On robust portfolio and naïve diversification: mixing ambiguous and unambiguous assets

Abstract

Effect of the availability of a riskless asset on the performance of naïve diversification strategies has been a controversial issue. Defining an investment environment containing both ambiguous and unambiguous assets, we investigate the performance of naïve diversification over ambiguous assets. For the ambiguous assets, returns follow a multivariate distribution involving distributional uncertainty. A nominal distribution estimate is assumed to exist, and the actual distribution is considered to be within a ball around this nominal distribution. Complete information is assumed for the return distribution of unambiguous assets. As the radius of uncertainty increases, the optimal choice on ambiguous assets is shown to converge to the uniform portfolio with equal weights on each asset. The tendency of the investor to avoid ambiguous assets in response to increasing uncertainty is proven, with a shift towards unambiguous assets. With an application on the $\textit{CVaR}$ risk measure, we derive rules for optimally combining uniform ambiguous portfolio with the unambiguous assets.

Appendix: Intermediate results and proofs

We begin by proving an upper bound for the absolute deviation in the risk measure caused by the difference in distributions P and Q, while holding the ambiguous portfolio selection vector w fixed. The bound is the product of a constant related to the risk measure $\mathcal {R}$, the norm of the ambiguous portfolio $\Vert w\Vert _{q}$, and the distance between the two measures P and Q, which is shown to be equal to the distance between marginals $P_{1}$ and $Q_{1}$.

Lemma 3

Let $\mathcal {R}:L^{p}\left( \varOmega ,\varSigma ,\mu \right) \rightarrow {\mathbb {R}}$ be a convex, law invariant risk measure with dual representation as discussed in Sect. 2, where $1\le p<\infty$ and $\frac{1}{p}+\frac{1}{q}=1$. Let $P=P_{1}\times P_{2}$ and $Q=Q_{1}\times P_{2}$ be product measures on $\left( {\mathbb {R}}^{N+L},\mathcal {B}\left( {\mathbb {R}}^{N+L}\right) \right)$, for arbitrary Borel probability measures $P_{1}$, $Q_{1}$ on $\left( {\mathbb {R}}^{N},\mathcal {B}\left( {\mathbb {R}}^{N}\right) \right)$ and $P_{2}$ on $\left( {\mathbb {R}}^{L},\mathcal {B}\left( {\mathbb {R}}^{L}\right) \right)$. We denote by $\mathcal {F^{\perp }}$ the largest sigma algebra independent from $\sigma (X^{P_{2}})$. Note that $\sigma (X^{P_{1}})\subset \mathcal {F}^{\perp }\perp \sigma (X^{P_{2}})$, since $X^{P_{1}}$ and $X^{P_{2}}$ are independent. Then,

$$\begin{aligned} d_{p}^{{\mathbb {R}}^{N+L}}\left( P,Q\right)= & {} d_{p}^{{\mathbb {R}}^{N}}\left( P_{1},Q_{1}\right) , \end{aligned}$$

and

$$\begin{aligned}&\left| \mathcal {R}\left( \left<X^{P_{1}},w\right>+\left<X^{P_{2}},v\right>\right) -\mathcal {R}\left( \left<X^{Q_{1}},w\right>+\left<X^{P_{2}},v\right>\right) \right| \nonumber \\&\quad \le {\displaystyle \sup _{Z:R\left( Z\right) <\infty }}\Vert \mathbb {E}[Z\vert \mathcal {F^{\perp }}]\Vert _{L^{q}}\Vert w\Vert _{q}\,d_{p}\left( P,Q\right) . \end{aligned}$$

(24)

Proof

Let $\hat{\pi }_{1}$ be the optimal transportation plan, i.e., the minimizing distribution giving the Kantorovich distance between $P_{1}$ and $Q_{1}$. We will define a transportation plan $\hat{\pi }$ on $({\mathbb {R}}^{N+L}\times {\mathbb {R}}^{N+L},\mathcal {B}({\mathbb {R}}^{N+L}\times {\mathbb {R}}^{N+L}))$ between P and Q as follows:

$$\begin{aligned} \hat{\pi }(A\times B\times C\times D) = \hat{\pi }_{1}(A\times C)\times P_{2}(B\cap D), \end{aligned}$$

(25)

where $A,\,C\in \mathcal {B}({\mathbb {R}}^{N})$ and $B,\,D\in \mathcal {B}({\mathbb {R}}^{L})$. The $\pi$-system defined as the product $A\times B\times C\times D$ of Borel sets generates $\mathcal {B}({\mathbb {R}}^{N+L}\times {\mathbb {R}}^{N+L})$, and (25) uniquely defines a measure on $({\mathbb {R}}^{N+L}\times {\mathbb {R}}^{N+L},\mathcal {B}({\mathbb {R}}^{N+L}\times {\mathbb {R}}^{N+L}))$. $\hat{\pi }$ is a transporation plan between P and Q, as projections $\hat{\pi }(A\times B\times {\mathbb {R}}^{N+L})=\hat{\pi }_{1}(A\times {\mathbb {R}}^{N})\times P_{2}(B)=P_{1}(A)\times P_{2}(B)$ and $\hat{\pi }({\mathbb {R}}^{N+L}\times C\times D)=\hat{\pi }_{1}({\mathbb {R}}^{N}\times C)\times P_{2}(D)=Q_{1}(C)\times P_{2}(D)$ coincide with P and Q, respectively on the $\pi$-system of measurable rectangles in $\mathcal {B}\left( {\mathbb {R}}^{N}\right) \times \mathcal {B}\left( {\mathbb {R}}^{L}\right)$. $\hat{\pi }$ is a horizontal transportation plan, in the sense that it adopts the plan indicated by $\hat{\pi }_{1}$ at each $\left( x_{N+1},\ldots ,x_{N+L}\right) \in {\mathbb {R}}^{L}$ to redistribute the weight between P and Q, and does not shift the distributional weight between two locations $(\bar{x}_{N+1},\ldots ,\bar{x}_{N+L})\ne (x_{N+1},\ldots ,x_{N+L})$ to reach Q from P.

We begin by showing $d_{p}^{{\mathbb {R}}^{N}}\left( P_{1},Q_{1}\right) \le d_{p}^{{\mathbb {R}}^{N+L}}\left( P,Q\right)$, denoting by $\pi$ the optimal transportation plan between P and Q, and by $\pi _{1}$ its projection $\pi (\cdot \times {\mathbb {R}}^{L}\times \cdot \times {\mathbb {R}}^{L})$ (that this is a transportation plan between $P_{1}$ and $Q_{1}$ can be checked as above):

$$\begin{aligned} d_{p}^{{\mathbb {R}}^{N+L}}\left( P,Q\right)= & {} \left( \int _{{\mathbb {R}}^{N+L}\times {\mathbb {R}}^{N+L}}\sum _{i=1}^{N+L}\left| x_{i}-y_{i}\right| {}^{p}d\pi \left( x,y\right) \right) {}^{\frac{1}{p}}\nonumber \\\ge & {} \left( \int _{{\mathbb {R}}^{N+L}\times {\mathbb {R}}^{N+L}}\sum _{i=1}^{N}\left| x_{i}-y_{i}\right| {}^{p}d\pi \left( x,y\right) \right) {}^{\frac{1}{p}}\nonumber \\= & {} \left( \int _{{\mathbb {R}}^{N}\times {\mathbb {R}}^{N}}\sum _{i=1}^{N}\left| x_{i}-y_{i}\right| {}^{p}d\pi _{1}\left( x,y\right) \right) {}^{\frac{1}{p}} \end{aligned}$$

(26)

$$\begin{aligned}\ge & {} \left( \int _{{\mathbb {R}}^{N}\times {\mathbb {R}}^{N}}\sum _{i=1}^{N}\left| x_{i}-y_{i}\right| {}^{p}d\hat{\pi }_{1}\left( x,y\right) \right) {}^{\frac{1}{p}}\nonumber \\= & {} d_{p}^{{\mathbb {R}}^{N}}\left( P_{1},Q_{1}\right) . \end{aligned}$$

(27)

In the above derivations, (26) follows since the integrand is constant with respect to

