A Neyman–Pearson problem with ambiguity and nonlinear pricing

Abstract

We consider a problem of the Neyman–Pearson type arising in the theory of portfolio choice in the presence of probability weighting, such as in markets with Choquet pricing (as in Araujo et al. in Econ Theory 49(1):1–35, 2011; Cerreia-Vioglio et al. in J Econ Theory 157(1):730–762, 2015; Chateauneuf and Cornet in Submodular financial markets with frictions. Working Paper, 2015; Chateauneuf et al. in Math Finance 6(3):323–330, 1996) and ambiguous beliefs about the payoffs of contingent claims (see Gilboa and Marinacci, in: Acemoglu, Arellano, Dekel (eds) Advances in economics and econometrics: theory and applications, tenth world congress of the econometric society, Cambridge University Press, Cambridge, 2013). Specifically, we consider a problem of optimal choice of a contingent claim so as to minimize a non-linear pricing functional (or a distortion risk measure), subject to a minimum expected performance measure (or a minimum expected return or utility), where expectations with respect to distorted probabilities are taken in the sense of Choquet. Such contingent claims are called cost-efficient. We give an analytical characterization of cost-efficient contingent claims under very mild assumptions on the probability weighting functions, thereby extending some of the results of Ghossoub (Math Financ Econ 10(1):87–111, 2016), and we provide examples of some special cases of interest. In particular, we show how a cost-efficient contingent claim exhibits a desirable monotonicity property: It is anti-comonotonic with the random mark-to-market value (or return, etc.) of the underlying financial position, and it is hence a hedge against such variability.

Appendices

Appendix A: Choquet integration and quantile-based risk measures

1.1 The Choquet integral

Consider a probability space $\left( S, \Sigma , P\right)$, and let $B\left( \Sigma \right)$ denote the linear space of all bounded, real-valued, and $\Sigma$-measurable functions on $\left( S,\Sigma \right)$.

Definition A.1

A (normalized) capacity on the measurable space $\left( S, \Sigma \right)$ is a set function $\nu : \Sigma \rightarrow \left[ 0,1\right]$ such that $\nu \left( \varnothing \right) =0$, $\nu \left( S\right) = 1$, and $\nu$ is monotone, that is, for any $A,B \in \Sigma , \ A \subseteq B \Rightarrow \nu \left( A\right) \leqslant \nu \left( B\right)$.

The capacity $\nu$ is said to be:

• (1) supermodular if $\nu \left( A \cup B\right) + \nu \left( A \cap B\right) \geqslant \nu \left( A\right) + \nu \left( B\right)$, for all $A,B \in \Sigma$; and,

• (2) submodular if $\nu \left( A \cup B\right) + \nu \left( A \cap B\right) \leqslant \nu \left( A\right) + \nu \left( B\right)$, for all $A,B \in \Sigma$.

Remark A.2

For instance, if $T : \left[ 0,1\right] \rightarrow \left[ 0,1\right]$ is an increasing function, such that $T(0)=0$ and $T(1)=1$, then the set function $\nu := T \circ P$ is a capacity on $\left( S, \Sigma \right)$ called a distorted probability measure. The function T is usually called a probability distortion. If, moreover, the distortion function T is convex (resp. concave), then the capacity $\nu = T \circ P$ is supermodular (resp. submodular) ([9, p. 287] or [14, Ex. 2.1]).

Definition A.3

A capacity $\nu$ on $\left( S,\Sigma \right)$ is said to be

• (1) Continuous from above if for any sequence $\{A_{n}\}_{n \geqslant 1}\subseteq \Sigma$ such that $A_{n+1}\subseteq A_{n}$ for each n,

  $$\begin{aligned} \underset{n \rightarrow +\infty }{\lim }\nu \left( A_{n}\right) = \nu \left( \bigcap _{n = 1}^{+\infty } A_{n}\right) ; \end{aligned}$$

• (2) Continuous from below if for any sequence $\{A_{n}\}_{n \geqslant 1}\subseteq \Sigma$ such that $A_{n+1}\supseteq A_{n}$ for each n,

  $$\begin{aligned} \underset{n \rightarrow +\infty }{\lim }\nu \left( A_{n}\right) = \nu \left( \bigcup _{n = 1}^{+\infty } A_{n}\right) ; \end{aligned}$$

• (3) Continuous if it is continuous both from above and below.

For instance, if $\nu$ is a distorted probability measure of the form $T \circ P$ where T is a continuous function, then $\nu$ is a continuous capacity.

Definition A.4

Let $\nu$ be a capacity on $\left( S, \Sigma \right)$. The Choquet integral of $Y \in B\left( \Sigma \right)$ with respect to $\nu$ is defined by

$$\begin{aligned} \int Y \ d\nu := \int _{0}^{+\infty } \nu \left( \{ s \in S: Y\left( s\right) \geqslant t \}\right) \ dt + \int _{-\infty }^{0} \left[ \nu \left( \{ s \in S: Y\left( s\right) \geqslant t \}\right) - 1\right] \ dt, \end{aligned}$$

where the integrals are taken in the sense of Riemann.

Definition A.5

Two functions $Y_{1},Y_{2} \in B\left( \Sigma \right)$ are said to be comonotonic (resp. anti-comonotonic) if

$$\begin{aligned} \Big [ Y_{1}\left( s\right) - Y_{1}\left( s^{\prime }\right) \Big ] \Big [ Y_{2}\left( s\right) - Y_{2}\left( s^{\prime }\right) \Big ] \geqslant 0 \ (\hbox {resp.} \leqslant 0), \hbox { for all } s, s^{\prime } \in S. \end{aligned}$$

Note that $Y_{1},Y_{2} \in B\left( \Sigma \right)$ are anti-comonotonic if and only if $Y_{1}$ and $-Y_{2}$ are comonotonic. Moreover, any $Y \in B\left( \Sigma \right)$ is both comonotonic and anti-comonotonic with any $c \in \mathbb {R}$. Also, if $Y_{1},Y_{2} \in B\left( \Sigma \right)$, and if $Y_{2}$ is of the form $Y_{2} = I \circ Y_{1}$, for some Borel-measurable function I, then $Y_{2}$ is comonotonic (resp. anti-comonotonic) with $Y_{1}$ if and only if the function I is nondecreasing (resp. nonincreasing).

Intuitively, two functions that are anti-comonotonic are a hedge against each other. The following result formalizes this fact.

