) =

Proof. If limx→∞ U(x)

x=k > 0, then for ∀ > 0 there exist a N∈R+such that (k−)x <

U(x)<(k+)xfor ∀x>N.

On one hand, if lim supt→1ϕ0(t) = 0, then for ∀n > 0, there exist y1∈(0,1) such that

ϕ0(y)<1

nfor ∀y∈(y1,1).

Let

Q(y) = (δ, y < y2,

b, y ≥y2,

where 0 < y2<1, and δis a constant satisfying x0+δϕ(0) >0 and U(δ)>−∞, and bis a

constant to be determined. Let b=x0−δ(ϕ(y2)−ϕ(0))

ϕ(1)−ϕ(y2), then Q(y) satisﬁes the constraint in (2.10),

thus Q(y) is a feasible solution. For y2≥max{ϕ−1(−x0+δϕ(0)

N−δ), y1}, we have b > N , and

U(Q(y))dy =U(δ)y2+U(b)(1 −y2)

>min{U(δ),0}+ (k−)x0+δϕ(0) −δϕ(y2)

ϕ(1) −ϕ(y2)(1 −y2)

>min{U(δ),0}+ (k−)(x0+δϕ(0)) 1−y2

−ϕ(y2)

≥min{U(δ),0}+ (k−)(x0+δϕ(0))n.

Thus supQR1

0U(Q(y))dy → ∞, when n→ ∞.This shows that the problem (2.10) is ill-posed.

On the other hand, if lim infy→1ϕ0(y)>0, then there exist land y3∈(0,1) such that ϕ0(y)> l

for ∀y∈[y3,1]. Thus, for ∀feasible Q(y), there exists a y4∈[0,1], such that Q(y)≥N+ 1,

∀y > y4(If Q(y)< N + 1 for ∀y∈[0,1], then we take y4= 1). Let y0=max{y3, y4}, then we

have

U(Q(y))dy =Zy0

U(Q(y))dy +Z1

U(Q(y))dy

< U(Q(y0))y0+k+

lZ1

Q(y)ϕ0(y)dy

≤U(Q(y0))y0+k+

lx0

<∞.

So, problem (2.10) is well-posed. 

Case 3: Rewriting problem (2.10) as

VA(x0) := sup

Q∈G Z1

U(Q(y))dy,

s.t. Z1

Q(y)ϕ0(y)dy =x0,

(A.1)

and replacing Uby its concave envelope ˆ

U, we get

V(x0) := sup

Q∈G Z1

U(Q(y))dy,

s.t. Z1

Q(y)ϕ0(y)dy =x0.

(A.2)

Let M0= inf{x∈[0, M ] : ˆ

U(x)> U(x)}, and here Mis the one mentioned in Assumption 2.1,

then

U(x)−(U(M)−U(M0)) ≤U(x)≤ˆ

U(x),∀x≥0,

i.e.

V(x0)−(U(M)−U(M0))Eρ≤VA(x0)≤ˆ

V(x0).

Consequently, we only need to consider the well-posedness problem of problem (A.2).

Let ξ= ˆϕ0(1 −Fρ(ρ)). Problem (A.2) is equivalent to

V(x0) = sup

E[ˆ

U(X)],

s.t. E[ξX ] = x0.

(A.3)

Thus, the well-posedness of problems (A.1) and (A.3) are equivalent.

The solution to problem (A.3) is

X∗= ( ˆ

U0)−1(λξ),(A.4)

where the inverse refers to the general inverse and λ > 0 is a constant satisfying E[ξX∗] = x0.

Taking the above analysis into account, the following two propositions hold true in our setup.

Proposition A.1 (Theorem 3.1 in Jin et al. (2008)) If E[( ˆ

U0)−1(λξ)ξ] = +∞for any λ > 0,

then V(x0)=+∞for ∀x0>0.

Here, this proposition leads to the necessity of Assumption 5.3 in Section 5.