$x_{N+1},\ldots ,x_{N+L}$, $y_{N+1},\ldots ,y_{N+L}$ and (27) follows due to the optimality of $\hat{\pi }_{1}$ among the transportation plans between $P_{1}$ and $Q_{1}$. The reverse, that is, $d_{p}^{{\mathbb {R}}^{N}}\left( P_{1},Q_{1}\right) \ge d_{p}^{{\mathbb {R}}^{N+L}}\left( P,Q\right)$, follows in a similar fashion. In the derivations, we evaluate the integral with respect to $\hat{\pi }$ separately over complementary sets $S=\left\{ \left( x_{1},x_{2},x_{3},x_{4}\right) \in {\mathbb {R}}^{N}\times {\mathbb {R}}^{L}\times {\mathbb {R}}^{N}\times {\mathbb {R}}^{L}:x_{2}=x_{4}\right\}$ and $\bar{S}=\left\{ \left( x_{1},x_{2},x_{3},x_{4}\right) \in {\mathbb {R}}^{N}\times {\mathbb {R}}^{L}\times {\mathbb {R}}^{N}\times {\mathbb {R}}^{L}:x_{2}\ne x_{4}\right\}$, which consist of ordered pairs in ${\mathbb {R}}^{N+L}\times {\mathbb {R}}^{N+L}$ that agree on the last L coordinates and those that disagree, respectively. S is closed, $\bar{S}$ is open and can be considered as a countable union of open sets of the form $O=O_{1}\times O_{2}\times O_{3}\times O_{4}$, where $O_{1},\,O_{3}\subset {\mathbb {R}}^{N}$ and $O_{2},\,O_{4}\subset {\mathbb {R}}^{L}$. $O\subset \bar{S}$ implies $O_{2}\cap O_{4}=\emptyset$, since otherwise $O\ne \emptyset$ would contain $\left( x_{1},x_{2},x_{3},x_{4}\right) \in {\mathbb {R}}^{N+L}\times {\mathbb {R}}^{N+L}$ such that $x_{2}=x_{4}$. By measure subadditivity, $\hat{\pi }\left( \bar{S}\right) =0$, i.e., $\bar{S}$ is a $\hat{\pi }$-negligible set, leading to following computations:

$$\begin{aligned} d_{p}^{{\mathbb {R}}^{N+L}}\left( P,Q\right)= & {} \left( \int _{{\mathbb {R}}^{N+L}\times {\mathbb {R}}^{N+L}}\sum _{i=1}^{N+L}\left| x_{i}-y_{i}\right| ^{p}d\pi \left( x,y\right) \right) ^{\frac{1}{p}}\nonumber \\\le & {} \left( \int _{{\mathbb {R}}^{N+L}\times {\mathbb {R}}^{N+L}}\sum _{i=1}^{N+L}\left| x_{i}-y_{i}\right| ^{p}d\hat{\pi }\left( x,y\right) \right) ^{\frac{1}{p}} \end{aligned}$$

(28)

$$\begin{aligned}= & {} \left( \int _{S}\sum _{i=1}^{N+L}\left| x_{i}-y_{i}\right| ^{p}d\hat{\pi }\left( x,y\right) +\int _{\bar{S}}\sum _{i=1}^{N+L}\left| x_{i}-y_{i}\right| ^{p}d\hat{\pi }\left( x,y\right) \right) ^{\frac{1}{p}}\nonumber \\= & {} \left( \int _{S}\sum _{i=1}^{N+L}\left| x_{i}-y_{i}\right| ^{p}d\hat{\pi }\left( x,y\right) \right) ^{\frac{1}{p}} \end{aligned}$$

(29)

$$\begin{aligned}= & {} \left( \int _{{\mathbb {R}}^{N+L}\times {\mathbb {R}}^{N+L}}\sum _{i=1}^{N}\left| x_{i}-y_{i}\right| ^{p}d\hat{\pi }\left( x,y\right) \right) ^{\frac{1}{p}} \nonumber \\= & {} \left( \int _{{\mathbb {R}}^{N}\times {\mathbb {R}}^{N}}\sum _{i=1}^{N}\left| x_{i}-y_{i}\right| ^{p}d\hat{\pi }_{1}\left( x,y\right) \right) ^{\frac{1}{p}}\end{aligned}$$

(30)

$$\begin{aligned}= & {} d_{p}^{{\mathbb {R}}^{N}}\left( P_{1},Q_{1}\right) . \end{aligned}$$

(31)

Here, (28) follows since $\pi$ is the optimal transportation plan for P and Q, (29) follows from the $\hat{\pi }$-neglibility of $\bar{S}$ and equality of $\left( x,y\right) \mapsto \sum _{i=1}^{N}\left| x_{i}-y_{i}\right| {}^{p}$ and $\left( x,y\right) \mapsto \sum _{i=1}^{N+L}\left| x_{i}-y_{i}\right| {}^{p}$ on S, (30) follows since $\sum _{i=1}^{N}\left| x_{i}-y_{i}\right| {}^{p}\mathbbm {1}_{S}=\sum _{i=1}^{N}\left| x_{i}-y_{i}\right| ^{p}$ almost everywhere. (31) holds since the integrand is constant with respect to $x_{N+1},\ldots ,x_{N+L}$, $y_{N+1},\ldots ,y_{N+L}$. We have that $d_{p}^{{\mathbb {R}}^{N}}(P_{1},Q_{1})=d_{p}^{{\mathbb {R}}^{N+L}}(P,Q)$.

Corresponding to the transportation plan $\hat{\pi }$, there exists a random variable $Y:\left( \varOmega ,\sigma ,\mu \right) \rightarrow {\mathbb {R}}^{N+L}\times {\mathbb {R}}^{N+L}$ with image measure $\mu \circ Y^{-1}=\hat{\pi }$. We denote the first $N+L$ components of Y by $X^{P}$, and the latter part by $X^{Q}$. It can be checked, as above, that the image measure of $X^{P}$ and $X^{Q}$ coincide with marginals P and Q, respectively. Note that $\mu \left( \left\{ \omega \in \varOmega :\,X^{Q_{2}}\left( \omega \right) \ne X^{P_{2}}\left( \omega \right) \right\} \}\right) =\hat{\pi }\left( \bar{S}\right) =0$, thus $X^{Q_{2}}=X^{P_{2}}$ almost everywhere, and we take $X^{Q}=\left[ \begin{array}{c} X^{Q_{1}}\\ X^{P_{2}} \end{array}\right]$ in our calculations. When Z is chosen from $\partial \mathcal {R}\left( \left<X^{P_{1}},w\right>+\left<X^{P_{2}},v\right>\right)$, we have:

$$\begin{aligned}&\mathcal {R}\left( \left<X^{P_{1}},w\right>+\left<X^{P_{2}},v\right>\right) -\mathcal {R}\left( \left<X^{Q_{1}},w\right>+\left<X^{P_{2}},v\right>\right) \nonumber \\&\quad \le \mathbb {E}\left[ \left( \left<X^{P_{1}},w\right>+\left<X^{P_{2}},v\right>\right) Z\right] -R\left( Z\right) -\mathbb {E}\left[ \left( \left<X^{Q_{1}},w\right>+\left<X^{P_{2}},v\right>\right) Z\right] +R\left( Z\right) \nonumber \\&\quad = \mathbb {E}\left[ \left<X^{P_{1}}-X^{Q_{1}},w\right>Z\right] \nonumber \\&\quad = \mathbb {E}\left[ \left<X^{P_{1}}-X^{Q_{1}},w\right>\mathbb {E}\left[ Z\vert \mathcal {F}^{\perp }\right] \right] \end{aligned}$$

(32)

$$\begin{aligned}&\quad \le \left\| \left<X^{P_{1}}-X^{Q_{1}},w\right> \right\| _{L^p} \left\| \mathbb {E}\left[ Z\vert \mathcal {F}^{\perp }\right] \right\| _{L^{q}} \end{aligned}$$

(33)

$$\begin{aligned}&\quad \le \left\| \left\| X^{P_{1}}-X^{Q_{1}}\right\| _{p} \left\| w\right\| _{q} \right\| _{L^p} \left\| \mathbb {E}\left[ Z\vert \mathcal {F}^{\perp }\right] \right\| _{L^{q}} \end{aligned}$$

(34)

$$\begin{aligned}&\quad = \Vert \mathbb {E}\left[ Z\vert \mathcal {F}^{\perp }\right] \Vert _{L^{q}}\Vert w\Vert _{q}\left( \int _{{\mathbb {R}}^{N+L}\times {\mathbb {R}}^{N+L}}\sum _{n=1}^{N}\left| x_{n}-y_{n}\right| ^{p}d\hat{\pi }\left( x,y\right) \right) ^{1/p}\\&\quad = \Vert \mathbb {E}\left[ Z\vert \mathcal {F}^{\perp }\right] \Vert _{L^{q}}\Vert w\Vert _{q}\left( \int _{{\mathbb {R}}^{N}\times {\mathbb {R}}^{N}}\sum _{n=1}^{N}\left| x_{n}-y_{n}\right| ^{p}d\hat{\pi }_{1}\left( x,y\right) \right) ^{1/p}\nonumber \\&\quad = \Vert \mathbb {E}\left[ Z\vert \mathcal {F}^{\perp }\right] \Vert _{L^{q}}\Vert w\Vert _{q}d_{p}^{{\mathbb {R}}^{N}}\left( P_{1},Q_{1}\right) \nonumber \\&\quad \le \sup _{Z:R\left( Z\right) <\infty }\Vert \mathbb {E}\left[ Z\vert \mathcal {F}^{\perp }\right] \Vert _{L^{q}}\Vert w\Vert _{q}d_{p}^{{\mathbb {R}}^{N+L}}\left( P,Q\right) .\nonumber \end{aligned}$$

