Benchmark Beating with the Increasing Convex Order^††thanks: Supported by the National Key R&D Program of China (2020YFA0712700) and NSFC (12071146).

Portfolio selection involves a trade-off between return and risk. In contrast to the expected utility theory which models return seeking and risk aversion of an investor with the monotonicity and the concavity of the utility function, the mean-variance theory of Markowitz (1952) models the return and the risk of a portfolio with its mean and variance. Thereby the trade-off between the return and the risk is explicitly formulated in Markowitz’s mean-variance theory.

In spite of the popularity of the mean-variance theory in both academia and industry, variance has been frequently criticized as a risk measure for its main shortcoming: it treats the volatile positive returns as a part of risk. While the return is measured by mean, some alternative risk measures have been proposed to replace variance in portfolio selection, e.g., value-at-risk (VaR) (Campbell et al. (2001), Alexander and Baptista (2002)), conditional VaR (Rockafellar and Uryasev (2000)), weighted VaR (He et al. (2015)), entropic VaR (Ahmadi-Javid and Fallah-Tafti (2019)), and expectile (Wagner and Uryasev (2019), Lin et al. (2021)); see He et al. (2015) and Lin et al. (2021) for reviews on the development along this line.

Instead of the risk aversion, we focus on the other side—the return seeking—of portfolio selection in this paper. In the mean-variance theory, the return of a portfolio is measured by its mean, which, unlike variance as a risk measure, has been seldom criticized. Moreover, a mean-variance efficient portfolio can be found out by minimizing the variance under the constraint that the return is no less than a benchmark level, which is modeled by a constant. We consider an extension of the mean constraint such that the benchmark is modeled by a random variable, which is much more flexible in modeling return seeking than a constant. To this end, we use a stochastic order.

Given a random variable $X_{0}$, which is called a benchmark, we say a random variable $X$ beats the benchmark $X_{0}$ and write $X\succeq_{\mathrm{icx}}X_{0}$ if $X$ dominates $X_{0}$ in the increasing convex order (ICX order), i.e.,

$$\mathrm{E}[f(X)]\geq\mathrm{E}[f(X_{0})]$$

(1.1)

for all increasing convex functions $f$ provided that the expectations are well defined. It is well known that $X\succeq_{\mathrm{icx}}X_{0}$ if and only if

$$\int_{t}^{1}Q_{X}(s)ds\geq\int_{t}^{1}Q_{X_{0}}(s)ds,\quad t\in(0,1),$$

(1.2)

where $Q_{X}$ and $Q_{X_{0}}$ are quantile functions of $X$ and $X_{0}$.

The financial meaning of (1.2) is clearer when $X_{0}$ is discretely distributed:

$$\mathbb{P}(X_{0}=a_{i})=p_{i},\quad 1\leq i\leq n,$$

(1.3)

where $n\geq 1$, $a_{1}<a_{2}<\dots<a_{n}$, $p_{i}\in(0,1]$ for all $i=1,\dots,n$, and $\sum_{i=1}^{n}p_{i}=1$. Let

$$q_{0}=0,\quad q_{i}=p_{1}+\dots+p_{i},\quad i=1,\dots,n-1.$$

It is easy to see that both sizes of (1.2) are concave w.r.t. $t$. Moreover, the right hand size of (1.2) is piecewise linear in the special case (1.3). Therefore, (1.2) is equivalent to

$$\int_{q_{i}}^{1}Q_{X}(s)ds\geq\int_{q_{i}}^{1}Q_{X_{0}}ds,\quad i=0,\dots,n-1.$$

(1.4)