Suppose that F(·) is the cumulative distribution function of ξ, then we have

Proposition A.2 (Theorem 5.4 in Jin et al. (2008))

1. If lim infx→+∞−xˆ

U00(x)

U0(x)>0, then the problem (A.3)is well-posed for ∀x0>0if and

only if E[ˆ

U(( ˆ

U0)−1(ξ))] <∞.

2. If lim supx→0+−xF 0(x)

F(x)<0, then the problem is well-posed for any x0>0if and only if

E[( ˆ

U0)−1(λξ)ξ]<+∞for some λ > 0.

B Proofs in Section 3.3

B.1 Proof of Theorem 3.1

Proof.

(1) First, we claim that for every Q∈ Q and λ > 0,

JD(Q, λ)≥JC(Q, λ),(B.5)

and moreover, the equality holds if and only if Qsatisﬁes (3.8).

In fact, recalling that ˆϕ(0) = ϕ(0),ˆϕ(1) = ϕ(1), then we have

JD(Q, λ)−JC(Q, λ) = λZ1

(ϕ0(x)−ˆϕ0(x))Q(x)dx

=λZ1

(ϕ0(x)−ˆϕ0(x))Q(x)dx −Q(0)λZ1

(ϕ0(x)−ˆϕ0(x))dx

=λZ1

(ϕ0(x)−ˆϕ0(x)) Zx

dQ(y)dx

=λZ1

( ˆϕ(x)−ϕ(x))dQ(x) (B.6)

≥0.

Thus, JD(Q, λ)≥JC(Q, λ) for any Q.

By using the continuity of ˆϕ−ϕ, we know that the set {x∈[0,1] : ˆϕ(x)> ϕ(x)}is an open

set, thus it can be written as the union of countable disjoint open intervals, i.e.

{x∈[0,1] : ˆϕ(x)> ϕ(x)}=

+∞

[

i=1

Ii,

where Iiis an open interval. Consequently, (B.6) can be rewritten as

JD(Q, λ)−JC(Q, λ) = λ

+∞

i=1 ZIi

( ˆϕ(x)−ϕ(x))dQ(x)

Recalling that ˆϕ(x)−ϕ(x)>0,∀x∈Ii, it is easy to show that JD(Q, λ) = JC(Q, λ) is equivalent

to ZIi

dQ(x) = 0, i = 1,2,· · · ,

Q(·) is a constant function on each Ii.

Now, for every interval (α, β)⊆ {x: ˆϕ(x)> ϕ(x)}, there must exist an i, such that (α, β)⊆Ii.

We then complete our proof of the claim.

Next, we prove the part 1 of the theorem. By taking supremum over (B.5), we have

VD(λ, x0)≥VC(λ, x0).

“If Part”: For any Q∈ YDsatisfying (3.8), the former claim implies

VD(λ, x0) = JD(Q, λ) = JC(Q, λ)≤VC(λ, x0).

We then get VC(λ, x0) = VD(λ, x0) immediately and Qis an optimal solution to problem C.

“Only If Part”: First, let

Q1(x) = ( ˆ

U0)−1

−(λˆϕ0(x)).(B.7)

Obviously, Q1(x)∈ YDsatisﬁes (3.8), which indicates that VC(λ, x0) = VD(λ, x0).

Now, if Q∈ YDis optimal to problem C, then Qmust satisfy (3.8), or otherwise we have

VD(λ, x0) = VC(λ, x0) = JC(Q, λ)< JD(Q, λ) = VD(λ, x0),

which leads to a contradiction.

(2) On one hand, if

U(Q(x)) = ˆ

U(Q(x)),∀y∈[0,1],

then

VB(λ, x0)≤VC(λ, x0) = JC(Q, λ)

=Z1

(ˆ

U(Q(y)) −λϕ0(y)Q(y))dy +λx0

=Z1

(U(Q(y)) −λϕ0(y)Q(y))dy +λx0

≤VB(λ, x0)

which shows that Q(·) must be the optimal solution to problem B.