(35)

Equality in (32) follows since $\left<X^{P_{1}}-X^{Q_{1}},w\right>$ is measurable with respect to $\mathcal {F}^{\perp }$, as $X^{P_{1}}$ and $X^{Q_{1}}$ are both independent from $X^{P_{2}}$, i.e., $\sigma (X^{P_{1}})\perp \sigma (X^{P_{2}})$ and $\sigma (X^{Q_{1}})\perp \sigma (X^{P_{2}})$, and $\sigma (X^{P_{1}})\vee \sigma (X^{Q_{1}})\subset \mathcal {F}^{\perp }$, where $\sigma \left( X^{P_{1}}\right) \vee \sigma \left( X^{Q_{1}}\right)$ is the smallest $\sigma$-algebra containing $\sigma \left( X^{P_{1}}\right) \cup \sigma \left( X^{Q_{1}}\right)$. Inequality (33) follows from the application of Hölder’s Inequality on the functions $\left<X^{P_{1}}-X^{Q_{1}},w\right>$ and $\mathbb {E}\left[ Z\vert \mathcal {F}^{\perp }\right]$, and (34) follows from Hölder’s Inequality on vectors $X^{P_{1}}-X^{Q_{1}}$ and w in ${\mathbb {R}}^{N}$. The arguments are based on a random variable Z specific to maximizing $\mathbb {E}\left[ \left( \left<X^{P_{1}},w\right>+\left<X^{P_{2}},v\right>\right) Z\right] -R\left( Z\right)$, but in the final step, by using the supremum over all $Z\in \partial \mathcal {R}\left( X\right)$ for X in the problem domain, this dependence is alleviated. Repeating the arguments for $\mathcal {R}\left( \left<X^{Q_{1}},w\right>+\left<X^{P_{2}},v\right>\right) -\mathcal {R}\left( \left<X^{P_{1}},w\right>+\left<X^{P_{2}},v\right>\right)$, we reach the result. $\square$

It is now proven that the absolute difference in risk measures due to different return distributions is bounded from above, by the product of a constant intrinsic to the risk measure $\mathcal {R}$, $\left\| w \right\| _{q}$, and the distance between the distributions. We will prove now that this bound is tight and attained, that is, given $\kappa >0$, for every distribution P, there is a distribution Q at distance $\kappa$ such that the difference in risk is equal to the bound in Lemma 3. In the two lemmas that follow, by restrictions on possibly $\mathcal {R}$, $P_{1}$ and $P_{2}$, it is possible to assume $\sup _{Z:R\left( Z\right) <\infty }\Vert \mathbb {E}\left[ Z\vert \mathcal {F}^{\perp }\right] \Vert _{L^{q}}$ is a constant C, and for two separate cases (the domain $L^{p}$ of $\mathcal {R}$ being defined for $p\in \left( 1,\infty \right)$ or $p=1$) it is proven that the bound stated in the previous lemma (that can now be pronounced as $C\kappa \Vert w\Vert _{q}$) is tight and attainable. Namely, for every distribution $P=P_{1}\times P_{2}$ and random variable $X^{P}$ with image measure P, there exists another random variable $X^{Q}$ whose image measure $Q=Q_{1}\times P_{2}$ is at a distance $d_{p}\left( P,Q\right) =\kappa$ to P, such that the difference in risk when investing $\left( w,v\right)$ is exactly $C\kappa \Vert w\Vert _{q}$ for P and Q. We begin with the case $p\in (1,\infty )$.

Lemma 4

Let the risk functional $\mathcal {R}$ be as defined above. Let $1<p<\infty$ and q be such that $\frac{1}{p}+\frac{1}{q}=1$. Let $P=P_{1}\times P_{2}$ be a probability measure on ${\mathbb {R}}^{N+L}$ with $P_{1}$, $P_{2}$ probability measures on ${\mathbb {R}}^{N}$, ${\mathbb {R}}^{L}$, respectively. Let $X^{P}=\left[ \begin{array}{c} X^{P_{1}}\\ X^{P_{2}} \end{array}\right]$ be a random variable with image measure P, and $\mathcal {F}^{\perp }\subset \varSigma$ be the largest $\sigma$-algebra independent from $\sigma \left( X^{P_{2}}\right)$. Assume that:

$$\begin{aligned} \Vert \mathbb {E}\left[ Z\vert \mathcal {F}^{\perp }\right] \Vert _{L^{q}}=C\text { for all }Z\in \bigcup _{X\in L^{p}}\partial \mathcal {R}\left( X\right) \text {with }R\left( Z\right) <\infty . \end{aligned}$$

Then it holds that for every $\kappa >0$ and every $\left( w,v\right) \in {\mathbb {R}}^{N}\times {\mathbb {R}}^{L}$, there are measures $Q_{1}$ on ${\mathbb {R}}^{N}$ and $Q=Q_{1}\times Q_{2}=Q_{1}\times P_{2}$ on ${\mathbb {R}}^{N+L}$ such that $d_{p}\left( P,Q\right) =\kappa$ and

$$\begin{aligned} \left| \mathcal {R}\left( \left<X^{Q_{1}},w\right>+\left<X^{Q_{2}},v\right>\right) -\mathcal {R}\left( \left<X^{P_{1}},w\right>+\left<X^{P_{2}},v\right>\right) \right| =C\kappa \Vert w\Vert _{q}, \end{aligned}$$

i.e., the bound of Lemma 3 holds with equality.

Proof

Fix a $Z\in \partial \mathcal {R}\left( \left<X^{P_{1}},w\right>+\left<X^{P_{2}},v\right>\right)$ with $R\left( Z\right) <\infty$. We set $\bar{Z}=\mathbb {E}\left[ Z\vert \mathcal {F}^{\perp }\right]$ and define a random variable $X^{Q}$ as follows:

$$\begin{aligned} X_{n}^{Q}= & {} X_{n}^{P}+c_{1}\left( n\right) \left| w_{n}\right| ^{\frac{q}{p}}\quad \text {with}\\ c_{1}\left( n\right)= & {} \frac{sign\left( w_{n}\right) sign(\bar{Z})c_{2}}{\Vert w\Vert _{q}^{q}}\left| \bar{Z}\right| ^{\frac{q}{p}} \end{aligned}$$

for $n\in \{1,\ldots ,N\}$ and a constant $c_{2}>0$. For $n\in \{N+1,\ldots ,N+L\}$, we let $X_{n}^{Q}=X_{n}^{P}$. Setting $c_{1}=\left| c_{1}\left( n\right) \right|$, it follows that:

$$\begin{aligned} c_{1}^{p}\left| w_{n}\right| ^{q}=\left| X_{n}^{Q}-X_{n}^{P}\right| ^{p},\quad \forall n:1\le n\le N. \end{aligned}$$

(36)

Also,

$$\begin{aligned} \left| \sum _{n=1}^{N}w_{n}\left( X_{n}^{Q}-X_{n}^{P}\right) \right| ^{p}= & {} \left| \sum _{n=1}^{N}w_{n}c_{1}\left( n\right) \left| w_{n}\right| ^{\frac{q}{p}}\right| ^{p} \nonumber \\= & {} \left| \sum _{n=1}^{N}\frac{sign(\bar{Z})c_{2}}{\Vert w\Vert _{q}^{q}}\left| \bar{Z}\right| ^{\frac{q}{p}}\left| w_{n}\right| ^{q}\right| ^{p} \nonumber \\= & {} \frac{c_{2}^{p}}{\Vert w\Vert _{q}^{pq}} \left| \bar{Z}\right| ^{q}\Vert w\Vert _{q}^{pq}=c_{2}^{p}\left| \bar{Z}\right| ^{q}. \end{aligned}$$

(37)