In (1.4), the constraint with $i=0$ amounts to $\mathrm{E}[X]\geq\mathrm{E}[X_{0}]$. For each $i=1,\dots,n-1$, $\int_{q_{i}}^{1}Q_{X}(s)ds$ is the average gain of $X$ in the best $(100\times q_{i})\%$ of cases and ignores the values of $X$ in the worst $(100\times(1-q_{i}))\%$ of cases. It is called expected upside, expected gain or tail gain; see Bacon (2013) for example.¹¹1In contrast, the expected shortfall or conditional value at risk with confidence level $\alpha$ is the average loss of $X$ in the worst $100\times\alpha\%$ of cases. The ratio of expected upside and expected shortfall gives the Rachev ratio in finance, see Biglova et al. (2004). Therefore (1.4) consists of a constraint on the mean of $X$ and some other constraints on the expected upsides of $X$. The ICX order imposes not only the usual mean constraint as in the mean-variance theory but also constraints on the right tail. The constraints on the right tail makes the ICX order appropriate for modeling benchmark beating (right-tail reward control), since by beating a benchmark we expect that $X$ is more profitable than the benchmark $X_{0}$.

For a constant $z$, $X\succeq_{\mathrm{icx}}z$ if and only if $\mathrm{E}[X]\geq z$. Therefore, the mean constraint $\mathrm{E}[X]\geq z$ can be regarded as beating constant $z$. The aim of the present paper is to extend the mean constraint in the mean-variance theory to the ICX order constraint. More precisely, we consider the following problem:

$$\operatorname*{minimize\,}_{X\in\mathscr{X}}\mathrm{Var}[X]\quad\text{subject to }X\succeq_{\mathrm{icx}}X_{0},$$

where $\mathscr{X}$ is a feasible set.

As initiating work, we study the problem of portfolio selection with ICX order constraint within a complete market. The original problem is generally non-convex since the set $\{X\mid X\succeq_{\mathrm{icx}}X_{0}\}$ is generally non-convex. But the set $\{Q_{X}\mid X\succeq_{\mathrm{icx}}X_{0}\}$ is always convex. Then by the recently developed quantile formulation (e.g., Schied (2004), Carlier and Dana (2006), Jin and Zhou (2008), and He and Zhou (2011)), we transform the original non-convex problem into a convex one and finally provide the optimal solution in closed form. Moreover, for a payoff $X$, let its performance of beating $X_{0}$ be defined as

$$\psi(X)\triangleq\sup\{m\in\mathbb{R}\mid X-m\succeq_{\mathrm{icx}}X_{0}\}\quad\text{with }\sup\emptyset=-\infty.$$

A payoff $X\in\mathscr{X}$ is called beating-performance-variance (BPV) efficient in $\mathscr{X}$ if there is no $Y\in\mathscr{X}$ such that

$$\psi(Y)\geq\psi(X)\quad\text{and}\quad\mathrm{Var}[Y]\leq\mathrm{Var}[X]$$

with at least one inequality holding strictly. We then also investigate the problem of BPV efficient portfolio and the BPV efficient portfolios are worked out in closed form. In the special case $X_{0}=0$, $\Psi(X)=\mathrm{E}[X]$ and hence BPV efficiency reduces to mean-variance efficiency.

The ICX order is closely related to another stochastic order, the increasing concave order (ICV order), which is also called the second-order stochastic dominance (SSD). A random random variable $X$ dominates $X_{0}$ in the ICV order and write $X\succeq_{\mathrm{icv}}X_{0}$ if (1.1) holds for all increasing concave functions $f$ provided that the expectations are well defined. It is well known that $X\succeq_{\mathrm{icv}}X_{0}$ if and only if

$$\int_{0}^{t}Q_{X}(s)ds\geq\int_{0}^{t}Q_{X_{0}}(s)ds,\quad t\in(0,1).$$

The financial meaning of ICX order is significantly different from that of ICV order. Actually, for a constant $z$, $X\succeq_{\mathrm{icx}}z$ amounts to $\mathrm{E}[X]\geq z$, whereas $X\succeq_{\mathrm{icv}}z$ amounts to $X\geq z$ a.s. Moreover, by Dentcheva and Ruszczyński (2003, Proposition 3.2), for a discretely distributed benchmark $X_{0}$ satisfying (1.3), $X\succeq_{\mathrm{icv}}X_{0}$ if and only if

$$X\geq a_{1}\text{ a.s. and }\mathrm{E}[(a_{i}-X)^{+}]\leq\mathrm{E}[(a_{i}-X_{0})^{+}],\ i=2,\dots,n.$$