On the other hand, it is easy to check that Q1deﬁned in (B.7) satisﬁes:

U(Q1(x)) = ˆ

U(Q1(x)),∀y∈[0,1].

Thus we have VC(λ, x0) = VB(λ, x0). Now, if Q(·)∈ YCis also optimal to problem B, but there

exists a point y0such that ˆ

U(Q(y0)) > U(Q(y0)), then the right-continuity of U, ˆ

U, Q implies

that there exist  > 0, δ > 0 such that

U(Q(y)) −U(Q(y)) > , ∀y∈[y0, y0+δ).

Hence

VC(λ, x0)−VB(λ, x0) = JC(Q, λ)−JB(Q, λ)

=Z1

(ˆ

U(Q(y)) −U(Q(y)))dy

≥Zy0+δ

(ˆ

U(Q(y)) −U(Q(y)))dy

> δ.

We arrive at a contradiction.

In conclusion, Q1is a common solution to problems D, C, B and we have VD(λ, x0) = VC(λ, x0) =

VB(λ, x0). This completes the proof. 

C Proofs in Section 4

C.1 Proof of Theorem4.3

Proof.

Because of the same solution structure in the two cases in this theorem, we only give the proof

to the case 1.

If x0≤xL, Theorem 4.2 implies that f(λ) = x0admits a unique root λ∗≥λ0, then λ∗ˆϕ0(y)≥

U0(ξ)∀y∈[0, c).Thus, Q∗(y) = 0 for y < c,and

ϕ0(y) = k−1

1exp ˜µ−a1˜σ−a2

1/2+(a1+ ˜σ)Φ−11−y

k1, y ≥c,

which shows that λ∗ˆϕ0(y) = U0(ξ), y ≥chas a unique solution

y∗= 1 −k1Φ ln k1u0(ξ)

λ∗+a1˜σ+a2

1/2−˜µ

a1+ ˜σ!.

Hence, we can obtain the explicit expression:

X∗

T=Q∗(1 −w(Fρ(ρ))) = (U0)−1(λ∗ϕ0(1 −w(Fρ(ρ))))1{1−w(Fρ(ρ))≥y∗}.

Recalling that for x≤b,

ϕ0(x) = F−1

ρ(w−1(1 −x))

w0(w−1(1 −x)) , w0(x) = k1exp{−a2

1/2−a1Φ−1(x)}.

Thus, recalling the notations c1, c2and c3in (4.14), we have

X∗

T= (U0)−1λ∗ρ

w0(Fρ(ρ))1{ρ≤c1}

= η+γw0(Fρ(ρ))

λ∗ρ1

1−γ!1{ρ≤c1}

= η+k1γ

λ∗1

1−γ

exp −1

1−γa2

1/2−a1˜µ

˜σρ−1

1−γ(1+ a1

˜σ)!1{ρ≤c1}

= (η+c2ρ−c3)1{ρ≤c1}.

Let

Zt:= ρ

ρt

then Ztis independent of Ftwith distribution function Φ ln x−At

Bt, where At, Btare deﬁned in

(4.11). As a result, we have

X∗

t=1

ρt

E[ρX∗

T|F] = ηE[Zt1{Zt≤c1/ρt}] + c2ρ−c3

tE[Z1−c3

t1{Zt≤c1/ρt}].

Recall the fact: if Y∼N(0,1), then E[eαY 1{Y≤β}] = eα2/2Φ(β−α). Then we can easily compute

X∗

t=ηexp{B2

t+A2

t}Φln c1/ρt−At

−Bt+c2exp{(1 −c3)At

+ (1 −c3)2B2

t/2}ρ−c3Φln c1/ρt−At

−(1 −c3)Bt.