$c_{2}$ is a parameter for adjusting the distance between the distributions so that $d_{p}\left( P,Q\right) =\kappa$, which is achieved unless $Z=0$ (which would require $C=0$, leading to a triviality where $\mathcal {R}$ is constant). Note that $X^{Q_{1}}$ is $\mathcal {F}^{\perp }$-measurable since $X^{P_{1}}$ is $\sigma \left( X^{P_{1}}\right)$-measurable ($\sigma \left( X^{P_{1}}\right) \subset \mathcal {F}^{\perp }$) and $sign(\bar{Z})\left| \bar{Z}\right| ^{\frac{q}{p}}$ is $\mathcal {F}^{\perp }$-measurable. The independence of $\mathcal {F}^{\perp }$ and $\sigma (X^{P_{2}})$ implies the product form of $Q=\mu \circ \left( X^{Q}\right) ^{-1}=\mu \circ \left( X^{Q_{1}}\right) ^{-1}\times \mu \circ \left( X^{P_{2}}\right) ^{-1}=Q_{1}\times P_{2}$. The result is obtained as follows:

$$\begin{aligned}&\left| \mathcal {R}\left( \left<X^{Q_{1}},w\right>+\left<X^{Q_{2}},v\right>\right) -\mathcal {R}\left( \left<X^{P_{1}},w\right>+\left<X^{P_{2}},v\right>\right) \right| \nonumber \\&\qquad \ge \mathbb {E}\left( \left<X^{Q_{1}},w\right>+\left<X^{Q_{2}},v\right>\right) -\mathbb {E}\left( \left<X^{P_{1}},w\right>+\left<X^{P_{2}},v\right>\right) \nonumber \\&\qquad = \mathbb {E}\left( \left<X^{Q_{1}}-X^{P_{1}},w\right>Z\right) \nonumber \\&\qquad = \mathbb {E}\left( \left( \sum _{n=1}^{N}\left( X_{n}^{Q}-X_{n}^{P}\right) w_{n}\right) \bar{Z}\right) \nonumber \\&\qquad = \mathbb {E}\left( \left| \sum _{n=1}^{N}\left( X_{n}^{Q}-X_{n}^{P}\right) w_{n} \right| \left| \bar{Z}\right| \right) \end{aligned}$$

(38)

$$\begin{aligned}&\qquad = \left( \mathbb {E}\left( \left| \sum _{n=1}^{N}\left( X_{n}^{Q}-X_{n}^{P}\right) w_{n} \right| ^{p}\right) \right) ^{1/p}\Vert \bar{Z}\Vert _{L^{q}} \end{aligned}$$

(39)

$$\begin{aligned}&\qquad = \left( \mathbb {E}\left( \sum _{n=1}^{N}\left| X_{n}^{Q}-X_{n}^{P}\right| \left| w_{n} \right| \right) ^{p}\right) ^{1/p}\Vert \bar{Z}\Vert _{L^{q}} \end{aligned}$$

(40)

$$\begin{aligned}&\qquad = \left( \mathbb {E}\left( \Vert X^{Q_{1}}-X^{P_{1}}\Vert _{p}\Vert w\Vert _{q}\right) ^{p}\right) ^{1/p}\Vert \bar{Z}\Vert _{L^{q}} \end{aligned}$$

(41)

$$\begin{aligned}&\qquad = \Vert \bar{Z}\Vert _{L^{q}}\Vert w\Vert _{q}\left( \int _{\varOmega } \sum _{n=1}^{N}\left| X_{n}^{Q}-X_{n}^{P}\right| ^{p}d\mu \right) ^{1/p}\end{aligned}$$

(42)

$$\begin{aligned}&\qquad \ge \Vert \bar{Z}\Vert _{L^{q}}\Vert w\Vert _{q}d_{p}^{{\mathbb {R}}^{N}} (P_{1},Q_{1})\\&\qquad = \Vert \bar{Z}\Vert _{L^{q}}\Vert w\Vert _{q}\kappa .\nonumber \end{aligned}$$

(43)

Transitions to (38) and (40) are possible due to the incorporation of $sign(w_{n})$ and sign(Z) in $X^{Q_1}-X^{P_1}$, which makes all terms in the sum non-negative. (39) and (41) are applications of Hölder’s Inequality, where conditions for equality in Hölder’s are assured by (37) and (36), respectively. Equation (39) is possible since $\left| \sum _{n=1}^{N}\left( X_{n}^{Q}-X_{n}^{P}\right) w_{n}\right| ^{p}$ is equal to $|\bar{Z}|^{q}$, when constant multipliers set aside. Similarly, (36) implies the condition for equality on transition to (41). Inequality in (43) follows since $\tilde{\pi }=\mu \circ \left( \left[ \begin{array}{c} X^{P_{1}}\\ X^{Q_{1}} \end{array}\right] \right) ^{-1}$ is a transportation plan between $P_{1}$ and $Q_{1}$, and the integral in (42) is equivalent to $\int _{{\mathbb {R}}^{N}\times {\mathbb {R}}^{N}}\sum _{n=1}^{N}\left| x_{n}-y_{n}\right| ^{p}d\tilde{\pi }\left( x,y\right)$, while $d_{p}(P_{1},Q_{1})=\kappa$ is given by the optimal transportation plan between $P_{1}$ and $Q_{1}$. $\square$

A similar result on the tightness of the bound given in Lemma 3 follows for the case $p=1$.

Lemma 5

Let $\mathcal {R}$ be a risk measure as defined above. Let $P=P_{1}\times P_{2}$ be a probability measure on ${\mathbb {R}}^{N+L}$ where $P_{1}$, $P_{2}$ are probability measures on ${\mathbb {R}}^{N}$, ${\mathbb {R}}^{L}$, respectively, and $X^{P}\in L^{1}\left( \varOmega ,\varSigma ,\mu \right)$ a random variable with image measure P. Let $\mathcal {F}^{\perp }\subset \varSigma$ be the largest $\sigma$-algebra independent from $\sigma \left( X^{P_{2}}\right)$. Assume:

$$\begin{aligned} \Vert Z\Vert _{L^{\infty }} = C,\;\mu \left( \left\{ \omega \in \varOmega :\left| Z\right| \left( \omega \right) \notin \left\{ 0,C\right\} \right\} \right) =0 \end{aligned}$$

(44)

for all $Z\in \partial \mathcal {R}\left( X\right)$, $X\in L^{1}\left( \varOmega ,\varSigma ,\mu \right)$. In addition, assume for all $\epsilon \in \left( 0,\frac{1}{2}\right)$ that there exists $B\in \mathcal {F}^{\perp }$ such that $\mu (B)>0$, and either

$$\begin{aligned} \mu \left( B\cap \left\{ \omega \in \varOmega :Z\left( \omega \right) =C\right\} \right) >\left( 1-\epsilon \right) \mu \left( B\right) \end{aligned}$$

or

$$\begin{aligned} \mu \left( B\cap \left\{ \omega \in \varOmega :Z\left( \omega \right) =-C\right\} \right) >\left( 1-\epsilon \right) \mu (B) \end{aligned}$$

holds. Then for every $\kappa >0$, there is a probability measure $Q=Q_{1}\times P_{2}$ on ${\mathbb {R}}^{N+L}$ with $Q_{1}$ a probability measure on ${\mathbb {R}}^{N}$ such that $d_{1}^{{\mathbb {R}}^{N+L}}\left( P,Q\right) =\kappa$ and

$$\begin{aligned} \left| \mathcal {R}\left( \left<X^{Q_{1}},w\right>+\left<X^{Q_{2}},v\right>\right) -\mathcal {R}\left( \left<X^{P_{1}},w\right>+\left<X^{P_{2}},v\right>\right) \right| = C\kappa \Vert w\Vert _{\infty }. \end{aligned}$$

Proof

Take $Z\in \partial \mathcal {R}\left( \left<X^{P_{1}},w\right>+\left<X^{P_{2}},v\right>\right)$. For $\epsilon \in \left( 0,\frac{1}{2}\right)$, we consider the case where there exists $B\in \mathcal {F}^{\perp }$ with $\mu \left( B\cap \left\{ \omega \in \varOmega :Z\left( \omega \right) =C\right\} \right) >\left( 1-\epsilon \right) \mu \left( B\right)$. For the alternative case, the result follows in a similar manner with a change of sign in $c_{1}\left( n\right)$ defined below. Let

$$\begin{aligned} X_{n}^{Q}= & {} X_{n}^{P}+c_{1}\left( n\right) \end{aligned}$$

(45)

$$\begin{aligned} c_{1}\left( n\right)= & {} \left\{ \begin{array}{ll} c_{2}sign\left( w_{n}\right) \mathbbm {1}_{B}, &{} \mathrm {if}\left| w_{n}\right| =\Vert w\Vert _{\infty }\\ 0, &{} \text{ otherwise } \end{array}\right. \end{aligned}$$

(46)

for $n\in \left\{ 1,\ldots ,N\right\}$ and a constant $c_{2}>0$. Again, $c_{2}$ is a constant for adjusting the distance between P and Q to $\kappa$. Let $X_{n}^{Q}=X_{n}^{P}$ for $n\in \left\{ N+1,\ldots ,N+L\right\}$. Let us label the number of entries in w that set its norm, i.e., $\left| \{n:\left| w_{n}\right| =\left\| w\right\| _{\infty }\}\right|$ by $\chi$.