Proposition A.6

(Prop. 4.5 of Denneberg [14]) Two functions $Y_{1},Y_{2}: S \rightarrow \mathbb {R}$ are anti-comonotonic if and only if there is a function $Z: S \rightarrow \mathbb {R}$, a non-decreasing function $u: \mathbb {R} \rightarrow \mathbb {R}$, and a nonincreasing function $v: \mathbb {R} \rightarrow \mathbb {R}$ such that

$$\begin{aligned} Y_{1} = u \circ Z \ \ \hbox {and} \ \ Y_{2} = v \circ Z. \end{aligned}$$

The Choquet integral with respect to a (countably additive) measure is the usual Lebesgue integral with respect to that measure [31, p. 59]. Unlike the Lebesgue integral, the Choquet integral is not an additive operator on $B\left( \Sigma \right)$. However, the Choquet integral is additive over comonotonic functions.

Proposition A.7

Let $\nu$ be a capacity on $\left( S, \Sigma \right)$.

• (1) Comonotonic Additivity: If $\phi _{1}, \phi _{2} \in B\left( \Sigma \right)$ are comonotonic, then $\int \left( \phi _{1} + \phi _{2}\right) \ d\nu = \int \phi _{1} \ d\nu + \int \phi _{2} \ d\nu$;

• (2) Monotonicity: If $\phi _{1}, \phi _{2} \in B\left( \Sigma \right)$ are such that $\phi _{1} \leqslant \phi _{2}$, then $\int \phi _{1} \ d\nu \leqslant \int \phi _{2} \ d\nu$;

• (3) Positive homogeneity: For all $\phi \in B\left( \Sigma \right)$ and all $c \geqslant 0$, $\int c\phi \ d\nu = c\int \phi \ d\nu$;

• (4) If $\nu$ is submodular, then for any $\phi _{1}, \phi _{2} \in B\left( \Sigma \right)$, $\int \left( \phi _{1} + \phi _{2}\right) \ d\nu \leqslant \int \phi _{1} \ d\nu + \int \phi _{2} \ d\nu$;

• (5) If $\nu$ is supermodular, then for any $\phi _{1}, \phi _{2} \in B\left( \Sigma \right)$, $\int \left( \phi _{1} + \phi _{2}\right) \ d\nu \geqslant \int \phi _{1} \ d\nu + \int \phi _{2} \ d\nu$.

We refer to Marinacci and Montrucchio [31] or Denneberg [14] for more about capacities and Choquet integrals.

1.2 Strongly diffuse capacities

If $\nu$ is a capacity on $\left( S, \Sigma \right)$ and X a random variable on $\left( S, \Sigma \right)$, then the set function $\nu \circ X^{-1}$ defined on the Borel $\sigma$-algebra on the range of X is a capacity.

Definition A.8

(Ghossoub [19]) Let $\nu$ be a capacity on $\left( S, \Sigma \right)$ and let X be a random variable on $\left( S, \Sigma \right)$. Then the capacity $\nu \circ X^{-1}$ is said to be:

• (1) Diffuse if $\nu \circ X^{-1} \Big (\{t\}\Big ) = 0$, for all $t \in \mathbb {R}$;

• (2) Strongly diffuse if $\nu \circ X^{-1}\Big (\left( a,b\right) \Big ) = \nu \circ X^{-1}\Big (\left[ a,b\right] \Big )$, for all $a,b \in \mathbb {R}$ such that $a \leqslant b$.

If $\nu \circ X^{-1}$ is strongly diffuse, we will also say that $\nu$ is strongly diffuse with respect to X (or simply, strongly diffuse if the context is clear). Strong diffuseness implies diffuseness. For capacities that are distortions of a probability measure, we have the following stronger result.

Proposition A.9

(Ghossoub [19]) Let $\nu$ be a capacity on $\left( S, \Sigma \right)$ and let X be a random variable on $\left( S, \Sigma \right)$, and suppose that $\nu$ is a distorted probability measure of the form $\nu = T \circ P$, for some probability measure P on $\left( S, \Sigma \right)$ and some distortion function $T: \left[ 0,1\right] \rightarrow \left[ 0,1\right]$, strictly increasing with $T\left( 0\right) = 0$ and $T\left( 1\right) = 1$. Then the following are equivalent.

• (1) $\nu \circ X^{-1}$ is strongly diffuse;

• (2) $\nu \circ X^{-1}$ is diffuse; and,

• (3) $P \circ X^{-1}$ is diffuse (i.e., nonatomic).

Definition A.10

Let $\nu$ be a capacity on the measurable space $\left( S, \Sigma \right)$ and let $\phi \in B\left( \Sigma \right)$. Define the upper-distribution of $\phi$ with respect to $\nu$ as the function

$$\begin{aligned} \begin{aligned} G_{\nu ,\phi }:&\ \mathbb {R} \rightarrow \left[ 0,1\right] \\&\ t \mapsto G_{\nu ,\phi }\left( t\right) := \nu \big ( \{s \in S: \phi \left( s\right) > t\} \big ). \end{aligned} \end{aligned}$$

If $\phi _{1}, \phi _{2} \in B\left( \Sigma \right)$, we write $\phi _{1} \overset{\nu }{\sim }\phi _{2}$ to mean that $\phi _{1}$ and $\phi _{2}$ have the same upper-distribution with respect to $\nu$. Then a map** $V: B\left( \Sigma \right) \rightarrow \mathbb {R}$ is said to be $\nu$-upper-law-invariant if for any $\phi _{1}, \phi _{2} \in B\left( \Sigma \right)$,

$$\begin{aligned} \phi _{1} \overset{\nu }{\sim }\phi _{2} \ \Longrightarrow \ V\left( \phi _{1}\right) = V\left( \phi _{2}\right) . \end{aligned}$$

The Choquet integral is an example of a $\nu$-upper-law-invariant functional on $B\left( \Sigma \right)$. Note that $G_{\nu ,\psi }$ is nonincreasing, and if $\nu$ is continuous from below then $G_{\nu ,\psi }$ is right-continuous [14, p. 46]. Moreover, if $\nu = T \circ P$, for some probability measure P on $\left( S, \Sigma \right)$ and some distortion function $T : \left[ 0,1\right] \rightarrow \left[ 0,1\right]$, then for any $\phi _{1}, \phi _{2} \in B\left( \Sigma \right)$, if $\phi _{1}$ and $\phi _{2}$ are identically distributed⁹ according to P, then they have the same upper-distribution with respect to $\nu$. Finally, if $\nu$ is a bone fide additive measure, then two functions have the same upper-distribution with respect to $\nu$ if and only if they are identically distributed according to $\nu$.