If we ﬁx t, then X∗

tis a decreasing function respect to ρt, which we note as f(ρt). We have

π∗

t=−µt−rt

σ2

tX∗(t)ρtf0(ρt)

=µt−rt

σ2

tX∗(t)η

exp{B2

t/2 + At}φln c1/ρt−At

−Bt

+c2c3exp{(1 −c3)At+ (1 −c3)2B2

t/2}ρ−c3Φln c1/ρt−At

−(1 −c3)Bt

+c2

exp{(1 −c3)At+ (1 −c3)2B2

t/2}ρ−c3φln c1/ρt−At

−(1 −c3)Bt.

This completes the proof. 

C.2 Proof of Lemma 4.2,4.3 and Theorem 4.4

Proof of Lemma 4.2:

(1) We only need to consider the following minimization problem:

inf

G(·)Z1

G(y)F−1

ρ(1 −y)dy

s.t. 1

αZα

G(y)dy =x1,

(C.8)

to judge whether the feasible set is empty or not.

Since G(y) is non-decreasing in y, the optimal solution to (C.8) must satisfy:

G(y) = G(α),∀y∈[α, 1],

then we can solve (C.8) in two steps.

Firstly, we ﬁx G(α), and solve

inf

G(·)≤G(α)Zα

G(y)F−1

ρ(1 −y)dy

s.t. 1

αZ1

G(y)dy =x1.

By using the Lagrange multiplier method, it turns out that

inf

G(·)≤G(α)Zα

G(y)(F−1

ρ(1 −y)−λ

α)dy

admits an optimal solution

G∗(y) = (0, y < c,

G(α), y ∈[c, α),

where c= 1 −Fρ(λ

α). Substituting the solution into the constraint, we obtain:

G(α) = αx1

α−c.

Secondly, we solve the initial problem (C.8) by minimizing:

inf

c∈[0,α)αx1R1

cF−1

ρ(1 −y)dy

α−c.

Recalling the deﬁnition in (4.28), we get the optimal value αRx1, and the optimal minimizer c∗.

Thus, x0≥αRx1holds, or equivalently the problem (4.23) admits no solution when x0< αRx1.

(2) Recalling the deﬁnition in (4.25), we have:

ϕλ(y2)−ϕλ(y1) = k(y2−y1).

Applying a total diﬀerential on each side, we obtain

ϕ0

λ(y2)dy2−ϕ0

λ(y1)dy1−(1 −y1

α)dλ =k(dy2−dy1)+(y2−y1)dk.

Thus dk

dλ =−1−y1

y2−y1

<0,if λ > 0,

and d(k+λ

α)

dλ =dk

dλ +1

α=

α−1

y2−y1

>0,if λ > 0.

Since (4.24) and (4.25) imply

y1= 1 −Fρk+λ

α,

y2= 1 −Fρ(k),

we can easily conclude that both kand y1are strictly decreasing, while y2is strictly increasing

w.r.t λ. In other words, λ, y2is strictly decreasing w.r.t. y1, while kis increasing w.r.t. y1.

Moreover, the continuity property is obvious since all the functions are diﬀerentiable.

Now, we want to ﬁnd the ranges for the above variables y1, y2, k, λ.λ≥0 is needed because

of the constraint of ES. One critical case is λ= 0, then y1=y2=α, k =F−1

ρ(1 −α).

The other condition ˆϕ0

λ(y)≥0 is needed in solving problem (4.21) too, otherwise the optimal

solution is G(y)=+∞. Now, we ﬁnd the other critical case: there exists y < 1, s.t. ˆϕ0

λ(y) = 0.

Thus, k= 0, y2= 1 and

F−1

ρ(1 −y1) = λ

α,

F−1

ρ(1 −s)ds =λ1−y1

α,

(C.9)

by the deﬁnitions in (4.24) and (4.25). Combining the above two equation, we have

g(y1) = F−1

ρ(1 −y1).

By comparing this equation with g0(c∗) = 0, we have that y1=c∗is the root to (C.9).

Consequently, we obtain the desired results: y1∈[c∗, α], y2∈[α, 1], λ ∈[0, αF −1

ρ(1 −c∗)] and

k∈[0, F −1

ρ(1 −α)]. This completes the proof. 