Via the inverse image of $X^{Q}$, we obtain a distribution $Q=\mu \circ (X^{Q})^{-1}$ on ${\mathbb {R}}^{N+L}$. Again, since $B\in \mathcal {F}^{\perp }$ and $\mathcal {F}^{\perp }$ and $\sigma (X^{P_{2}})$ are independent, the product form of $Q=Q_{1}\times Q_{2}$ follows. We have:

$$\begin{aligned}&\mathcal {R}\left( \left<X^{Q_{1}},w\right>+\left<X^{Q_{2}},v\right>\right) -\mathcal {R}\left( \left<X^{P_{1}},w\right>+\left<X^{P_{2}},v\right>\right) \nonumber \\&\quad \ge \mathbb {E}\left( \left<X^{Q_{1}}-X^{P_{1}},w\right>Z\right) \end{aligned}$$

(47)

$$\begin{aligned}&\quad =\mathbb {E}\left( \left<X^{Q_{1}}-X^{P_{1}},w\right>Z\mathbbm {1}_{B}\right) \nonumber \\&\quad =\mathbb {E}\left( \left<X^{Q_{1}}-X^{P_{1}},w\right>Z\mathbbm {1}_{\left\{ Z>0\right\} \cap B}\right) +\mathbb {E}\left( \left<X^{Q_{1}}-X^{P_{1}},w\right>Z\mathbbm {1}_{\left\{ Z\le 0\right\} \cap B}\right) \nonumber \\&\quad \ge \mathbb {E}\left( \left<X^{Q_{1}}-X^{P_{1}},w\right>Z\mathbbm {1}_{\left\{ Z>0\right\} \cap B}\right) -c_{2}C\chi \left\| w\right\| _{\infty }\mu \left( \left\{ Z\le 0\right\} \cap B\right) \nonumber \\&\quad \ge \mathbb {E}\left( \left<X^{Q_{1}}-X^{P_{1}},w\right>Z\mathbbm {1}_{\left\{ Z>0\right\} \cap B}\right) -\epsilon c_{2}C\chi \left\| w\right\| _{\infty }\mu \left( B\right) \nonumber \\&\quad =\mathbb {E}\left( \left| \left<X^{Q_{1}}-X^{P_{1}},w\right>\mathbbm {1}_{\left\{ Z>0\right\} \cap B}\right| \left| Z\right| \right) -\epsilon c_{2}C\chi \left\| w\right\| _{\infty }\mu \left( B\right) \nonumber \\&\quad =\mathbb {E}\left( \left| \left<X^{Q_{1}}-X^{P_{1}},w\right>\mathbbm {1}_{\left\{ Z>0\right\} \cap B}\right| \right) \left\| Z\right\| _{L^{\infty }}-\epsilon c_{2}C\chi \left\| w\right\| _{\infty }\mu \left( B\right) \end{aligned}$$

(48)

$$\begin{aligned}&\quad =c_{2}\chi \left\| w\right\| _{\infty }\mu \left( \left\{ Z>0\right\} \cap B\right) \left\| Z\right\| _{L^{\infty }}-\epsilon c_{2}C\chi \left\| w\right\| _{\infty }\mu \left( B\right) \nonumber \\&\quad > \left( 1-\epsilon \right) c_{2}\chi \left\| w\right\| _{\infty }\mu \left( B\right) C-\epsilon c_{2}C\chi \left\| w\right\| _{\infty }\mu \left( B\right) \nonumber \\&\quad = \left( 1-2\epsilon \right) c_{2}\chi \left\| w\right\| _{\infty }\mu \left( B\right) C\nonumber \\&\quad =\left( 1-2\epsilon \right) \mathbb {E}\left( \int _{\varOmega }\sum _{n=1}^{N}\left| c_{1}(n)\right| d\mu \right) \left\| w\right\| _{\infty }C\nonumber \\&\quad =\left( 1-2\epsilon \right) \mathbb {E}\left( \Vert X^{Q_{1}}-X^{P_{1}}\Vert _{1}\right) \left\| w\right\| _{\infty }C\nonumber \\&\quad \ge \left( 1-2\epsilon \right) d_{1}^{{\mathbb {R}}^{N}}\left( P_{1},Q_{1}\right) \left\| w\right\| _{\infty }C \end{aligned}$$

(49)

$$\begin{aligned}&\quad =\left( 1-2\epsilon \right) \kappa \left\| w\right\| _{\infty }C \\&\quad = \left( 1-2\epsilon \right) \kappa \left\| w\right\| _{\infty }C.\nonumber \end{aligned}$$

(50)

As $X^{Q_{2}}=X^{P_{2}}$, and Z is possibly not in $\partial \mathcal {R}\left( \left<X^{Q_{1}},w\right>+\left<X^{Q_{2}},v\right>\right)$, (47) follows. Equality (48) is assured since the $L^1$ function has value 0 on $\left\{ Z\ne \left\| Z\right\| _{L^\infty }\right\}$. Equation (49) follows since $X^{Q_{1}}$ and $X^{P_{1}}$ jointly have an image measure which is a transportation between $Q_{1}$ and $P_{1}$ (which is not necessarily optimal), and (50) since $d_{1}^{{\mathbb {R}}^{N}}\left( P_{1},Q_{1}\right) =d_{1}^{{\mathbb {R}}^{N+L}}\left( P,Q\right) =\kappa$. With the above holding for all $\epsilon \in \left( 0,\frac{1}{2}\right)$, the result is established. $\square$

The perturbation used in Lemma 4 to obtain $X^{Q}$ from $X^{P}$ was based on the conditional expectation $\mathbb {E}\left[ Z\vert \mathcal {F}^{\perp }\right]$ of the random variable $Z \in \partial \mathcal {R}\left( \left<X^{P_{1}},w\right>+\left<X^{P_{2}},v\right>\right)$, whereas in Lemma 5 we used the characteristic function of the set B defined in the assumptions. However, the two results are analogous, and agree the figure in Lemma 3, since the assumptions in Lemma 5 imply that $\left\| Z\right\| _{\infty }=\left\| \mathbb {E}\left[ Z\vert \mathcal {F}^{\perp }\right] \right\| _{\infty }$ for all $Z\in \partial \mathcal {R}\left( X\right)$, $X\in L^{1}\left( \varOmega ,\varSigma ,\mu \right)$. To observe this, one can check that the value of $\mathbb {E}\left[ Z\vert \mathcal {F}^{\perp }\right]$ on the set B, as referred to in Lemma 5, is either inside $\left( \left( 1-2\epsilon \right) C,C\right]$ or $\left[ -C,-\left( 1-2\epsilon \right) C\right)$.

With the proofs of Lemmas 4 and 5, we show that the bound in Lemma 3 is tight and attained, and also we are ready to prove Proposition 1.

Proof of Proposition 1

With fixed portfolio selection $\left( w,v\right)$, the maximum (absolute) difference in the risk measure due to distributions P and Q ($d_p\left( P,Q\right) \le \kappa$) is $C\kappa \Vert w\Vert _{q}$, by Lemma 3. By assumption, $\mathcal {R}$ and P satisfy the assumptions in Lemma 4 or 5, as necessitated by the domain of the risk measure, i.e., the value of p. Moreover, the deviation in the risk measure due to the perturbed measure $Q=Q_{1}\times P_{2}$ in the proofs of Lemmas 4 and 5 is in the positive direction, therefore we can write:

$$\begin{aligned}&\sup _{\bar{P}\in \tilde{\mathcal {B}}_{\kappa }(P)}\mathcal {R}(\langle X^{\bar{P}_{1}},w\rangle +\langle X^{\bar{P}_{2}},v\rangle ) \nonumber \\&\qquad = \mathcal {R}(\langle X^{Q_{1}},w\rangle +\langle X^{Q_{2}},v\rangle ) \end{aligned}$$

(51)

$$\begin{aligned}&\qquad = \mathcal {R}(\langle X^{P_{1}},w\rangle +\langle X^{P_{2}},v\rangle )+C\kappa \Vert w\Vert _{q}. \end{aligned}$$

(52)

$\square$

Proof of Lemma 1

Taking arbitrary $Z\in \partial \mathcal {R}\left( \left\langle X^{P_{1}},w^{1}\right\rangle +\left\langle X^{P_{2}},v\right\rangle \right)$,

$$\begin{aligned}&\mathcal {R}\left( \left\langle X^{P_{1}},w^{1}\right\rangle +\left\langle X^{P_{2}},v\right\rangle \right) -\mathcal {R}\left( \left\langle X^{P_{1}},w^{2}\right\rangle +\left\langle X^{P_{2}},v\right\rangle \right) \nonumber \\&\qquad \le \mathbb {E}\left[ \left( \left\langle X^{P_{1}},w^{1}\right\rangle +\left\langle X^{P_{2}},v\right\rangle \right) Z\right] -\mathbb {E}\left[ \left( \left\langle X^{P_{1}},w^{2}\right\rangle +\left\langle X^{P_{2}},v\right\rangle \right) Z\right] \nonumber \\&\qquad =\mathbb {E}\left[ \left\langle X^{P_{1}},w^{1}-w^{2}\right\rangle Z\right] \nonumber \\&\qquad =\mathbb {E}\left[ \left\langle X^{P_{1}},w^{1}-w^{2}\right\rangle \bar{Z}\right] \nonumber \\&\qquad =\mathbb {E}\left[ \left\langle X^{P_{1}},w^{1}-w^{2}\right\rangle \mathbbm {1}_{\{\bar{Z}\ne 0\}}\bar{Z}\right] \end{aligned}$$