1.3 Robust representations of the Choquet integral

Let $ba\left( \Sigma \right)$ denote the linear space of all bounded finitely additive set functions on $\left( S,\Sigma \right)$, endowed with the usual mixing operations. When endowed with the variation norm $\Vert . \Vert _{v}$, $ba\left( \Sigma \right)$ is a Banach space. By a classical result [15, IV.5.1], $\left( ba\left( \Sigma \right) , \Vert .\Vert _{v}\right)$ is isometrically isomorphic to the norm-dual of the Banach space $\left( B\left( \Sigma \right) ,\Vert .\Vert _{sup} \right)$ via the duality $<\phi , \lambda \>= \int \phi \ d \lambda , \ \forall \lambda \in ba\left( \Sigma \right) , \ \forall \phi \in B\left( \Sigma \right)$. Consequently, we can endow $ba\left( \Sigma \right)$ with the weak$^{*}$ topology $\sigma \left( ba\left( \Sigma \right) , B\left( \Sigma \right) \right)$. If $ca\left( \Sigma \right)$ denotes the collection of all countably additive elements of $ba\left( \Sigma \right)$, then $ca\left( \Sigma \right)$ is a $\Vert .\Vert _{v}$-closed linear subspace of $ba\left( \Sigma \right)$. Hence, $ca\left( \Sigma \right)$ is $\Vert .\Vert _{v}$-complete, i.e., $\left( ca\left( \Sigma \right) , \Vert .\Vert _{v}\right)$ is a Banach space. By a classical result of Huber and Strassen [25] and Schmeidler [37, 38], we have the following representations of the Choquet integral with respect to a given capacity.

Proposition A.11

Let $\nu$ be a capacity on $\left( S, \Sigma \right)$.

• (1) If $\nu$ is supermodular (e.g., a convex distortion of a probability measure), then there exists a non-empty, convex, and weak$^{*}$-compact collection $\Pi \subset ca\left( \Sigma \right)$ of probability measures, called the core of $\nu$ such that for all $Y \in B\left( \Sigma \right)$,

  $$\begin{aligned} \int Y d\nu = \underset{\mu \in \Pi }{\min }\int Y d\mu . \end{aligned}$$

• (2) If $\nu$ is submodular (e.g., a concave distortion of a probability measure), then there exists a non-empty, convex, and weak$^{*}$-compact collection $\mathcal {A} \subset ca\left( \Sigma \right)$ of probability measures, called the anti-core of $\nu$ such that for all $Y \in B\left( \Sigma \right)$,

  $$\begin{aligned}\int Y d\nu = \underset{\mu \in \mathcal {A}}{\max }\int Y d\mu .\end{aligned}$$

1.4 Quantiles and quantile-based risk measures

For $Y \in B\left( \Sigma \right)$, let $F_{Y}\left( t\right) := P\left( \{s \in S : Y\left( s\right) \leqslant t\}\right)$ denote the cumulative distribution function of Y with respect to the probability measure P, and let $F^{-1}_{Y}\left( t\right)$ denote the left-continuous inverse of the $F_{Y}$ (i.e., the quantile function of Y), defined by

$$\begin{aligned} F^{-1}_{Y}\left( t\right) := \inf \Big \{z \in \mathbb {R} \ \Big | \ F_{Y}\left( z\right) \geqslant t\Big \}, \ \forall t \in \left[ 0, 1\right] . \end{aligned}$$

If the space $\left( S, \Sigma , P\right)$ is nonatomic, then there exists a random variable U that is uniformly distributed on $\left( 0,1\right)$ [18, Proposition A.27]. For all $Y \in B\left( \Sigma \right)$ and all distortion functions T, we then have

$$\begin{aligned} \int Y dT \circ P = \int T^{\prime }\left( 1-U\right) F^{-1}_{Y}\left( U\right) dP = \int _{0}^{1} T^{\prime }\left( 1-t\right) F^{-1}_{Y}\left( t\right) dt. \end{aligned}$$

Letting $k\left( t\right) = T^{\prime }\left( 1-t\right)$, it follows that

$$\begin{aligned} \int Y dT \circ P = \int _{0}^{1} k\left( t\right) F^{-1}_{Y}\left( t\right) dt, \end{aligned}$$

(A.1)

and $\int _{0}^{1} k\left( t\right) dt = 1$. Risk measures $\rho : B\left( \Sigma \right) \rightarrow \mathbb {R}$ of the form given in Eq. (A.1) are referred to as quantile-based risk measures (Kusuoka [29]). By Proposition A.11, we obtain the following result.

Proposition A.12

Let $\rho : B\left( \Sigma \right) \rightarrow \mathbb {R}$ be a quantile-based risk measures of the form $\rho \left( Y\right) = \int _{0}^{1} k\left( t\right) F^{-1}_{Y}\left( t\right) dt$, where $\int _{0}^{1} k\left( t\right) dt = 1$.

• (1) If k is increasing, then there exists a non-empty, convex, and weak$^{*}$-compact collection of probability measures $\mathcal {A} \subset ca\left( \Sigma \right)$ such that $\rho$ has the following robust representation:

  $$\begin{aligned} \rho \left( Y\right) = \underset{\mu \in \mathcal {A}}{\max }\int Y d\mu , \hbox { for all } Y \in B\left( \Sigma \right) . \end{aligned}$$

• (2) If k is decreasing, then there exists a non-empty, convex, and weak$^{*}$-compact collection of probability measures $\Pi \subset ca\left( \Sigma \right)$ such that $\rho$ has the following robust representation:

  $$\begin{aligned} \rho \left( Y\right) = \underset{\mu \in \Pi }{\min }\int Y d\mu , \hbox { for all } Y \in B\left( \Sigma \right) . \end{aligned}$$

Remark A.13

Given a quantile-based risk measures of the form $\rho \left( Y\right) = \int _{0}^{1} k\left( t\right) F^{-1}_{Y}\left( t\right) dt$, where $\int _{0}^{1} k\left( t\right) dt = 1$, defining an absolutely continuous function $T: \left[ 0,1\right] \rightarrow \left[ 0,1\right]$ by $T^{\prime }\left( 1-t\right) = k\left( t\right)$ yields a representation of quantile-based risk measures as Choquet integrals with respect to the distorted probability measure $T \circ P$.

Appendix B: Proof of Theorem 3.1

1.1 Quantile re-formulation

For $Y \in B\left( \Sigma \right)$, let $F_{Y}\left( t\right) := P\{s \in S : Y\left( s\right) \leqslant t\}$ denote the cumulative distribution function of Y with respect to the probability measure P, and let $F^{-1}_{Y}\left( t\right)$ denote the left-continuous inverse of the $F_{Y}$ (i.e., the quantile function of Y), defined by

$$\begin{aligned} F^{-1}_{Y}\left( t\right) := \inf \Big \{z \in \mathbb {R} \ \Big | \ F_{Y}\left( z\right) \geqslant t\Big \}, \ \forall t \in \left[ 0, 1\right] . \end{aligned}$$

