Strongly consistent multivariate conditional risk measures

Abstract

We consider families of strongly consistent multivariate conditional risk measures. We show that under strong consistency these families admit a decomposition into a conditional aggregation function and a univariate conditional risk measure as introduced Hoffmann et al. (Stoch Process Appl 126(7):2014–2037, 2016). Further, in analogy to the univariate case in Föllmer (Stat Risk Model 31(1):79–103, 2014), we prove that under law-invariance strong consistency implies that multivariate conditional risk measures are necessarily multivariate conditional certainty equivalents.

Appendices

Auxiliary results

Note that the strict antitonicity of $\rho _\mathcal {G}$ implies that the inverse function $f_{\rho _\mathcal {G}}^{-1}$ in Definition 2.2 is well-defined. Indeed let $\beta \in {{\mathrm{Im\,}}}f_{\rho _\mathcal {G}}$ and $\alpha _1,\alpha _2\in L_{}^\infty (\mathcal {G})$ such that $f_{\rho _\mathcal {G}}(\alpha _1)=\beta =f_{\rho _\mathcal {G}}(\alpha _2)$. Suppose that $\mathbb {P}(A)>0$ where $A:=\{\alpha _1>\alpha _2\}\in \mathcal {G}$. Then by strict antitonicity and $\mathcal {G}$-locality we obtain that

$$\begin{aligned} \beta \mathbbm {1}_A+\rho _\mathcal {G}(\mathbf {0}_d)\mathbbm {1}_{A^C}&=\rho _\mathcal {G}(\alpha _1\mathbf {1}_d)\mathbbm {1}_A+\rho _\mathcal {G}(\mathbf {0}_d)\mathbbm {1}_{A^C}=\rho _\mathcal {G}(\alpha _1\mathbf {1}_d\mathbbm {1}_A)\\&\le \rho _\mathcal {G}(\alpha _2\mathbf {1}_d\mathbbm {1}_A)=\rho _\mathcal {G}(\alpha _2\mathbf {1}_d)\mathbbm {1}_A+\rho _\mathcal {G}(\mathbf {0}_d)\mathbbm {1}_{A^C}\\&=\beta \mathbbm {1}_A+\rho _\mathcal {G}(\mathbf {0}_d)\mathbbm {1}_{A^C}, \end{aligned}$$

and the inequality is strict with positive probability which is a contradiction. Thus we have that $\mathbb {P}(\alpha _1>\alpha _2)=0$. The same argument for $\{\alpha _1<\alpha _2\}$ yields $\alpha _1=\alpha _2$ $\mathbb {P}$-a.s.

Next we will show that properties of $\rho _\mathcal {G}$ transfer to $f_{\rho _\mathcal {G}}$ and $f^{-1}_{\rho _\mathcal {G}}$. Since the domain of $f_{\rho _\mathcal {G}}^{-1}$ might be only a subset of $L_{}^\infty (\mathcal {G})$, we need to adapt the definition of the Lebesgue property for $f^{-1}_{\rho _\mathcal {G}}$ in the following way: If $(\beta _n)_{n\in \mathbb {N}}\subset {{\mathrm{Im\,}}}f_{\rho _\mathcal {G}}$ is a sequence which is lower- and upper-bounded by some $\underline{\beta },\overline{\beta }\in {{\mathrm{Im\,}}}f_{\rho _\mathcal {G}}$, i.e. $\underline{\beta }\le \beta _n\le \overline{\beta }$ for all $n\in \mathbb {N}$, and such that $\beta _n\rightarrow \beta$ $\mathbb {P}$-a.s., then $f_{\rho _\mathcal {G}}^{-1}(\beta _n)\rightarrow f_{\rho _\mathcal {G}}^{-1}(\beta )$ $\mathbb {P}$-a.s. Note that this alternative definition of the Lebesgue property is equivalent to Definition 2.1 (iv) if the domain is $L_{}^\infty (\mathcal {G})$. The properties ’strict antitonicity’ and ’locality’ of $f_{\rho _\mathcal {G}}$ or $f^{-1}_{\rho _\mathcal {G}}$ are defined analogous to Definition 2.1 (ii) and (iii).

Lemma A.1

Let $f_{\rho _\mathcal {G}}$ and $f^{-1}_{\rho _\mathcal {G}}$ be as in Definition 2.2. Then $f_{\rho _\mathcal {G}}$ and $f^{-1}_{\rho _\mathcal {G}}$ are strictly antitone, $\mathcal {G}$-local and fulfill the Lebesgue property.

Proof

For $f_{\rho _\mathcal {G}}$ the statement follows immediately from the definition and the corresponding properties of $\rho _\mathcal {G}$. Concerning the properties of $f^{-1}_{\rho _\mathcal {G}}$, we start by proving strict antitonicity. Let $\beta _1,\beta _2\in {{\mathrm{Im\,}}}f_{\rho _\mathcal {G}}$ such that $\beta _1\ge \beta _2$ and $\mathbb {P}(\beta _1>\beta _2)>0$. Suppose that $\mathbb {P}(A)>0$ where $A:=\left\{ f^{-1}_{\rho _\mathcal {G}}(\beta _1)>f^{-1}_{\rho _\mathcal {G}}(\beta _2)\right\} \in \mathcal {G}$. Then