Proof of Lemma 4.3:

(1) We only prove this property for function f. The monotonic side is easy to check, since

(U0)−1(·) is a non-increasing function.

For any λ, λ1satisfying λ1k < U 0(ξ), let ˆϕ0

λ(y0) = U0(ξ)

λ1i.e.

y0= 1 −Fρλ

α+U0(ξ)

λ1,(C.10)

then y0is continuous w.r.t. λor λ1and

f(λ, λ1) = 1

αZα

(U0)−1(λ1ˆϕ0

λ(y))dy

αZy1

(U0)−1λ1F−1

ρ(1 −y)−λλ1

αdy + (U0)−1(λ1k)1−y1

α.

Here (U0)−1is continuous on (0, U0(ξ)) by the deﬁnition of U,y1is continuous w.r.t. λand y0is

continuously w.r.t. λor λ1, so it is easy to get the continuity results.

Moreover, the statements (2) (3) can be easily deduced from the statement (1). This completes

the proof. 

Proof of the Theorem 4.4:

(1) If x0>∆(x1), then we can remove the ES constraint and obtain the optimal solution to

problem (4.23), i.e. let λ= 0 and we can solve g(0, λ1) = x0to get the root λ∗

1and the optimal

solution G∗

0,λ∗

1(y).

By Lemma 4.3,x0>∆(x1) is equivalent to λ∗

1< f1(0, x1), or equivalently, f(0, λ1)> x1.

Hence the ES constraint holds, which means that G∗

0,λ∗

1(y) is optimal.

(2) It is easy to check that:

g(c∗, f1(c∗, x1)) = αRx1, g(0, f1(0, x1)) = ∆(x1).

When x1≥ξ(1 −c∗

α), f1(·, x1) is well-deﬁned on [0, c∗], thus by Lemma 4.3, when x0∈

[αRx1,∆(x1)], x0=g(λ, f1(λ, x1)) must admit at least one solution λ∗and (λ∗, f1(λ∗, x1)) is

a root to the equation system (4.29). Thus problem (4.23) must admit an optimal solution.

(3) By deﬁnition of y1, for any solution in (4.26), we have G(y1)≥ξ, thus

x1=1

αZα

Gλ,λ1(y)dy ≥1

αξ(α−y1).

Now, if 0 < x1< ξ(1 −c∗

α), then y1≥α(1 −x1

ξ)> c∗, which implies that λmust be restricted

in [0,¯

λ].

By Lemma 4.3,g(·, f1(·, x1)) must admit its maximum and its minimum, since it is continuous

on its domain [0,¯

λ]. Obviously, its maximum is ∆(x1). Now, set

L(x1) = min

λ∈[0,¯

λ]g(λ, f1(λ, x1)),

then the range of g(·, f1(·, x1)) is [L(x1),∆(x1)]. Accordingly, g(λ, f1(λ, x1)) = x0admits at least

one root when x0∈[L(x1),∆(x1)] and admits no root when x0∈(αRx1, L(x1)).

In conclusion,

•if x0=αRx1, then problem (4.23) admits a unique feasible solution which is also optimal.

•if x0∈x0∈[L(x1),∆(x1)], then problem (4.23) admits at least one optimal solution in the

form (4.26).

•if x0∈(αRx1, L(x1)), then the strong duality does not hold, and our method fails to ﬁnd

the optimal solution.

This completes the proof. 

C.3 Proof of Theorem 4.5

Before starting our proof, we ﬁrst introduce a useful result. Fix x0, if we consider the function ϕ

in problem (4.31) as a parameter, and denote the optimal value function as W(ϕ), then we have

the following result.

Proposition C.3 If two increasing and absolutely continuous functions f1, f2satisfy: f1≤f2

and f1(0) = f2(0), f1(1) = f2(1), then

W(f1)≤W(f2).