Let $U := F_{X}\left( X\right)$. By assumption of nonatomicity of $P \circ X^{-1}$, U is a uniformly distributed random variable on $\left( 0,1\right)$ [18, Lemma A.21]. For all $Y \in B\left( \Sigma \right)$, the fact that $\phi$ is increasing and U is uniformly distributed on $\left( 0,1\right)$ implies that

$$\begin{aligned} \begin{aligned} \int \phi \left( Y\right) d T_{2}\circ P&= \int T_{2}^{\prime }\left( 1-U\right) F^{-1}_{\phi \left( Y\right) }\left( U\right) dP = \int T_{2}^{\prime }\left( 1-U\right) \phi \left( F^{-1}_{Y}\left( U\right) \right) dP \\&= \int _{0}^{1} T_{2}^{\prime }\left( 1-t\right) \phi \left( F^{-1}_{Y}\left( t\right) \right) dt = \int _{0}^{1} T_{2}^{\prime }\left( t\right) \phi \left( F^{-1}_{Y}\left( 1-t\right) \right) dt. \\ \end{aligned} \end{aligned}$$

Moreover,

$$\begin{aligned} \int Y dT_{1} \circ P= & {} \int T_{1}^{\prime }\left( 1-U\right) F^{-1}_{Y}\left( U\right) dP \\= & {} \int _{0}^{1} T_{1}^{\prime }\left( 1-t\right) F^{-1}_{Y}\left( t\right) dt = \int _{0}^{1} T_{1}^{\prime }\left( t\right) F^{-1}_{Y}\left( 1-t\right) dt, \end{aligned}$$

and $0 \leqslant Y \leqslant N$ whenever $0 \leqslant F_{Y}^{-1}\left( t\right) \leqslant N$, for all $t \in \left( 0,1\right) .$

Let $\mathcal {Q}$ denote the collection of all quantile functions, and let $\mathcal {Q}^{*}$ denote the collection of all quantile functions f that satisfy $0 \leqslant f\left( t\right) \leqslant N$, for all $t \in \left( 0,1\right)$. Then

$$\begin{aligned} Q = \Big \{f: \left( 0,1\right) \rightarrow \mathbb {R} \ \Big | \ f \hbox { is nondecreasing and left-continuous}\Big \}, \end{aligned}$$

and

$$\begin{aligned} \mathcal {Q}^{*} = \Big \{ f \in \mathcal {Q}: 0 \leqslant f\left( t\right) \leqslant N, \hbox { for each } 0< t < 1\Big \}. \end{aligned}$$

(B.1)

Consider the following problem:

Problem B.1

For $P_{0}$ as in Assumption 2.4,

$$\begin{aligned} \inf _{f \in \mathcal {Q}^{*}} \left\{ \int _{0}^{1} T_{1}^{\prime }\left( 1-t\right) f\left( t\right) dt : \int _{0}^{1} T_{2}^{\prime }\left( 1-t\right) \phi \left( f\left( t\right) \right) dt \geqslant P_{0} \right\} . \end{aligned}$$

(B.2)

Lemma B.2

If $f^{*}$ is optimal for Problem (B.1), then $Y^{*} := f^{*}\left( 1-F_{X}\left( X\right) \right)$ is optimal for Problem (2.6) and anti-comonotonic with X.

Proof

Let $f^{*}$ be optimal for Problem (B.1). Then, by definition of $\mathcal {Q}^{*}$, $f^{*}$ is the quantile function of some $Z \in B\left( \Sigma \right)$ such that $0 \leqslant Z \leqslant N$. Therefore, since $U := F_{X}\left( X\right)$ is a uniformly distributed random variable on $\left( 0,1\right)$, $Y^{*} = f^{*}\left( 1-U\right) = F_{Z}^{-1}\left( 1-U\right)$ is the nondecreasing equimeasurable rearrangement of Z with respect to X, and hence $0 \leqslant Y^{*} \leqslant N$ and $F_{Y^{*}} = F_{Z}$ (see Ghossoub [19] and references therein). Thus, by law-invariance of the Choquet integral with respect to a distortion of P,

$$\begin{aligned} \int \phi \left( Y^{*}\right) \ dT_{2} \circ P = \int \phi \left( Z\right) \ dT_{2} \circ P= \int _{0}^{1} T_{2}^{\prime }\left( 1-t\right) \phi \left( f\left( t\right) \right) dt \geqslant P_{0}, \end{aligned}$$

where the last inequality follows from the feasibility of $f^{*}$ for Problem (B.1). Hence, $Y^{*}$ is feasible for Problem (2.6).

To show optimality of $Y^{*}$ for Problem (2.6), let Y by any other feasible solution for Problem (2.6) and $F_{Y}^{-1}$ its quantile function. Then $F_{Y}^{-1}$ is feasible for Problem (B.1), and hence

$$\begin{aligned} \int Y d T_{1}\circ P= & {} \int _{0}^{1} T_{1}^{\prime }\left( 1-t\right) F^{-1}_{R}\left( t\right) dt \\\geqslant & {} \int _{0}^{1} T_{1}^{\prime }\left( 1-t\right) f^{*}\left( t\right) dt =\int Z d T_{1}\circ P =\int Y^{*} d T_{1}\circ P. \end{aligned}$$

Therefore, $Y^{*}$ is optimal or Problem (2.6). $\square$

Now, letting $v\left( t\right) = T_{1}^{-1}\left( t\right)$ and using the change of variable $z = v^{-1}\left( t\right)$ gives, for all $f \in \mathcal {Q}^{**}$,

$$\begin{aligned} \int _{0}^{1} T_{1}^{\prime }\left( t\right) f\left( 1-t\right) dt= & {} \int _{0}^{1} f\left( 1-t\right) dT_{1}\left( t\right) \\= & {} \int _{0}^{1} f\left( 1-t\right) dv^{-1}\left( t\right) =\int _{0}^{1} f\left( 1-v\left( z\right) \right) dz =\int _{0}^{1} q\left( t\right) dt, \end{aligned}$$

where $q\left( t\right) := f\left( 1-v\left( t\right) \right)$, for all $t \in \left( 0,1\right)$. Moreover,

$$\begin{aligned} \begin{aligned} \int _{0}^{1} T_{2}^{\prime }\left( t\right) \phi \left( f\left( 1-t\right) \right) dt&=\int _{0}^{1} T_{2}^{\prime }\left( v\left( z\right) \right) \phi \left( f\left( 1-v\left( z\right) \right) \right) dv\left( z\right) \\&=\int _{0}^{1} T_{2}^{\prime }\left( T_{1}^{-1}\left( z\right) \right) \phi \left( q\left( z\right) \right) v^{\prime }\left( z\right) dz\\&=\int _{0}^{1} \frac{T_{2}^{\prime }\left( T_{1}^{-1}\left( t\right) \right) }{T_{1}^{\prime }\left( T_{1}^{-1}\left( t\right) \right) } \phi \left( q\left( t\right) \right) dt. \\ \end{aligned} \end{aligned}$$