(1.5)

The constraints in (1.5) pay attention to the left-tail risk of $X$ measured by $\operatorname*{ess\,inf}X$ and $\mathrm{E}[(a_{i}-X)^{+}$ ($i=2,\dots,n$). Therefore, the ICV order is appropriate for modeling risk control (left-tail risk control). In a series of papers, Dentcheva and Ruszczyński (2003, 2004, 2006a, 2006b) introduced a stochastic optimization model with ICV order constraints. As an application, they also investigated the problem of portfolio selection with risk control by ICV order instead of variance. For further developments along this line, see Rudolf and Ruszczyński (2008), Fábián et al. (2011) and Dentcheva et al. (2016). In particular, Wang and Xia (2021) investigate the problem of expected utility maximization with the ICV order constraint.

The rest of this paper is organized as follows. Sections 2 and 3 introduce the basic model and the quantile formulation, respectively. Section 4 establishes the duality between the primal and the dual problems. Section 5 investigates the dual problem and gives the dual optimizer in closed form. Section 6 studies the primal optimizer. Section 7 presents the optimal solution for the special case of two-point distributed benchmark. Section 8 discusses some problems including BPV efficient portfolios, multi-benchmark beating, and mean-variance portfolio selection with ICX order constraint, The Appendix collects some proofs.

Consider a complete nonatomic probability space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$. Let $L^{2}$ (resp., $L^{\infty}$) denote all squarely integrable (resp., essentially bounded) $\mathcal{F}$-measurable random variables. The (upper) quantile function $Q_{X}$ of a random variable $X$ is defined by

$$Q_{X}(t)\triangleq\inf\{x\in\mathbb{R}\mid\mathbb{P}(X\leq x)>t\},\quad t\in[0,1).$$

By convention, we extend $Q_{X}$ by letting $Q_{X}(1)\triangleq Q_{X}(1-)=\lim_{t\uparrow 1}Q_{X}(t)$ and $Q_{X}(0-)\triangleq 0$ if needed. For more details about quantile functions, see, e.g., Föllmer and Schied (2016, Appendix A.3).

Let $X$ and $Y$ be two random variables. We say that $X$ dominates $Y$ in the increasing convex order (ICX order) and write $X\succeq_{\mathrm{icx}}Y$ if $\mathrm{E}[f(X)]\geq\mathrm{E}[f(Y)]$ for all increasing convex functions $f$ provided that the expectations exist. It is well known that²²2Throughout the paper, for $t_{1}<t_{2}$, we use $\int_{t_{1}}^{t_{2}}$ to denote the integration on the open interval $(t_{1},t_{2})$, that is, $\int_{t_{1}}^{t_{2}}=\int_{(t_{1},t_{2})}$.

	$\displaystyle X\succeq_{\mathrm{icx}}Y\Longleftrightarrow$	$\displaystyle\int_{t}^{1}Q_{X}(s)ds\geq\int_{t}^{1}Q_{Y}(s)ds\text{ for all }t\in[0,1]$		(2.1)
	$\displaystyle\Longleftrightarrow$	$\displaystyle\int_{[0,1]}Q_{X}(s)dw(s)\geq\int_{[0,1]}Q_{Y}(s)dw(s)\text{ for all }w\in\mathscr{W}^{\mathrm{icx}},$

where

$$\mathscr{W}^{\mathrm{icx}}\triangleq\{w:[0,1]\to[0,\infty)\mid w\text{ is increasing and convex with }w(0)=0\}.$$