(53)

$$\begin{aligned}&\qquad \le \mathbb {E}\left[ \left| \left\langle X^{P_{1}},w^{1}-w^{2}\right\rangle \mathbbm {1}_{\{\bar{Z}\ne 0\}}\right| ^{p}\right] {}^{\frac{1}{p}}\Vert \bar{Z}\Vert _{L^{q}} \end{aligned}$$

(54)

$$\begin{aligned}&\qquad \le \mathbb {E}\left[ \left( \Vert X^{P_{1}}\Vert _{p}\Vert w^{1}-w^{2}\Vert _{q}\right) ^{p}\mathbbm {1}_{\{\bar{Z}\ne 0\}}\right] {}^{\frac{1}{p}}\Vert \bar{Z}\Vert _{L^{q}} \end{aligned}$$

(55)

$$\begin{aligned}&\qquad =\mathbb {E}\left[ \Vert X^{P_{1}}\Vert _{p}^{p}\mathbbm {1}_{\{\bar{Z}\ne 0\}}\right] {}^{\frac{1}{p}}\Vert w^{1}-w^{2}\Vert _{q}C. \end{aligned}$$

(56)

Inequalities (54) and (55) follow due to Hölder’s Inequality applied on functions $\left\langle X^{P_{1}},w^{1}-w^{2}\right\rangle \mathbbm {1}_{\{\bar{Z}\ne 0\}}\in L^{p}$ and $\bar{Z}\in L^{q}$, and vectors in ${\mathbb {R}}^{N}$, respectively. The result, (56), follows due to the assumptions in Lemmas 4 and 5 that imply $\Vert \bar{Z}\Vert _{L^{q}}=C$ for $Z\in \partial \mathcal {R}\left( \left\langle X^{P_{1}},w^{1}\right\rangle +\left\langle X^{P_{2}},v\right\rangle \right)$. $\square$

Proof of Lemma 2

$\kappa \ge \frac{\Vert w-w^{u,v}\Vert _{q}}{\Vert w\Vert _{q}-\Vert w^{u,v}\Vert _{q}}\mathbb {E}\left[ \Vert X^{P_{1}}\Vert _{p}^{p}\mathbbm {1}_{\{\bar{Z}\ne 0\}}\right] ^{\frac{1}{p}}$ implies, by multiplying both sides by $C(\Vert w\Vert _{q}-\Vert w^{u,v}\Vert _{q})$:

$$\begin{aligned} C\kappa \left( \Vert w\Vert _{q}-\Vert w^{u,v}\Vert _{q}\right)\ge & {} C\left( \Vert w\Vert _{q}-\Vert w^{u,v}\Vert _{q}\right) \frac{\Vert w-w^{u,v}\Vert _{q}}{\Vert w\Vert _{q}-\Vert w^{u,v}\Vert _{q}}\mathbb {E}\left[ \Vert X^{P_{1}}\Vert _{p}^{p}\mathbbm {1}_{\{\bar{Z}\ne 0\}}\right] {}^{\frac{1}{p}}\\= & {} C\Vert w-w^{u,v}\Vert _{q}\mathbb {E}\left[ \Vert X^{P_{1}}\Vert _{p}^{p}\mathbbm {1}_{\{\bar{Z}\ne 0\}}\right] ^{\frac{1}{p}}\\\ge & {} \mathcal {R}\left( \langle X^{P_{1}},w^{u,v}\rangle +\langle X^{P_{2}},v\rangle \right) -\mathcal {R}\left( \langle X^{P_{1}},w\rangle +\langle X^{P_{2}},v\rangle \right) , \end{aligned}$$

where the last inequality follows by Lemma 1. Regrouping terms above, we have:

$$\begin{aligned} \mathcal {R}\left( \langle X^{P_{1}},w\rangle +\langle X^{P_{2}},v\rangle \right) +C\kappa \Vert w\Vert _{q} \ge \mathcal {R}\left( \langle X^{P_{1}},w^{u,v}\rangle +\langle X^{P_{2}},v\rangle \right) +C\kappa \Vert w^{u,v}\Vert _{q}, \end{aligned}$$

for all $w\in B$ ($\Vert w\Vert _{q}\ne \Vert w^{u,v}\Vert _{q}$ is assumed, which holds true for $w\ne w^{u,v}$ on a hyperplane of fixed $\langle \mathbbm {1},w\rangle =\langle \mathbbm {1},w^{u,v}\rangle$, since $w^{u,v}$ uniquely minimizes $\Vert \cdot \Vert _{q}$ on that hyperplane). $\square$

Proof of Proposition 2

Case $\left\langle \mathbbm {1},v\right\rangle =1$, $1\le p<\infty$

In this case $w^{u,v}=0$, hence $\Vert w-w^{u,v}\Vert _{q}=\Vert w\Vert _{q}-\Vert w^{u,v}\Vert _{q}=\Vert w\Vert _{q}$. Thus the condition in Lemma 2 is satisfied by all $w\ne 0$ if

$$\begin{aligned} \kappa \ge \mathbb {E}\left[ \Vert X^{P_{1}}\Vert _{p}^{p}\mathbbm {1}_{\{\bar{Z}\ne 0\}}\right] ^{\frac{1}{p}}, \end{aligned}$$

and the optimality of $w^{u,v}=0$ follows.

Case $\left\langle \mathbbm {1},v\right\rangle >1$, $p=1$

In this case, $1-\left\langle \mathbbm {1},v\right\rangle <0$, and $w^{u,v}=\frac{1-\left\langle \mathbbm {1},v\right\rangle }{N}\mathbbm {1}$ is composed of negative entries. For any w such that $\langle \mathbbm {1},w\rangle =1-\left\langle \mathbbm {1},v\right\rangle$, $w\ne w^{u,v}$, we let $n^{*}=\arg \max _{1\le n\le N}\left| w_{n}-\frac{1-\left\langle \mathbbm {1},v\right\rangle }{N}\right|$. If $w_{n^{*}}<\frac{1-\left\langle \mathbbm {1},v\right\rangle }{N}$, $\Vert w-w^{u,v}\Vert _{\infty }=\left| w_{n^{*}}-\frac{1-\left\langle \mathbbm {1},v\right\rangle }{N}\right| =\Vert w\Vert _{\infty }-\Vert w^{u,v}\Vert _{\infty }$. Otherwise, $w_{n*}>\frac{1-\left\langle \mathbbm {1},v\right\rangle }{N}$, $w_{n^{*}}=\max _{1\le n\le N}w_{n}$, and

$$\begin{aligned} \min _{1\le n\le N}w_{n}\le & {} \frac{1-\left\langle \mathbbm {1},v\right\rangle }{N}-\frac{w_{n^{*}}-\frac{1-\left\langle \mathbbm {1},v\right\rangle }{N}}{N-1}\\= & {} \frac{1-\left\langle \mathbbm {1},v\right\rangle }{N}-\frac{\Vert w-w^{u,v}\Vert _{\infty }}{N-1}. \end{aligned}$$

Since $\min _{1\le n\le N}w_{n}\ge -\Vert w\Vert _{\infty }$, $\frac{1-\left\langle \mathbbm {1},v\right\rangle }{N}-\frac{\Vert w-w^{u,v}\Vert _{\infty }}{N-1}\ge -\Vert w\Vert _{\infty }$, and noting that $\frac{1-\left\langle \mathbbm {1},v\right\rangle }{N}=-\Vert w^{u,v}\Vert _{\infty }$, we have:

$$\begin{aligned} \Vert w\Vert _{\infty }-\Vert w^{u,v}\Vert _{\infty } \ge \frac{\Vert w-w^{u,v}\Vert _{\infty }}{N-1}, \end{aligned}$$

equivalently:

$$\begin{aligned} \frac{\Vert w-w^{u,v}\Vert _{\infty }}{\Vert w\Vert _{\infty }-\Vert w^{u,v}\Vert _{\infty }} \le N-1. \end{aligned}$$

If $\kappa \ge \left( N-1\right) \mathbb {E}\left[ \Vert X^{P_{1}}\Vert _{1}\mathbbm {1}_{\{\bar{Z}\ne 0\}}\right]$, then $\kappa \ge \frac{\Vert w-w^{u,v}\Vert _{\infty }}{\Vert w\Vert _{\infty }-\Vert w^{u,v}\Vert _{\infty }}\mathbb {E}\left[ \Vert X^{P_{1}}\Vert _{1}\mathbbm {1}_{\{\bar{Z}\ne 0\}}\right]$ holds for all $w\ne w^{u,v}$ such that $\left\langle \mathbbm {1},w\right\rangle =1-\left\langle \mathbbm {1},v\right\rangle$, and $w^{u,v}$ is optimal to (10)–(11).