Now, define the set $\mathcal {Q}^{**}$ by:

$$\begin{aligned} \mathcal {Q}^{**}:= & {} \Big \{q: \left( 0,1\right) \rightarrow \mathbb {R} \ \Big | \ q \hbox { is nonincreasing } \hbox { and left-continuous, } \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \hbox {and } 0 \leqslant q\left( t\right) \leqslant N, \hbox { for each } 0< t < 1\Big \},\nonumber \\ \end{aligned}$$

(B.3)

and consider the following problem:

Problem B.3

For $P_{0}$ as in Assumption 2.4,

$$\begin{aligned} \inf _{q \in \mathcal {Q}^{**}} \left\{ \int _{0}^{1} q\left( t\right) dt: \int _{0}^{1} \frac{T_{2}^{\prime }\left( T_{1}^{-1}\left( t\right) \right) }{T_{1}^{\prime }\left( T_{1}^{-1}\left( t\right) \right) } \phi \left( q\left( t\right) \right) dt \geqslant P_{0} \right\} . \end{aligned}$$

Lemma B.4

If $q^{*}$ is optimal for Problem (B.3), then the function $f^{*}$ defined by $f^{*}\left( t\right) := q^{*}\left( T_{1}\left( 1-t\right) \right)$ is optimal for Problem (B.1). Moreover, $Y^{*} := f^{*}\left( 1-F_{X}\left( X\right) \right) = q^{*} \left( T_{1} \left( F_{X}\left( X\right) \right) \right)$ is optimal for Problem (2.6) and anti-comonotonic with X.

Proof

Suppose $q^{*}$ is optimal for Problem (B.3), and let $f^{*}\left( t\right) := q^{*}\left( T_{1}\left( 1-t\right) \right)$. Then

• $0 \leqslant f^{*} \leqslant N$ since $q^{*} \in \mathcal {Q}^{**}$;

• By continuity of $T_{1}$ and left-continuity of $q^{*}$, $f^{*}$ is left-continuous;

• Since $q^{*}$ is nonincreasing and $T_{1}$ is increasing, $f^{*}$ is nondecreasing;

• Morevoer,

  $$\begin{aligned} \int _{0}^{1} T_{2}^{\prime }\left( 1-t\right) \phi \left( f^{*}\left( t\right) \right) dt= & {} \int _{0}^{1} T_{2}^{\prime }\left( t\right) \phi \left( f^{*}\left( 1-t\right) \right) dt\\= & {} \int _{0}^{1} \frac{T_{2}^{\prime }\left( T_{1}^{-1}\left( z\right) \right) }{T_{1}^{\prime }\left( T_{1}^{-1}\left( z\right) \right) } \phi \left( f^{*}\left( 1-v\left( z\right) \right) \right) dz\\= & {} \int _{0}^{1} \frac{T_{2}^{\prime }\left( T_{1}^{-1}\left( z\right) \right) }{T_{1}^{\prime }\left( T_{1}^{-1}\left( z\right) \right) } \phi \left( q^{*}\left( T_{1}\left( 1-\left( 1-v\left( z\right) \right) \right) \right) \right) dz\\= & {} \int _{0}^{1} \frac{T_{2}^{\prime }\left( T_{1}^{-1}\left( z\right) \right) }{T_{1}^{\prime }\left( T_{1}^{-1}\left( z\right) \right) } \phi \left( q^{*}\left( T_{1}\left( v\left( z\right) \right) \right) \right) dz\\= & {} \int _{0}^{1} \frac{T_{2}^{\prime }\left( T_{1}^{-1}\left( t\right) \right) }{T_{1}^{\prime }\left( T_{1}^{-1}\left( t\right) \right) } \phi \left( q^{*}\left( t\right) \right) dt \geqslant P_{0}, \end{aligned}$$

where the last equality follows from the feasibility of $q^{*}$ for Problem (B.3).

Therefore, $f^{*}$ is feasible for Problem (B.1). To show optimality of $f^{*}$ for Problem (B.1), let f be any other feasible solution for Problem (B.1) and define q by $q\left( t\right) := f\left( 1-v\left( t\right) \right)$. Then:

• $0 \leqslant q \leqslant N$ since $f^{*} \in \mathcal {Q}^{*}$;

• q is left-continuous and nonincreasing, by the Inverse Function Theorem;

• Moreover,

  $$\begin{aligned} \begin{aligned} \int _{0}^{1} \frac{T_{2}^{\prime }\left( T_{1}^{-1}\left( t\right) \right) }{T_{1}^{\prime }\left( T_{1}^{-1}\left( t\right) \right) } \phi \left( q\left( t\right) \right) dt&=\int _{0}^{1} \frac{T_{2}^{\prime }\left( T_{1}^{-1}\left( t\right) \right) }{T_{1}^{\prime }\left( T_{1}^{-1}\left( t\right) \right) } \phi \left( f\left( 1-v\left( t\right) \right) \right) dt\\&=\int _{0}^{1} T^{\prime }_{2}\left( t\right) \phi \left( f\left( 1-t\right) \right) dt \\&=\int _{0}^{1} T^{\prime }_{2}\left( 1-t\right) \phi \left( f\left( t\right) \right) dt \geqslant P_{0}, \end{aligned} \end{aligned}$$

where the last equality follows from the feasibility of f for Problem (B.1).

Hence, q is feasible for Problem (B.3), and so $\int _{0}^{1} q^{*}\left( t\right) dt \leqslant \int _{0}^{1} q\left( t\right) dt.$ But,

$$\begin{aligned} \begin{aligned} \int _{0}^{1} T^{\prime }\left( 1-t\right) f^{*}\left( t\right) dt&= \int _{0}^{1}T^{\prime }\left( t\right) f^{*}\left( 1-t\right) dt \\&= \int _{0}^{1} f^{*}\left( 1-v\left( t\right) \right) dt = \int _{0}^{1} q^{*}\left( T_{1}\left( v\left( t\right) \right) \right) dt = \int _{0}^{1} q^{*}\left( t\right) dt, \end{aligned} \end{aligned}$$

and $\int _{0}^{1} T^{\prime }\left( 1-t\right) f\left( t\right) dt = \int _{0}^{1} q\left( t\right) dt$. Therefore,

$$\begin{aligned} \int _{0}^{1} T^{\prime }\left( 1-t\right) f^{*}\left( t\right) dt \leqslant \int _{0}^{1} T^{\prime }\left( 1-t\right) f\left( t\right) dt. \end{aligned}$$

Hence, $f^{*}$ is optimal for Problem (B.1), and the rest follows from Lemma B.2. $\square$

1.2 Solving problem (B.3)

In light of Lemma B.4, we turn our attention to solving Problem (B.3). In order to do that, we will use a similar methodology to the one used by Xu [40], but modified and adapted to the present setting.