By convention, we set $w(0-)\triangleq 0$ for $w\in\mathscr{W}^{\mathrm{icx}}$. We can identify each $w\in\mathscr{W}^{\mathrm{icx}}$ as a finite measure on the measurable space $\left([0,1],\mathcal{B}_{[0,1]}\right)$, where $\mathcal{B}_{[0,1]}$ denotes all Borel subsets of $[0,1]$. For each $w\in\mathscr{W}^{\mathrm{icx}}$, the measure of $\{0\}$ is $0$ and the measure of $\{1\}$ is $w(1)-w(1-)$. Furthermore, for each $w\in\mathscr{W}^{\mathrm{icx}}$ and $X\in L^{2}$, by the convexity of $w$ and $Q_{X}\in L^{2}([0,1))$, we know that $\int_{[0,1]}Q_{X}(s)dw(s)=\int_{(0,1]}Q_{X}(s)dw(s)$ is well defined and $\int_{[0,1]}Q_{X}(s)dw(s)>-\infty$. For more details about the increasing convex order, see, e.g., Shaked and Shanthikumar (2007, Chapter 4).

Consider an arbitrage-free market. Assume that the market is complete and has a unique stochastic discount factor (SDF) $\rho\in L^{2}$ with $\mathbb{P}(\rho>0)=1$ and variance $\mathrm{Var}[\rho]>0$.

Let us consider an investor with initial capital $\mathit{x}$, who wants to minimize the risk measured by the payoff’s variance $\mathrm{Var}[X]$ while beating a benchmark payoff $X_{0}\in L^{\infty}$ in the sense of the ICX order. The problem of the investor is thus

$$\begin{split}&\operatorname*{minimize\,}_{X\in L^{2}}\mathrm{Var}[X]\\ &\text{subject to }\mathrm{E}[\rho X]\leq x,\;X\succeq_{\mathrm{icx}}X_{0}.\end{split}$$

(2.2)

Let

$$\mathscr{X}_{\mathrm{icx}}(x,X_{0})\triangleq\{X\in L^{2}\>|\>\mathrm{E}[\rho X]\leq x\text{ and }X\succeq_{\mathrm{icx}}X_{0}\}.$$

A payoff $X$ is called variance-minimal in $\mathscr{X}_{\mathrm{icx}}(x,X_{0})$ if it solves problem (2.2).

From (2.1), the advantage of ICX order is obvious: it pays more attention to the right tail (the gain part) than to the left tail (the loss part) of a random variable. This advantage makes the ICX order appropriate for modeling benchmark beating, since by beating a benchmark we expect that the payoff $X$ has more profit than the benchmark $X_{0}$. The mean, however, pays equal attention to the right and the left tails.

In general, the set $\mathscr{X}_{\mathrm{icx}}(x,X_{0})$ is not convex, which leads to a difficulty of the problem. The appropriate technique for overcoming this difficulty is the well-developed “quantile formulation”: changing the decision variable of the problem from the random variable $X$ to its quantile function $Q_{X}$; see Schied (2004), Carlier and Dana (2006), Jin and Zhou (2008), and He and Zhou (2011). This formulation recovers the implicit convexity (in terms of quantile functions) of $\mathscr{X}_{\mathrm{icx}}(x,X_{0})$.

Let

$$\mathscr{Q}\triangleq\left\{Q:[0,1)\rightarrow\mathbb{R}\>\bigg{|}\>Q\>\text{is increasing, right-continuous and}\>\int_{0}^{1}Q^{2}(s)ds<\infty\right\}.$$

Obviously, $\mathscr{Q}$ is the set of quantile functions of random variables $\mathit{X}\in L^{2}$, that is,

$$\mathscr{Q}=\{Q_{X}\>|\>X\in L^{2}\}.$$

For any $Q_{1},Q_{2}\in\mathscr{Q}$, we write $Q_{1}\succeq_{\mathrm{icx}}Q_{2}$ if

$$\int_{t}^{1}Q_{1}(s)ds\geq\int_{t}^{1}Q_{2}(s)ds\quad\text{ for all }t\in[0,1].$$

Obviously,

$$Q_{1}\succeq_{\mathrm{icx}}Q_{2}\Longleftrightarrow\int_{[0,1]}Q_{1}(s)dw(s)\geq\int_{[0,1]}Q_{2}(s)dw(s)\text{ for all }w\in\mathscr{W}^{\mathrm{icx}}.$$