Moreover if Q(y)is an optimal solution to problem (4.31)with the parameter f2and Q(y)is

constant on any open interval which is a subset of {y:f1(y)< f2(y)}, then Q(y)is an optimal

solution to problem (4.31)with the parameter f1and

U(f1) = U(f2).

Proof.

We let Q(y) solve the problem (4.31) with the parameter f1. By noticing that

Q(y)(f0

2(y)−f0

1(y))dy =Z1

0Zy

dQ(z)(f0

2(y)−f0

2(y))dy =Z1

(f1(y)−f2(y))dQ(y)≤0,

and we denote the above value as −, where ≥0. Now, let

Q1(y) = Q(y) + 

−ϕ(0),

then Q1(y) is feasible to problem (4.31) with the parameter f2, and

W(f1) = Z1

Q(y)dy ≤Z1

Q1(y)dy ≤W(f2).

Let = 0, the equality term is obvious. Thus we complete the proof. 

Proof of Theorem 4.5:By Proposition C.3,W(ϕ)≤W( ˆϕ). If k= 0, then the case 2 in

Appendix A shows that our problem is ill-posed, and the result holds naturally. Hence we just

consider k > 0.

For any feasible solution Q(y) to problem (4.31) with parameter ˆϕ,

x0=Z1

Q(y) ˆϕ0(y)dy ≥kZ1

Q(y)dy.

By taking supreme of the right hand, we get

W( ˆϕ)≤x0

Thus

W(ϕ)≤x0

Now, for any strictly increasing sequence {cn},cn↑1, deﬁne

fn(y) = (ˆϕ(y), y < cn,

−ˆϕ(cn)

1−cn(y−cn) + ˆϕ(cn), y ≥cn.

Then,

fn↑ˆϕ, W (fn)↑W( ˆϕ).

By using the conclusion in the former special case, we have

W(fn) = 1−cn

−ϕ(cn)x0↑x0

with optimal solution

Qn(y) = x0

−ϕ(cn).

Hence we get

W( ˆϕ) = x0

(1) If c= 1, then ˆϕis not aﬃne on the neighborhood of point 1, then there exists a δ > 0 such

that ˆϕ(y) = ϕ(y),∀y∈(1 −δ, 1). Consequently, there exists a N∈N∗, such that cn>1−δfor

any n > N, which indicates that Qnis also a feasible solution to problem (4.31) with parameter

ϕ. So we must have

W( ˆϕ) = lim

n→+∞W(fn)≤W(ϕ),

which leads to

W(ϕ) = x0

Now, we will show that this value is unattainable.

For any feasible Qto the initial problem (4.31), there must exist an a∈(1 −δ, 1) with

Q(a) = b > 0,Ra

0Q(y)dy =: I1> 1and ˆϕ0(a)> k +2. Let I2:= R1

aQ(y)dy, then

x0=Z1

Q(y)ϕ0(y)dy ≥Z1

Q(y) ˆϕ0(y)dy

=Za

Q(y) ˆϕ0(y)dy +Z1

Q(y) ˆϕ0(y)dy

≥ˆϕ0(a)I1+kI2.

Hence, we have

k≥I2+ˆϕ0(a)

kI1=I1+I2+ˆϕ0(a)−k

kI1>Z1

Q(y)dy +12

which implies that the maximum x0

kcannot be attained.

(2) c < 1, then ˆϕ(c) = ϕ(c), and

Q∗(y) = x0

−ϕ(c)1{y≥c}

is an optimal solution which can reach its maximum x0

k. And if ˆϕ(y)> ϕ(y),∀y∈(c, 1), by the

conclusion in case (4.34), the optimal solution is unique. This completes the proof. 

D Proofs in Section 5

D.1 Proof of Theorem 5.6

Proof.

(1) k0= 0,¯

M < +∞. For any λ > 0, ( ˆ

U0)−1(λˆϕ0(x)+) <¯

Mwhich indicates f(λ+) <+∞

for any λ > 0, we then get λ0= 0 immediately. Now, let λ= 0, then ( ˆ

U0)−1(λˆϕ0(x)+) = ¯

Mfor

every x∈[0,1], which implies f(λ0+) = ¯

ME[ρ].