Now, define the function $\psi : \left[ 0,1\right] \rightarrow \mathbb {R}^{+}$ by

$$\begin{aligned} \psi \left( t\right) := \int _{0}^{t} \frac{T_{2}^{\prime }\left( T_{1}^{-1}\left( x\right) \right) }{T_{1}^{\prime }\left( T_{1}^{-1}\left( x\right) \right) } dx = \int _{0}^{T_{1}^{-1}\left( t\right) } T_{2}^{\prime }\left( x\right) dx = T_{2}\left( T_{1}^{-1}\left( t\right) \right) , \end{aligned}$$

(B.4)

so that $\psi ^{\prime }\left( t\right) = \frac{T_{2}^{\prime }\left( T_{1}^{-1}\left( t\right) \right) }{T_{1}^{\prime }\left( T_{1}^{-1}\left( t\right) \right) }$.

Lemma B.5

Let $\delta$ be the concave envelope of the function $\psi$ on $\left[ 0,1\right]$. Then for any $q \in \mathcal {Q}^{**}$,

$$\begin{aligned} \int _{0}^{1} \left( \phi \circ q\right) \left( t\right) \psi ^{\prime }\left( t\right) dt \leqslant \int _{0}^{1} \left( \phi \circ q\right) \left( t\right) \delta ^{\prime }\left( t\right) dt. \end{aligned}$$

Proof

Let $\delta$ be the concave envelope of the function $\psi$ on $\left[ 0,1\right]$. Since $\phi$ is increasing and each $q \in \mathcal {Q}^{**}$ is nonincreasing, and since $\delta \left( t\right) \geqslant \psi \left( t\right)$, for all $t \in \left[ 0,1\right]$, it follows that¹⁰ for all $q \in \mathcal {Q}^{**}$,

$$\begin{aligned} \int _{0}^{1} \left[ \delta \left( y\right) - \psi \left( y\right) \right] d \phi \circ q\left( y\right) \leqslant 0. \end{aligned}$$

Therefore, since $\psi \left( 0\right) = \delta \left( 0\right)$ and $\psi \left( 1\right) = \delta \left( 1\right)$, Fubini’s Theorem¹¹ gives

$$\begin{aligned} \begin{aligned} 0&\leqslant \int _{0}^{1} \left[ \left( \delta \left( 1\right) - \psi \left( 1\right) \right) - \left( \delta \left( y\right) - \psi \left( y\right) \right) \right] d \phi \circ q\left( y\right) \\&= \int _{0}^{1} \int _{y}^{1} \left[ \delta ^{\prime }\left( x\right) - \psi ^{\prime }\left( x\right) \right] dx \ d \phi \circ q\left( y\right) \\&= \int _{0}^{1} \int _{0}^{x} \left[ \delta ^{\prime }\left( x\right) - \psi ^{\prime }\left( x\right) \right] d \phi \circ q\left( y\right) \ dx = \int _{0}^{1} \left[ \int _{0}^{x} d \phi \circ q\left( y\right) \right] \left[ \delta ^{\prime }\left( x\right) - \psi ^{\prime }\left( x\right) \right] dx \\&= \int _{0}^{1} \left[ \left( \phi \circ q\right) \left( x\right) - \left( \phi \circ q\right) \left( 0\right) \right] \left[ \delta ^{\prime }\left( x\right) - \psi ^{\prime }\left( x\right) \right] dx \\&= \int _{0}^{1} \delta ^{\prime }\left( x\right) \left( \phi \circ q\right) \left( x\right) dx - \int _{0}^{1} \psi ^{\prime }\left( x\right) \left( \phi \circ q\right) \left( x\right) dx - \left( \phi \circ q\right) \left( 0\right) \left[ \delta \left( 1\right) - \delta \left( 0\right) \right] \\&\quad + \left( \phi \circ q\right) \left( 0\right) \left[ \psi \left( 1\right) - \psi \left( 0\right) \right] \\&= \int _{0}^{1} \delta ^{\prime }\left( x\right) \left( \phi \circ q\right) \left( x\right) dx - \int _{0}^{1} \psi ^{\prime }\left( x\right) \left( \phi \circ q\right) \left( x\right) dx - \left( \phi \circ q\right) \left( 0\right) \left[ \psi \left( 1\right) - \psi \left( 0\right) \right] \\&\quad + \left( \phi \circ q\right) \left( 0\right) \left[ \psi \left( 1\right) - \psi \left( 0\right) \right] \\&= \int _{0}^{1} \delta ^{\prime }\left( x\right) \left( \phi \circ q\right) \left( x\right) dx - \int _{0}^{1} \psi ^{\prime }\left( x\right) \left( \phi \circ q\right) \left( x\right) dx. \end{aligned} \end{aligned}$$

That is, $\int _{0}^{1} \left( \phi \circ q\right) \left( t\right) \psi ^{\prime }\left( t\right) dt \leqslant \int _{0}^{1} \left( \phi \circ q\right) \left( t\right) \delta ^{\prime }\left( t\right) dt$. $\square$

Now consider the following problem:

Problem B.6

For $P_{0}$ as in Assumption 2.4,

$$\begin{aligned} \inf _{q \in \mathcal {Q}^{**}} \left\{ \int _{0}^{1} q\left( t\right) dt: \int _{0}^{1} \delta ^{\prime }\left( t\right) \phi \left( q\left( t\right) \right) dt \geqslant P_{0} \right\} . \end{aligned}$$

We first solve Problem (B.6) and then show that the solution is also optimal for Problem (B.3).

Lemma B.7

If $q^{*} \in \mathcal {Q}^{**}$ satisfies:

• (1) $\int _{0}^{1} \delta ^{\prime }\left( t\right) \left( \phi \circ q^{*}\right) \left( t\right) dt = P_{0}$; and,

• (2) There exists some $\lambda \geqslant 0$ such that for all $t \in \left( 0,1\right)$,

  $$\begin{aligned} q^{*}\left( t\right) = \underset{0 \leqslant y \leqslant N}{{{\mathrm{\arg \min }}}}\left[ y - \lambda \delta ^{\prime }\left( t\right) \phi \left( y\right) \right] , \end{aligned}$$

Then $q^{*}$ is optimal for Problem (B.6).