For any $Q\in\mathscr{Q}$, we have

$\displaystyle\begin{aligned} &Q\succeq_{\mathrm{icx}}Q_{0}\Longleftrightarrow\inf_{w\in\mathscr{W}^{\mathrm{icx}}}\left(\int_{[0,1]}(Q(s)-Q_{0}(s))dw(s)\right)=0,\\ &Q\not\succeq_{\mathrm{icx}}Q_{0}\Longleftrightarrow\inf_{w\in\mathscr{W}^{\mathrm{icx}}}\left(\int_{[0,1]}(Q(s)-Q_{0}(s))dw(s)\right)=-\infty.\end{aligned}$

(3.1)

For notational simplicity, we write $\mathit{Q_{\mathrm{0}}}$ instead of $\mathit{Q_{X_{\mathrm{0}}}}$. Let

$$\mathscr{Q}_{\mathrm{icx}}(x,Q_{0})\triangleq\left\{Q\in\mathscr{Q}\>\Big{|}\>\int_{0}^{1}Q(s)Q_{\rho}(1-s)ds\leq x\>\mathrm{and}\>Q\succeq_{\mathrm{icx}}Q_{0}\right\}.$$

Obviously, the set $\mathscr{Q}_{\mathrm{icx}}(x,Q_{0})$ is convex and closed in $L^{2}([0,1))$.

With abuse of notation, we let

$$\mathrm{E}[Q]\triangleq\int_{0}^{1}Q(s)ds\ \mbox{ and }\mathrm{Var}[Q]\triangleq\int_{0}^{1}Q^{2}(s)ds-\left(\int_{0}^{1}Q(s)ds\right)^{2},\quad Q\in\mathscr{Q}.$$

A quantile function $Q\in\mathscr{Q}_{\mathrm{icx}}(x,Q_{0})$ is called variance-minimal in $\mathscr{Q}_{\mathrm{icx}}(x,Q_{0})$ if it solves the following problem:

$$\begin{split}&\operatorname*{minimize\,}_{Q\in\mathscr{Q}}\mathrm{Var}[Q]\\ &\text{subject to }\int_{0}^{1}Q(s)Q_{\rho}(1-s)ds\leq x,\;Q\succeq_{\mathrm{icx}}Q_{0}.\end{split}$$

(3.2)

Let $v^{\circ}(x)$ denote the minimal value of problem (3.2). Obviously, $v^{\circ}$ is convex.

The next proposition shows that finding variance-minimal payoffs is equivalent to finding variance-minimal quantile functions.

Hereafter, we focus on studying variance-minimal quantile functions.

The following theorem implies that $\mathscr{Q}_{\mathrm{icx}}(x,Q_{0})\neq\emptyset$ for any $x\in\mathbb{R}$.

The following proposition shows that the problem is trivial when $Q_{0}(1)\mathrm{E}[\rho]\leq x$.

Hereafter, we always make the following assumption unless otherwise stated.

The next theorem establishes the existence and uniqueness of the variance-minimal solution.

It is well known that $\mathrm{Var}[Q]$ is convex in $Q$. By Theorem 4.1, there exists some $Q\in\mathscr{Q}$ such that $Q\succeq_{\mathrm{icx}}Q_{0}$ and $\int_{0}^{1}Q(s)Q_{\rho}(1-s)ds<x$. Therefore, the Slater condition holds for the budget constraint in (3.2). Then by the standard results in convex optimization theory (see, e.g., Luenberger (1969, Sections 8.3–8.5)), we have the following proposition.