The conditions on k0,¯

Mare equal to U(x) = U(¯

M),for every x∈[¯

M, +∞), which shows that

VA(x0)≤U(¯

M).

For any feasible Qwith Q(x)≥¯

Mand R1

0U(Q(x))dx =U(¯

M), Qis optimal to Problem A.

(2) k0= 0,¯

M= +∞. If l0= 0 and ¯c < 1, then, for any λ > 0, λˆϕ0(x)=0,∀x∈[¯c, 1]. As a

result,

(ˆ

U0)−1(λˆϕ0(x)+) = +∞,∀x∈[¯c, 1],

and

f(λ+)≥Z1

¯c

(ˆ

U0)−1(λˆϕ0(x)+)dx = +∞.

Thus, λ0= +∞, and Problem A is ill-posed.

If l0= 0 and ¯c= 1, then the exact value of λ0depends on the precise expressions of ˆ

Uand ˆϕ.

But the strong duality does not hold.

If l0>0, for any λ > 0, λˆϕ0(x)≥λl0, then

(ˆ

U0)−1(λˆϕ0(x)+) ≤(ˆ

U0)−1(λl0)<+∞.

Consequently,

f(λ+) ≤(ˆ

U0)−1(λl0)E[ρ]<+∞

for all λ > 0, which implies λ0= 0. Moreover, the conditions k0= 0 and ¯

M= +∞means

that, for any large n, there exist a δ > 0, such that ( ˆ

U0)−1(x)≥nif x∈(0, δ). Similarly,

there exists 0 < δ1<1 such that ˆϕ0(x)< l0+ 1 on (δ1,1). Now, if we choose λ < δ

l0+1 , we get

f(λ+) ≥(1 −δ1)n, hence we have f(0+) = +∞.

(3) k0>0, l0= 0. For any λ > 0, there exist 0 < δ < 1 such that ˆϕ0(x)<k0

λfor x∈(δ, 1),

hence

f(λ+) ≥Z1

(ˆ

U0)−1(k0−)dx = +∞.

As a result, λ0= +∞, and Problem A is ill-posed.

(4) k0>0, l0>0, it is not hard to get λ0=k0

l0.

If ¯

M= +∞, and f(λ0+) <+∞, then there is no Q∗

λ∈ YBwhich solves the constraint (5.6),

thus the strong duality does not hold.

If ¯

M < +∞and ¯c= 1, then f(λ0+) ≤¯

ME[ρ]. For any x0> f (λ0+), there is no Q∗

λ∈ YB

solves the constraint (5.6), hence the strong duality does not hold.

If ¯

M < +∞and ¯c < 1, let

Q∗(x) = ((ˆ

U0)−1(λ0ˆϕ0(x)+), x < ¯c,

M+x0−f(λ0+)

−ϕ(¯c), x ≥¯c,

then Q∗(x) solves the constraint (5.6). Moreover, if l0∈ L1, such Q∗can be determined uniquely.

This completes the proof. 

D.2 Proof of Theorem 5.7

Proof. The proof can be divided into separate discussions for the following many cases that

exhaust all possibilities:

Case 1: If λ /∈Λ, ﬁrstly, we want to show that f(λ) is continuous at λ. Obviously, we can

ﬁnd the explicit left or right limit of Q∗

λ. In fact, for sequence 1/n ↓0, Q∗

λ+1/n ↑(ˆ

U0)−1(λˆϕ0(·)+)

and Q∗

λ−1/n ↓(ˆ

U0)−1(λˆϕ0(·)−). Thus, by using the monotone convergence theorem, we have

lim

n→+∞f(λ+ 1/n) = f(λ+),

lim

n→+∞f(λ−1/n) = f(λ−).