Proof

Let $q^{*} \in \mathcal {Q}^{**}$ be such that the two conditions above are satisfied. Then $q^{*}$ is feasible for Problem (B.6). To show optimality, let $q \in \mathcal {Q}^{**}$ be any feasible solution for Problem (B.6). Then, by definition of $q^{*}$, it follws that for each t,

$$\begin{aligned} q^{*}\left( t\right) - q\left( t\right) \leqslant \lambda \delta ^{\prime }\left( t\right) \left[ \phi \left( q^{*}\left( t\right) \right) - \phi \left( q\left( t\right) \right) \right] . \end{aligned}$$

Hence,

$$\begin{aligned} \int _{0}^{1} q^{*}\left( t\right) dt - \int _{0}^{1} q\left( t\right) dt\leqslant & {} \lambda \left[ \int _{0}^{1} \delta ^{\prime }\left( t\right) \phi \left( q^{*}\left( t\right) \right) dt - \int _{0}^{1} \delta ^{\prime }\left( t\right) \phi \left( q\left( t\right) \right) dt\right] \\= & {} \lambda \left( P_{0} - \int _{0}^{1} \delta ^{\prime }\left( t\right) \phi \left( q\left( t\right) \right) dt\right) \leqslant 0. \end{aligned}$$

Therefore, $\int _{0}^{1} q^{*}\left( t\right) dt \leqslant \int _{0}^{1} q\left( t\right) dt$. $\square$

Lemma B.8

For each $\lambda > 0$, define the function $q^{*}_{\lambda }$ by

$$\begin{aligned} q^{*}_{\lambda }\left( t\right) := \min \left[ N, \max \left( 0, \left( \phi ^{\prime }\right) ^{-1}\left( \frac{1}{\lambda \delta ^{\prime }\left( t\right) }\right) \right) \right] . \end{aligned}$$

(B.5)

Then:

• (1) For each $\lambda > 0$, $q^{*}_{\lambda } \in \mathcal {Q}^{**}$;

• (2) There exists $\lambda ^{*} \geqslant 0$ such that $\int _{0}^{1} \delta ^{\prime }\left( t\right) \left( \phi \circ q^{*}_{\lambda ^{*}}\right) \left( t\right) dt = P_{0}$; and

• (3) For all $t \in \left( 0,1\right)$, $q^{*}_{\lambda ^{*}}\left( t\right) = \underset{0 \leqslant y \leqslant N}{{{\mathrm{\arg \min }}}}\left[ y - \lambda ^{*} \delta ^{\prime }\left( t\right) \phi \left( y\right) \right] .$

Proof

Follows from convexity of $\phi$ and the monotonicity and continuity properties of $\phi$ and $\delta ^{\prime }$, as well as from Assumption 2.4 and the Intermediate Value Theorem. $\square$

Therefore, lemmata B.5, B.7, and B.8 imply that for any $\lambda > 0$ and any $q \in \mathcal {Q}^{**}$,

$$\begin{aligned} \begin{aligned} \int _{0}^{1}\left[ q\left( t\right) - \lambda \psi ^{\prime }\left( t\right) \phi \left( q\left( t\right) \right) \right] dt&= \int _{0}^{1}q\left( t\right) dt - \lambda \int _{0}^{1} \psi ^{\prime }\left( t\right) \phi \left( q\left( t\right) \right) dt\\&\geqslant \int _{0}^{1}q\left( t\right) dt - \lambda \int _{0}^{1} \delta ^{\prime }\left( t\right) \phi \left( q\left( t\right) \right) dt \\&= \int _{0}^{1}\left[ q\left( t\right) dt - \lambda \delta ^{\prime }\left( t\right) \phi \left( q\left( t\right) \right) \right] dt \\&\geqslant \int _{0}^{1}\left[ q^{*}_{\lambda }\left( t\right) dt - \lambda \delta ^{\prime }\left( t\right) \phi \left( q^{*}_{\lambda }\left( t\right) \right) \right] dt, \end{aligned} \end{aligned}$$

where $q^{*}_{\lambda }$ is as in Eq. (B.5). Now, for all $\lambda > 0$, since $q^{*}_{\lambda }$ is monotone, it is differentiable a.e., and we have:

$$\begin{aligned} q^{*}_{\lambda }\left( t\right) = \left\{ \begin{array}{ll} 0 &{} \quad \hbox {if }\ \ \left( \phi ^{\prime }\right) ^{-1}\left( \frac{1}{\lambda \delta ^{\prime }\left( t\right) }\right) \leqslant 0,\\ \left( \phi ^{\prime }\right) ^{-1}\left( \frac{1}{\lambda \delta ^{\prime }\left( t\right) }\right) &{} \quad \hbox {if }\ \ 0< \left( \phi ^{\prime }\right) ^{-1}\left( \frac{1}{\lambda \delta ^{\prime }\left( t\right) }\right) < N,\\ N &{} \quad \hbox {if }\ \ \left( \phi ^{\prime }\right) ^{-1}\left( \frac{1}{\lambda \delta ^{\prime }\left( t\right) }\right) \geqslant N,\\ \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} dq^{*}_{\lambda }\left( t\right) = \left\{ \begin{array}{l l} 0 &{} \quad \hbox {if }\ \ \left( \phi ^{\prime }\right) ^{-1}\left( \frac{1}{\lambda \delta ^{\prime }\left( t\right) }\right) \leqslant 0,\\ \xi _{\lambda }\left( t\right) d\delta ^{\prime }\left( t\right) &{} \quad \hbox {if }\ \ 0< \left( \phi ^{\prime }\right) ^{-1}\left( \frac{1}{\lambda \delta ^{\prime }\left( t\right) }\right) < N,\\ 0 &{} \quad \hbox {if }\ \ \left( \phi ^{\prime }\right) ^{-1}\left( \frac{1}{\lambda \delta ^{\prime }\left( t\right) }\right) \geqslant N,\\ \end{array} \right. \end{aligned}$$

(B.6)

where $\xi _{\lambda }\left( t\right) := \frac{- \left( \left( \phi ^{\prime }\right) ^{-1}\right) ^{\prime }\left( \frac{1}{\lambda \delta ^{\prime }\left( t\right) }\right) }{\lambda \left( \delta ^{\prime }\right) ^{2}\left( t\right) }$.

Now, define the subsets $\mathcal {A}$ and $\mathcal {B}$ of $\left[ 0,1\right]$ by:

$$\begin{aligned} \mathcal {A}:= & {} \left\{ t \in \left[ 0,1\right] : \delta \left( t\right) = \psi \left( t\right) \right\} \ \hbox {and }\\ \mathcal {B}:= & {} \left\{ t \in \left[ 0,1\right] : \delta \left( t\right) \ne \psi \left( t\right) \right\} = \left\{ t \in \left[ 0,1\right] : \delta \left( t\right) > \psi \left( t\right) \right\} . \end{aligned}$$

Then for any $\lambda > 0$,

$$\begin{aligned} \begin{aligned} \int _{0}^{1} \left[ \delta \left( t\right) - \psi \left( t\right) \right] d \phi \circ q^{*}_{\lambda }\left( t\right)&= \int _{0}^{1} \phi ^{\prime }\left( q^{*}_{\lambda }\left( t\right) \right) \left[ \delta \left( t\right) - \psi \left( t\right) \right] dq^{*}_{\lambda }\left( t\right) \\&= \int _{\mathcal {A}} \phi ^{\prime }\left( q^{*}_{\lambda }\left( t\right) \right) \left[ \delta \left( t\right) - \psi \left( t\right) \right] dq^{*}_{\lambda }\left( t\right) \\&\quad + \int _{\mathcal {B}} \phi ^{\prime }\left( q^{*}_{\lambda }\left( t\right) \right) \left[ \delta \left( t\right) - \psi \left( t\right) \right] dq^{*}_{\lambda }\left( t\right) \\&= \int _{\mathcal {B}} \phi ^{\prime }\left( q^{*}_{\lambda }\left( t\right) \right) \left[ \delta \left( t\right) - \psi \left( t\right) \right] dq^{*}_{\lambda }\left( t\right) . \\ \end{aligned} \end{aligned}$$