We now consider, for any fixed $\lambda>0$, the following problem:

$$\begin{split}&\operatorname*{minimize\,}_{Q\in\mathscr{Q}}\quad\mathrm{Var}[Q]+\lambda\int_{0}^{1}Q(s)Q_{\rho}(1-s)ds\\ &\text{subject to }\quad Q\succeq_{\mathrm{icx}}Q_{0},\end{split}$$

(4.1)

which is equivalent to

$$\begin{split}&\operatorname*{minimize\,}_{Q\in\mathscr{Q}}\quad\min_{\beta\in\mathbb{R}}\int_{0}^{1}(Q(s)-\beta)^{2}ds+\lambda\int_{0}^{1}Q(s)Q_{\rho}(1-s)ds\\ &\text{subject to }\quad Q\succeq_{\mathrm{icx}}Q_{0}.\end{split}$$

(4.2)

To solve problem (4.1) or (4.2), we first consider, for any fixed $\lambda>0$ and $\beta\in\mathbb{R}$, the following problem:

$$\begin{split}&\operatorname*{minimize\,}_{Q\in\mathscr{Q}}\quad\int_{0}^{1}(Q(s)-\beta)^{2}ds+\lambda\int_{0}^{1}Q(s)Q_{\rho}(1-s)ds\\ &\text{subject to }\quad Q\succeq_{\mathrm{icx}}Q_{0}.\end{split}$$

(4.3)

Let $v(\beta,\lambda)$ denote the optimal value of problem (4.3). Let

	$\displaystyle L(Q,w;\beta,\lambda)\triangleq$	$\displaystyle\int_{0}^{1}(Q(s)-\beta)^{2}ds+\lambda\int_{0}^{1}Q(s)Q_{\rho}(1-s)ds$
		$\displaystyle-\left(\int_{[0,1]}Q(s)dw(s)-\int_{[0,1]}Q_{0}(s)dw(s)\right),\quad Q\in\mathscr{Q},\;w\in\mathscr{W}^{\mathrm{icx}}.$

In view of (3.1), we know that

$\displaystyle v(\beta,\lambda)=\inf_{Q\in\mathscr{Q}}\sup_{w\in\mathscr{W}^{\mathrm{icx}}}L(Q,w;\beta,\lambda).$

Moreover, we have the following proposition.

Similarly to Lemma B.1(a), problem (4.3) has a unique optimal solution $Q^{*}_{\beta,\lambda}$. Then by Proposition 4.7, there exists some $w^{*}_{\beta,\lambda}\in\mathscr{W}^{\mathrm{icx}}$ such that $(Q^{*}_{\beta,\lambda},w^{*}_{\beta,\lambda})$ is a saddle point of $L(\cdot\;,\cdot\;;\beta,\lambda)$. In this case, $w^{*}_{\beta,\lambda}$ is an optimal solution to the following dual optimization problem:

$$\operatorname*{maximize\,}_{w\in\mathscr{W}^{\mathrm{icx}}}\inf_{Q\in\mathscr{Q}}L(Q,w;\beta,\lambda),$$

(4.4)

which will be discussed in the next section.

Note that each $w\in\mathscr{W}^{\mathrm{icx}}$ is continuous on $[0,1)$ and can be discontinuous at the right end-point $1$. The next lemma shows that the dual optimizer is continuous on $[0,1]$.

The next lemma shows that $w$ solves the dual optimization problem only if $w^{\prime}\in L^{2}([0,1))$.

Let

	$\displaystyle\mathscr{W}^{\mathrm{icx}}_{c,2}\triangleq$	$\displaystyle\{w\in\mathscr{W}^{\mathrm{icx}}\mid w\text{ is continuous and }w^{\prime}\in L^{2}([0,1))\}$
	$\displaystyle=$	$\displaystyle\left\{w:[0,1]\to[0,\infty)\;\left|\;\begin{aligned} &w\text{ is increasing, continuous, and convex,}\\ &w^{\prime}\in L^{2}([0,1)),w(0)=0\end{aligned}\right.\right\}.$

Then by Lemmas 5.1 and 5.2, the dual optimization problem (4.4) is equivalent to

$$\operatorname*{maximize\,}_{w\in\mathscr{W}^{\mathrm{icx}}_{c,2}}\inf_{Q\in\mathscr{Q}}L(Q,w;\beta,\lambda).$$

(5.1)

To solve problem (5.1), we first consider, for any given $w\in\mathscr{W}^{\mathrm{icx}}_{c,2}$, the inner optimization problem

$$\operatorname*{minimize\,}_{Q\in\mathscr{Q}}L(Q,w;\beta,\lambda).$$

(5.2)

The next lemma completely solves the inner optimization problem for $w\in\mathscr{W}^{\mathrm{icx}}_{c,2}$.