Case $\left\langle \mathbbm {1},v\right\rangle <1$, $p=1$

In this case, $w^{u,v}=\frac{1-\left\langle \mathbbm {1},v\right\rangle }{N}\mathbbm {1}$ is composed of positive entries. Similar to the case where $\left\langle \mathbbm {1},v\right\rangle >1$, we set $n^{*}=\arg \max _{1\le n\le N}\left| w_{n}-\frac{1-\left\langle \mathbbm {1},v\right\rangle }{N}\right|$. If $w_{n^{*}}>\frac{1-\left\langle \mathbbm {1},v\right\rangle }{N}$, $\Vert w-w^{u,v}\Vert _{\infty }=\left( w_{n^{*}}-\frac{1-\left\langle \mathbbm {1},v\right\rangle }{N}\right) =\Vert w\Vert _{\infty }-\Vert w^{u,v}\Vert _{\infty }$. Otherwise, $w_{n^{*}}<\frac{1-\left\langle \mathbbm {1},v\right\rangle }{N}$, $w_{n^{*}}=\min _{1\le n\le N}w_{n}$, and

$$\begin{aligned} \max _{1\le n\le N}w_{n}\ge & {} \frac{1-\left\langle \mathbbm {1},v\right\rangle }{N}+\frac{w_{n^{*}}-\frac{1-\left\langle \mathbbm {1},v\right\rangle }{N}}{N-1}\\= & {} \frac{1-\left\langle \mathbbm {1},v\right\rangle }{N}+\frac{\Vert w\Vert _{\infty }-\Vert w^{u,v}\Vert _{\infty }}{N-1}. \end{aligned}$$

Since $\Vert w\Vert _{\infty }\ge \max _{1\le n\le N}w_{n}$, and $\Vert w^{u,v}\Vert _{\infty }=\frac{1-\left\langle \mathbbm {1},v\right\rangle }{N}$, we can write:

$$\begin{aligned} \Vert w\Vert _{\infty }-\Vert w^{u,v}\Vert _{\infty }\ge & {} \frac{\Vert w-w^{u,v}\Vert _{\infty }}{N-1}\\ N-1\ge & {} \frac{\Vert w-w^{u,v}\Vert _{\infty }}{\Vert w\Vert _{\infty }-\Vert w^{u,v}\Vert _{\infty }}. \end{aligned}$$

Again, if $\kappa \ge \left( N-1\right) \mathbb {E}\left[ \Vert X^{P_{1}}\Vert _{1}\mathbbm {1}_{\{\bar{Z}\ne 0\}}\right]$, the condition for Lemma 2 is satisfied in the feasible region and $w^{u,v}$ is optimal to (10)–(11).

Case $\left\langle \mathbbm {1},v\right\rangle \ne 1$, $p=2$

Let $f_{2},\ldots ,f_{N}$ be unit vectors orthogonal to each other and $w^{u,v}$. Then a unique selection of $c_{2},\ldots ,c_{N}\in {\mathbb {R}}$ gives $w=w^{u,v}+\sum _{i=2}^{N}c_{i}f_{i}$ (since $w^{u,v}=\frac{1-\left\langle \mathbbm {1},v\right\rangle }{N}\mathbbm {1}$, $f_{2},\ldots ,f_{N}$ are orthogonal to $w^{u,v}$, and $\left\langle \mathbbm {1},w\right\rangle =1-\left\langle \mathbbm {1},v\right\rangle$, $\left\langle \mathbbm {1},w^{u,v}\right\rangle =\left\langle \mathbbm {1},w\right\rangle$ implies that the coefficient of $w^{u,v}$ in w is equal to 1). Then:

$$\begin{aligned} \frac{\Vert w-w^{u,v}\Vert _{2}}{\Vert w\Vert _{2}-\Vert w^{u,v}\Vert _{2}}= & {} \frac{\Vert w-w^{u,v}\Vert _{2}}{\left( \frac{\left( 1-\left\langle \mathbbm {1},v\right\rangle \right) ^{2}}{N}+\sum _{i=2}^{N}c_{i}^{2}\right) ^{\frac{1}{2}}-\frac{\left| 1-\left\langle \mathbbm {1},v\right\rangle \right| }{\sqrt{N}}}\\= & {} \frac{\Vert w-w^{u,v}\Vert _{2}\left[ \left( \frac{\left( 1-\left\langle \mathbbm {1},v\right\rangle \right) ^{2}}{N}+\sum _{i=2}^{N}c_{i}^{2}\right) ^{\frac{1}{2}}+\frac{\left| 1-\left\langle \mathbbm {1},v\right\rangle \right| }{\sqrt{N}}\right] }{\frac{\left( 1-\left\langle \mathbbm {1},v\right\rangle \right) ^{2}}{N}+\sum _{i=2}^{N}c_{i}^{2}-\frac{\left( 1-\left\langle \mathbbm {1},v\right\rangle \right) ^{2}}{N}}\\= & {} \frac{\Vert w-w^{u,v}\Vert _{2}\left[ \left( \frac{\left( 1-\left\langle \mathbbm {1},v\right\rangle \right) ^{2}}{N}+\sum _{i=2}^{N}c_{i}^{2}\right) ^{\frac{1}{2}}+\frac{\left| 1-\left\langle \mathbbm {1},v\right\rangle \right| }{\sqrt{N}}\right] }{\Vert w-w^{u,v}\Vert _{2}^{2}}\\= & {} \frac{\left( \frac{\left( 1-\left\langle \mathbbm {1},v\right\rangle \right) ^{2}}{N}+\sum _{i=2}^{N}c_{i}^{2}\right) ^{\frac{1}{2}}+\frac{\left| 1-\left\langle \mathbbm {1},v\right\rangle \right| }{\sqrt{N}}}{\Vert w-w^{u,v}\Vert _{2}}\\= & {} \left( \frac{\left( 1-\left\langle \mathbbm {1},v\right\rangle \right) ^{2}}{N\Vert w-w^{u,v}\Vert _{2}^{2}}+1\right) ^{\frac{1}{2}}+\frac{\left| 1-\left\langle \mathbbm {1},v\right\rangle \right| }{\sqrt{N}\Vert w-w^{u,v}\Vert _{2}}. \end{aligned}$$

Defining the set B as $B:\left\{ w\in {\mathbb {R}}^{N}:\Vert w-w^{u,v}\Vert _{2}\ge D\right\}$, the above equality implies that inside the set B,

$$\begin{aligned} \frac{\Vert w-w^{u,v}\Vert _{2}}{\Vert w\Vert _{2}-\Vert w^{u,v}\Vert _{2}}\le & {} \left( \frac{\left( 1-\left\langle \mathbbm {1},v\right\rangle \right) ^{2}}{ND^{2}}+1\right) ^{\frac{1}{2}}+\frac{\left| 1-\left\langle \mathbbm {1},v\right\rangle \right| }{\sqrt{N}D}. \end{aligned}$$

If the value of $\kappa$ satisfies

$$\begin{aligned} \kappa\ge & {} \left[ \left( \frac{\left( 1-\left\langle \mathbbm {1},v\right\rangle \right) ^{2}}{ND^{2}}+1\right) ^{\frac{1}{2}}+\frac{\left| 1-\left\langle \mathbbm {1},v\right\rangle \right| }{\sqrt{N}D}\right] \mathbb {E}\left[ \Vert X^{P_{1}}\Vert _{2}^{2}\mathbbm {1}_{\{\bar{Z}\ne 0\}}\right] ^{\frac{1}{2}} \end{aligned}$$

then

$$\begin{aligned} \kappa\ge & {} \frac{\Vert w-w^{u,v}\Vert _{2}}{\Vert w\Vert _{2}-\Vert w^{u,v}\Vert _{2}}\mathbb {E}\left[ \Vert X^{P_{1}}\Vert _{2}^{2}\mathbbm {1}_{\{\bar{Z}\ne 0\}}\right] ^{\frac{1}{2}} \end{aligned}$$

for $w\in B$, and by Lemma 2, a solution with better objective value than $w^{u,v}$ can only be inside $\left\{ w\in {\mathbb {R}}^{N}:\Vert w-w^{u,v}\Vert _{2}<D\right\}$.