But, since $\delta$ is affine on $\mathcal {B}$, it follows from Eq. (B.6) that $dq^{*}_{\lambda }\left( t\right) = 0$ on $\mathcal {B}$. Consequently,

$$\begin{aligned} \int _{0}^{1} \left[ \delta \left( t\right) - \psi \left( t\right) \right] d \phi \circ q^{*}_{\lambda }\left( t\right) = 0. \end{aligned}$$

Therefore, applying Fubini’s theorem, as in the proof of Lemma B.5, gives

$$\begin{aligned} \begin{aligned} 0&=\int _{0}^{1} \left[ \delta \left( t\right) - \psi \left( t\right) \right] d \phi \circ q^{*}_{\lambda }\left( t\right) = \int _{0}^{1} \left( \phi \circ q^{*}_{\lambda }\right) ^{\prime }\left( t\right) \left[ \delta \left( t\right) - \psi \left( t\right) \right] dt \\&= - \int _{0}^{1} \left( \phi \circ q^{*}_{\lambda }\right) ^{\prime }\left( t\right) \left[ \left( \delta \left( 1\right) - \psi \left( 1\right) \right) - \left( \delta \left( t\right) - \psi \left( t\right) \right) \right] dt, \end{aligned} \end{aligned}$$

and hence

$$\begin{aligned} 0= & {} \int _{0}^{1} \left( \phi \circ q^{*}_{\lambda }\right) ^{\prime }\left( t\right) \left[ \int _{t}^{1}\left( \delta ^{\prime }\left( x\right) - \psi ^{\prime }\left( x\right) \right) dx\right] dt \\= & {} \int _{0}^{1} \int _{t}^{1} \left( \phi \circ q^{*}_{\lambda }\right) ^{\prime }\left( t\right) \left[ \delta ^{\prime }\left( x\right) - \psi ^{\prime }\left( x\right) \right] dx \ dt\\= & {} \int _{0}^{1} \int _{0}^{x} \left( \phi \circ q^{*}_{\lambda }\right) ^{\prime } \left( t\right) \left[ \delta ^{\prime }\left( x\right) - \psi ^{\prime }\left( x\right) \right] dt \ dx \\= & {} \int _{0}^{1} \left[ \int _{0}^{x} \left( \phi \circ q^{*}_{\lambda }\right) ^{\prime } \left( t\right) dt\right] \left[ \delta ^{\prime }\left( x\right) - \psi ^{\prime }\left( x\right) \right] dx\\= & {} \int _{0}^{1} \left[ \left( \phi \circ q^{*}_{\lambda }\right) \left( x\right) - \left( \phi \circ q^{*}_{\lambda }\right) \left( 0\right) \right] \left[ \delta ^{\prime }\left( x\right) - \psi ^{\prime }\left( x\right) \right] dx \\= & {} \int _{0}^{1} \delta ^{\prime }\left( x\right) \left( \phi \circ q^{*}_{\lambda }\right) \left( x\right) dx - \int _{0}^{1} \psi ^{\prime }\left( x\right) \left( \phi \circ q^{*}_{\lambda }\right) \left( x\right) dx - \left( \phi \circ q^{*}_{\lambda }\right) \left( 0\right) \left[ \delta \left( 1\right) - \delta \left( 0\right) \right] \\&+ \left( \phi \circ q^{*}_{\lambda }\right) \left( 0\right) \left[ \psi \left( 1\right) - \psi \left( 0\right) \right] \\= & {} \int _{0}^{1} \delta ^{\prime }\left( x\right) \left( \phi \circ q^{*}_{\lambda }\right) \left( x\right) dx - \int _{0}^{1} \psi ^{\prime }\left( x\right) \left( \phi \circ q^{*}_{\lambda }\right) \left( x\right) dx - \left( \phi \circ q^{*}_{\lambda }\right) \left( 0\right) \left[ \psi \left( 1\right) - \psi \left( 0\right) \right] \\&+ \left( \phi \circ q^{*}_{\lambda }\right) \left( 0\right) \left[ \psi \left( 1\right) - \psi \left( 0\right) \right] \\= & {} \int _{0}^{1} \delta ^{\prime }\left( x\right) \left( \phi \circ q^{*}_{\lambda }\right) \left( x\right) dx - \int _{0}^{1} \psi ^{\prime }\left( x\right) \left( \phi \circ q^{*}_{\lambda }\right) \left( x\right) dx. \end{aligned}$$

Consequently, for each $\lambda > 0$,

$$\begin{aligned} \int _{0}^{1} \delta ^{\prime }\left( t\right) \left( \phi \circ q^{*}_{\lambda }\right) \left( t\right) dt = \int _{0}^{1} \psi ^{\prime }\left( t\right) \left( \phi \circ q^{*}_{\lambda }\right) \left( t\right) dt, \end{aligned}$$

and so, for all $q \in \mathcal {Q}^{**}$,

$$\begin{aligned} \begin{aligned} \int _{0}^{1}\left[ q\left( t\right) - \lambda \psi ^{\prime }\left( t\right) \phi \left( q\left( t\right) \right) \right] dt&= \int _{0}^{1}q\left( t\right) dt - \lambda \int _{0}^{1} \psi ^{\prime }\left( t\right) \phi \left( q\left( t\right) \right) dt \\&\geqslant \int _{0}^{1}q\left( t\right) dt - \lambda \int _{0}^{1} \delta ^{\prime }\left( t\right) \phi \left( q\left( t\right) \right) dt\\&= \int _{0}^{1}\left[ q\left( t\right) dt - \lambda \delta ^{\prime }\left( t\right) \phi \left( q\left( t\right) \right) \right] dt \\&\geqslant \int _{0}^{1}\left[ q^{*}_{\lambda }\left( t\right) dt - \lambda \delta ^{\prime }\left( t\right) \phi \left( q^{*}_{\lambda }\left( t\right) \right) \right] dt \\&= \int _{0}^{1}\left[ q^{*}_{\lambda }\left( t\right) dt - \lambda \psi ^{\prime }\left( t\right) \phi \left( q^{*}_{\lambda }\left( t\right) \right) \right] dt. \end{aligned} \end{aligned}$$

Therefore, $q^{*}_{\lambda ^{*}}$ is optimal for Problem (B.3). Lemma B.4 concludes the proof of Theorem 3.1. $\square$