For every $w\in\mathscr{W}^{\mathrm{icx}}_{c,2}$, by Lemma 5.3 and by plugging (5.3) into $L(Q,w;\beta,\lambda)$, we know that

		$\displaystyle\min_{Q\in\mathscr{Q}}L(Q,w;\beta,\lambda)$
	$\displaystyle=$	$\displaystyle-{1\over 4}\int_{0}^{1}(w^{\prime}(s)-\lambda Q_{\rho}(1-s))^{2}ds+\int_{0}^{1}(Q_{0}(s)-\beta)w^{\prime}(s)ds+\lambda\beta\mathrm{E}[\rho]$
	$\displaystyle=$	$\displaystyle\int_{0}^{1}\left(-{1\over 4}(w^{\prime}(s))^{2}+\left({1\over 2}\lambda Q_{\rho}(1-s)+Q_{0}(s)-\beta\right)w^{\prime}(s)\right)ds-{1\over 4}\lambda^{2}\mathrm{E}[\rho^{2}]+\lambda\beta\mathrm{E}[\rho].$

Therefore, the dual optimization problem (4.4) reduces to

$$\operatorname*{maximize\,}_{w\in\mathscr{W}^{\mathrm{icx}}_{c,2}}\int_{0}^{1}\left(-{1\over 2}(w^{\prime}(s))^{2}-\left(2\beta-\lambda Q_{\rho}(1-s)-2Q_{0}(s)\right)w^{\prime}(s)\right)ds,$$

or, equivalently,

$$\operatorname*{maximize\,}_{w\in\mathscr{W}^{\mathrm{icx}}_{c,2}}\int_{0}^{1}-{1\over 2}(w^{\prime}(s))^{2}ds-\int_{0}^{1}w^{\prime}(s)dH_{\beta,\lambda}(s),$$

(5.4)

where

$$H_{\beta,\lambda}(s)\triangleq\int_{0}^{s}(2\beta-\lambda Q_{\rho}(1-t)-2Q_{0}(t))dt,\quad s\in[0,1].$$

(5.5)

Problem (5.4) can be reformulated as

$$\begin{split}&\operatorname*{maximize\,}_{G\in\mathscr{Q}}\int_{0}^{1}-{1\over 2}(G(s))^{2}ds-\int_{0}^{1}G(s)dH_{\beta,\lambda}(s)\\ &\text{subject to }\quad G(s)\geq 0,s\in[0,1).\end{split}$$

(5.6)

Problem (5.6) is similar to a problem arising in rank-dependent utility maximization. If $H_{\beta,\lambda}$ is concave, then its right derivative $H_{\beta,\lambda}^{\prime}$ is decreasing. In this case, the solution to problem (5.6) is given by the pointwise optimizer $G^{*}(s)=(-H_{\beta,\lambda}^{\prime}(s))^{+}$. In general, $H_{\beta,\lambda}$ is potentially non-concave, and therefore $(-H_{\beta,\lambda}^{\prime}(s))^{+}$ is potentially not increasing, which violates the constraint $G\in\mathscr{Q}$. Thus, the problem cannot be solved straightforwardly by pointwise optimization in general. To overcome such an obstacle, in the context of rank-dependent utility maximization and by calculus of variations, Xia and Zhou (2016) showed that the solution is given explicitly by the concave envelope. Then Xu (2016) provided the concave envelope relaxation approach to the solution, which is more straightforward. For more applications of concave envelope relaxation, see Rogers (2009), Wei (2018), and Wang and Xia (2021).