Case $p\notin \{1,2\}$, $\left\langle \mathbbm {1},v\right\rangle \ne 1$

In this case, we show that for an increasing sequence $\kappa _{n}$, the optimal solution gets to fall inside a smaller neighborhood surrounding $w^{u,v}$ as $\kappa _{n}\longrightarrow \infty$. We define the set

$$\begin{aligned} \begin{aligned} A_{n}=&\{w\in {\mathbb {R}}^{N}:\langle \mathbbm {1},w\rangle =1-\left\langle \mathbbm {1},v\right\rangle ,\\&\mathcal {R}(\langle X^{P_{1}},w^{u,v}\rangle +\left\langle X^{P_{2}},v\right\rangle )+C\Vert w^{u,v}\Vert _{q}\kappa _{n}\ge \mathcal {R}(\langle X^{P_{1}},w\rangle +\left\langle X^{P_{2}},v\right\rangle )+C\Vert w\Vert _{q}\kappa _{n}\}. \end{aligned} \end{aligned}$$

Since C and $\kappa _{n}$ are positive, $\mathcal {R}\left( \langle X^{P_{1}},\cdot \rangle +\left\langle X^{P_{2}},v\right\rangle \right)$ and $\Vert \cdot \Vert _{q}$ are convex functions of w, $A_{n}$ is a closed and convex set; and since $\Vert w\Vert _{q}-\Vert w^{u,v}\Vert _{q}>0$ for $w\ne w^{u,v}$ with $\langle \mathbbm {1},w\rangle =1-\left\langle \mathbbm {1},v\right\rangle$, $A_{n}$ is monotone decreasing and $\cap _{n=1}^{\infty }A_{n}=\left\{ w^{u,v}\right\}$. $A_{n}$ is bounded, since, $\mathcal {R}(\left\langle X^{P_{1}},w\right\rangle +\left\langle X^{P_{2}},v\right\rangle )$ is bounded from below (the nominal problem (5)–(6) is well-posed), and thus $\Vert w\Vert _{q}$ is bounded from above. Being closed and bounded, compactness of $A_{n}$ follows.

Let $B_{n}^{\epsilon }=A_{n}\cap \left\{ w\in {\mathbb {R}}^{N}:\Vert w-w^{u,v}\Vert _{q}\ge \epsilon \right\}$. $\cap _{n=1}^{\infty }B_{n}^{\epsilon }=\emptyset$, since $\cap _{n=1}^{\infty }A_{n}=\left\{ w^{u,v}\right\}$ and $w^{u,v}\notin B_{n}^{\epsilon },\;\forall n\in \mathbb {N}$. Compactness of $B_{n}^{\epsilon }$, $n\in \mathbb {N}$, implies that there is $N^{\epsilon }\in \mathbb {N}$ such that $B_{N^{\epsilon }}^{\epsilon }=\emptyset$, that is, the optimal solution is inside $\left\{ w\in {\mathbb {R}}^{N}:\Vert w-w^{u,v}\Vert _{q}<\epsilon \right\}$ for $\kappa \ge \kappa _{N^{\epsilon }}$. $\square$

Proof of Proposition 3

Case $p=1$

When $\kappa$ exceeds (14), $w^{u,v}$ is the optimal solution for the inner problem for all $v\in {\mathbb {R}}^{L}$, and we are seeking the solution of the problem (17). For some $s\ne 0$, let us fix the total allocation to ambiguous assets as s, i.e., $\langle \mathbbm {1},w^{u,v}\rangle =s$. The objective value for a solution $v\in {\mathbb {R}}^{L}$ with $\langle \mathbbm {1},v\rangle =1-s$ is $\mathcal {R}(\frac{s}{N}\left\langle X^{P_{1}},\mathbbm {1}\right\rangle +\left\langle X^{P_{2}},v\right\rangle )+C\kappa \frac{\left| s\right| }{N}$. Here, the term on the left remains constant as $\kappa \rightarrow \infty$, and the term on the right tends to infinity. Let $\tilde{v}\in {\mathbb {R}}^{L}$, $\left\langle \mathbbm {1},\tilde{v}\right\rangle =1$. The objective value of $\tilde{v}$ in (17) is $\mathcal {R}\left( \left\langle X^{P_{2}},\tilde{v}\right\rangle \right)$. $\tilde{v}$ has the same objective value in (17) for all $\kappa$, and when $\kappa$ exceeds $\frac{N}{C\left| s\right| }\left[ \mathcal {R}(\langle X^{P_{2}},\tilde{v}\rangle )-\mathcal {R}(\frac{s}{N}\left\langle X^{P_{1}},\mathbbm {1}\right\rangle +\left\langle X^{P_{2}},v\right\rangle )\right]$, $v\in {\mathbb {R}}^{L}$ with $\langle \mathbbm {1},v\rangle =1-s$ is suboptimal.

Case $p=2$

Again, for $\tilde{v}\in {\mathbb {R}}^{L}$ with $\langle \mathbbm {1},\tilde{v}\rangle =1$, the objective value in (16) is $\mathcal {R}(\langle X^{P_{2}},\tilde{v}\rangle )$. For $v\in {\mathbb {R}}^{L}$ with $\langle \mathbbm {1},v\rangle =1-s$, $s\ne 0$, the objective value is greater than

$$\begin{aligned} g\left( v\right)&=\mathcal {R}\left( \frac{s}{N}\left\langle X^{P_{1}},\mathbbm {1}\right\rangle +\left\langle X^{P_{2}},v\right\rangle \right) \\&\qquad +C\kappa \left| s\right| \left( \frac{1}{\sqrt{N}}-D\left( \frac{1}{\kappa }\sup _{v \in {\mathbb {R}}^L}\mathbb {E}\left( \Vert X^{P_{1}}\Vert _{2}^{2}\mathbbm {1}_{\{Z_{v}\ne 0\}}\right) {}^{\frac{1}{2}}+1\right) \right) . \end{aligned}$$

There exists $k\in \mathbb {N}$ such that for all $\kappa >k$, D can be picked small enough to assure $\frac{1}{\sqrt{N}}-D\left( \frac{1}{\kappa }\sup _{v\in {\mathbb {R}}^{L}}\mathbb {E}\left( \Vert X^{P_{1}}\Vert _{2}^{2}\mathbbm {1}_{\{Z_{v}\ne 0\}}\right) {}^{\frac{1}{2}}+1\right) >\delta$ for some $\delta >0$. Then, as in the previous case, $g\left( v\right)$ and hence the objective value of v tends to infinity as $\kappa \rightarrow \infty$ while that of $\tilde{v}$ stays constant.

Case $p\notin \left\{ 1,2\right\}$

When $p\notin \left\{ 1,2\right\}$, the threshold of uncertainty $\kappa _{\epsilon }$ for $w^{*,v}$ to be inside $\left\{ w\in {\mathbb {R}}^{N}:\Vert w-w^{u,v}\Vert _{q}<\epsilon \right\}$ is not determined by a function of $v\in {\mathbb {R}}^{L}$ or $\epsilon$, therefore, it is not possible to define a lower bound function such as $g\left( v\right)$ for $f^{*}\left( v\right)$, the objective value of $v\in {\mathbb {R}}^{L}$ (paired with $w^{*,v}$) in the outer problem (16). The situation is otherwise similar to the case $p=2$. Let us take arbitrary $v\in {\mathbb {R}}^{L}$ such that $\left\langle \mathbbm {1},v\right\rangle =1-s$, $s\ne 0$, and again, note that for $\tilde{v}\in {\mathbb {R}}^{L}$ with $\langle \mathbbm {1},\tilde{v}\rangle =1$, the objective value is constant as the parameter $\kappa$ increases. With similar calculations to the case $p=2$, one can show that the objective value $\mathcal {R}\left( \left\langle X^{P_{1}},w^{*,v}\right\rangle +\left\langle X^{P_{2}},v\right\rangle \right) +C\kappa \Vert w^{*,v}\Vert _{q}$ for v is larger than

$$\begin{aligned}&f\left( v\right) -\epsilon C\left( \sup _{v\in {\mathbb {R}}^{L}}\mathbb {E}\left[ \Vert X^{P_{1}}\Vert _{p}^{p}\mathbbm {1}_{\{Z_{v}\ne 0\}}\right] {}^{\frac{1}{p}}+\kappa \right) \\&\quad = \mathcal {R}\left( \frac{s}{N}\left\langle X^{P_{1}},\mathbbm {1}\right\rangle +\left\langle X^{P_{2}},v\right\rangle \right) +C\kappa \frac{\left| s\right| }{N^{1-\frac{1}{q}}}-\epsilon C\left( \sup _{v\in {\mathbb {R}}^{L}}\mathbb {E}\left[ \Vert X^{P_{1}}\Vert _{p}^{p}\mathbbm {1}_{\{Z_{v}\ne 0\}}\right] {}^{\frac{1}{p}}+\kappa \right) , \end{aligned}$$

given $\kappa \ge \kappa _{\epsilon }$. Picking $\epsilon <\frac{\left| s\right| }{N^{1-\frac{1}{q}}}$, the value of the above term increases once $\kappa$ exceeds $\kappa _{\epsilon }$ and tends to infinity with $\kappa \rightarrow \infty$. Thus, the arbitrary solution with total allocation $s\ne 0$ turns suboptimal as $\kappa \rightarrow \infty$. $\square$