Now, ( ˆ

U0)−1is jumped on K, which indicates that ( ˆ

U0)−1(λˆϕ0(·)−)>(ˆ

U0)−1(λˆϕ0(·)+) on {x:

λˆϕ0(x)∈ K},λ /∈Λ, and that {x:λˆϕ0(x)∈ K} is a countable set with measure 0. Hence we get

f(λ−)−f(λ+) = 0,

which implies that f(λ) is continuous at λ. Since Q∗

λis uniquely determined if λ /∈Λ, we get

that λ /∈Λ is equivalent to f(λ)/∈Sλ∈Λ0[f(λ+), f (λ−)].

Thus, for any x0/∈Sλ∈Λ[f(λ+), f (λ−)], we can ﬁnd a unique λ∗, so that f(λ∗) = x0. Therefore,

Q∗

λ∗is the unique optimal solution to problem A.

Case 2: If λ∈Λ, then, we can ﬁnd the corresponding k∈ K, l ∈ L and (a, b),(α, β). Recalling

(5.1), we can easily obtain:

Q∗

λ−(x) = ((ˆ

U0)−1(λˆϕ0(x)−), x /∈[α, β),

b, x ∈[α, β),

Q∗

λ+(x) = ((ˆ

U0)−1(λˆϕ0(x)−), x /∈[α, β),

a, x ∈[α, β).

Hence we have

f(λ−)−f(λ+) = Z1

(Q∗

λ−(x)−Q∗

λ+(x))ϕ0(x)dx

=Zβ

(b−a)ϕ0(x)dx

= (b−a)(ϕ(β)−ϕ(α))

>0.

Thus, f(λ) is jumped at λand λ∈Λ is equivalent to x0∈[f(λ+), f (λ−)]. We will discuss this

situation in four cases.

Case I: If k∈ K1, l ∈ L1, i.e. λ∈Λ1, then only when x0=f(λ+) or f(λ−), we can choose

Q∗

λin (5.2) which satisﬁes the constraint in (5.6), while x0∈(f(λ+), f(λ−)), the optimal λ∗is

not exist. Thus, the strong duality (3.10) does not hold.

Case II: If k∈ K1, l ∈ L2, then Q∗

λis given by (5.3), or equivalently,

Q∗

λ=Q∗

λ++ (b−a)1[c,β).

Thus,

Q∗

λ(x)ϕ0(x)dx =f(λ+) + (b−a)(ϕ(β)−ϕ(c)).

Let the right hand to be x0, then we must have

c=ϕ−1ϕ(β)−x0−f(λ+)

b−a.

Only in this situation, the optimal solution Q∗

λsatisﬁes the constraint (5.6). Thus, problem A

admits a unique solution.

Case III: If k∈ K2, l ∈ L1, thus Q∗

λis deﬁned in (5.4), or equivalently,

Q∗

λ(x) = Q∗

λ++ (c−a)1[α,β).

Thus,

Q∗

λ(x)ϕ0(x)dx =f(λ+) + (c−a)(ϕ(β)−ϕ(α)).

Let the right hand be x0, then we can get

c=a+x0−f(λ+)

ϕ(β)−ϕ(α).

Only in this situation Q∗

λsatisﬁes the constraint (5.6) which gives the evidence that Problem

Aadmits a unique solution.

Case IV: If k∈ K2, l ∈ L2i.e. λ∈Λ2, then the optimal solution to problem B is described

in (5.5), or equivalently,

Q∗

λ(x) = Q∗

λ++ (h(x)−a)1[α,β).

Let h(x) deﬁned in (5.5) satisfy:

(h(x)−a)ϕ0(x)dx =x0−f(λ+),

then Q∗

λsatisﬁes the constraint (5.6). This indicates that such Q∗

λis the optimal solution to

Problem A.

It is easy to show, only when x0=f(λ+) or f(λ−), h(x) is unique. Thus, if x0∈(f(λ+), f (λ−)),

then Problem A may admit inﬁnite number of optimal solutions.

In conclusion, combining Case 1 and Case 2, we will get the desired results. This completes

the proof. 

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