Let $\invbreve{H}_{\beta,\lambda}$ denotes the concave envelope of $H_{\beta,\lambda}$, that is, $\invbreve{H}_{\beta,\lambda}$ is the smallest concave function on $[0,1]$ that is no less than $H_{\beta,\lambda}$:

$$\invbreve{H}_{\beta,\lambda}(s)\triangleq\inf\{H(s)\,|\,H\mbox{ is concave and }H\geq H_{\beta,\lambda}\mbox{ on }[0,1]\},\quad s\in[0,1].$$

By Lemma A.2,

$$\invbreve{H}_{\beta,\lambda}(0)=H_{\beta,\lambda}(0)=0,\ \invbreve{H}_{\beta,\lambda}(1)=H_{\beta,\lambda}(1)=2\beta-\lambda\mathrm{E}[\rho]-2\mathrm{E}[X_{0}],\ \invbreve{H}_{\beta,\lambda}\geq H,$$

$\invbreve{H}_{\beta,\lambda}$ is continuous on $[0,1]$, and the right derivative $\invbreve{H}^{\prime}_{\beta,\lambda}$ is flat on $[\invbreve{H}_{\beta,\lambda}>H_{\beta,\lambda}]$.

We replace $H_{\beta,\lambda}$ in (5.6) with its concave envelope $\invbreve{H}_{\beta,\lambda}$ and consider the problem

$$\begin{split}&\operatorname*{maximize\,}_{G\in\mathscr{Q}}\int_{0}^{1}-{1\over 2}(G(s))^{2}ds-\int_{0}^{1}G(s)d\invbreve{H}_{\beta,\lambda}(s)\\ &\text{subject to }\quad G(s)\geq 0,s\in[0,1).\end{split}$$

(5.7)

We have the following lemma for problems (5.6) and (5.7).

From Lemma 5.4, we have the following theorem, which provides the solution to the dual optimization problem in closed form.

The next theorem provides, for every $(\beta,\lambda)$, the optimal solution to problem (4.3) in closed form, which is an immediate consequence of combining Theorem 5.5, Lemma 5.3, and Proposition 4.7.

Recall that, for any $\beta$ and $\lambda$,

$$H_{\beta,\lambda}(s)=2\beta s-N_{\lambda}(s),\quad s\in[0,1],$$

where

$$N_{\lambda}(s)\triangleq\int_{0}^{s}(\lambda Q_{\rho}(1-t)+2Q_{0}(t))dt,\quad s\in[0,1].$$

Let $\breve{N}_{\lambda}$ denote the convex envelope of $N_{\lambda}$. Then

$$\invbreve{H}_{\beta,\lambda}(s)=2\beta s-\breve{N}_{\lambda}(s),\quad s\in[0,1]$$

and

$$\invbreve{H}^{\prime}_{\beta,\lambda}=2\beta-\breve{N}^{\prime}_{\lambda}.$$

Therefore,

$$Q^{*}_{\beta,\lambda}(s)=\beta+{\left(\breve{N}_{\lambda}^{\prime}(s)-2\beta\right)^{+}-\lambda Q_{\rho}((1-s)-)\over 2},\quad s\in[0,1),$$

(6.1)

By Theorem 6.1,

	$\displaystyle v(\beta,\lambda)=$	$\displaystyle\int_{0}^{1}(Q^{*}_{\beta,\lambda}(s)-\beta)^{2}ds+\lambda\int_{0}^{1}Q^{*}_{\beta,\lambda}(s)Q_{\rho}(1-s)ds$
	$\displaystyle=$	$\displaystyle{1\over 4}\int_{0}^{1}\left(\left(\breve{N}_{\lambda}^{\prime}(s)-2\beta\right)^{+}\right)^{2}ds+\lambda\beta\mathrm{E}[\rho]-{\lambda^{2}\over 4}\mathrm{E}[\rho^{2}].$

The function $x\mapsto(x^{+})^{2}$ is convex and continuously differentiable. So is $v$ with respect to $\beta$ and

$${\partial v(\beta,\lambda)\over\partial\beta}=\lambda\mathrm{E}[\rho]-\int_{0}^{1}\left(\breve{N}_{\lambda}^{\prime}(s)-2\beta\right)^{+}ds\triangleq h(\beta,\lambda).$$

(6.2)