$$\begin{aligned} \beta _1\mathbbm {1}_A+f_{\rho _\mathcal {G}}(0)\mathbbm {1}_{A^C}&=f_{\rho _\mathcal {G}}\left( f^{-1}_{\rho _\mathcal {G}}(\beta _1)\right) \mathbbm {1}_A+f_{\rho _\mathcal {G}}(0)\mathbbm {1}_{A^C}=f_{\rho _\mathcal {G}}\left( f^{-1}_{\rho _\mathcal {G}}(\beta _1)\mathbbm {1}_A\right) \\&\le f_{\rho _\mathcal {G}}\left( f^{-1}_{\rho _\mathcal {G}}(\beta _2)\mathbbm {1}_A\right) =\beta _2\mathbbm {1}_A+f_{\rho _\mathcal {G}}(0)\mathbbm {1}_{A^C}, \end{aligned}$$

and the inequality is strict on a set with positive probability since $f_{\rho _\mathcal {G}}$ is strictly antitone. This of course contradicts $\beta _1\ge \beta _2$. Hence $f^{-1}_{\rho _\mathcal {G}}(\beta _1)\le f^{-1}_{\rho _\mathcal {G}}(\beta _2)$. Moreover, as

$$\begin{aligned} \mathbb {P}(\beta _1>\beta _2)=\mathbb {P}\left( f_{\rho _\mathcal {G}}\left( f^{-1}_{\rho _\mathcal {G}}(\beta _1)\right)>f_{\rho _\mathcal {G}}\left( f^{-1}_{\rho _\mathcal {G}}(\beta _2)\right) \right) >0 \end{aligned}$$

we must have $f^{-1}_{\rho _\mathcal {G}}(\beta _1)\ne f^{-1}_{\rho _\mathcal {G}}(\beta _2)$ with positive probability, i.e.

$$\begin{aligned} \mathbb {P}\left( f^{-1}_{\rho _\mathcal {G}}(\beta _1)<f^{-1}_{\rho _\mathcal {G}}(\beta _2)\right) >0. \end{aligned}$$

Now we show that $f^{-1}_{\rho _\mathcal {G}}$ is $\mathcal {G}$-local. Let $\beta _1,\beta _2\in {{\mathrm{Im\,}}}f_{\rho _\mathcal {G}}$ as well as $A\in \mathcal {G}$ be arbitrary. Further let $\alpha _i=f^{-1}_{\rho _\mathcal {G}}(\beta _i),i=1,2$, i.e. $f_{\rho _\mathcal {G}}(\alpha _i)=\beta _i$. Then we have that

$$\begin{aligned} f_{\rho _\mathcal {G}}(\alpha _1\mathbbm {1}_A+\alpha _2\mathbbm {1}_{A^C})=f_{\rho _\mathcal {G}}(\alpha _1)\mathbbm {1}_A+f_{\rho _\mathcal {G}}(\alpha _2)\mathbbm {1}_{A^C}=\beta _1\mathbbm {1}_A+\beta _2\mathbbm {1}_{A^C}. \end{aligned}$$

Thus $f^{-1}_{\rho _\mathcal {G}}(\beta _1\mathbbm {1}_A+\beta _2\mathbbm {1}_{A^C})=\alpha _1\mathbbm {1}_A+\alpha _2\mathbbm {1}_{A^C}.$

Finally for the Lebesgue property let $\underline{\beta },\overline{\beta }\in {{\mathrm{Im\,}}}f_{\rho _\mathcal {G}}$ and let $(\beta _n)_{n\in \mathbb {N}}\subset {{\mathrm{Im\,}}}f_{\rho _\mathcal {G}}$ be a sequence with $\underline{\beta }\le \beta _n\le \overline{\beta }$ for all $n\in \mathbb {N}$ and $\beta _n\rightarrow \beta$ $\mathbb {P}$-a.s. Consider the bounded sequences $\beta ^u_n:=\sup _{k\ge n}\beta _k$ and $\beta ^d_n:=\inf _{k\ge n}\beta _k$, $n\in \mathbb {N}$ which converge monotonically almost surely to $\beta$, i.e. $\beta ^u_n\downarrow \beta$ $\mathbb {P}$-a.s. and $\beta ^d_n\uparrow \beta$ $\mathbb {P}$-a.s. Since $\underline{\beta }\le \beta _n^u\le \overline{\beta }$ for all $n\in \mathbb {N}$ which by antitonicity of $f^{-1}_{\rho _\mathcal {G}}$ yields $f^{-1}_{\rho _\mathcal {G}}(\overline{\beta })\le f^{-1}_{\rho _\mathcal {G}}(\beta _n^u) \le f^{-1}_{\rho _\mathcal {G}}(\underline{\beta })$, we observe that the sequence $\left( f^{-1}_{\rho _\mathcal {G}}(\beta _n^u)\right) _{n\in \mathbb {N}}$ is uniformly bounded in $L_{}^\infty (\mathcal {G})$. Note that by the same argumentation also the sequences $\left( f^{-1}_{\rho _\mathcal {G}}(\beta _n^d)\right) _{n\in \mathbb {N}}$ and $\left( f^{-1}_{\rho _\mathcal {G}}(\beta _n)\right) _{n\in \mathbb {N}}$ are uniformly bounded in $L_{}^\infty (\mathcal {G})$. Next we will show that $\beta ^u_n\in {{\mathrm{Im\,}}}f_{\rho _\mathcal {G}}$ for all $n\in \mathbb {N}$. Fix $n\in \mathbb {N}$ and set recursively

$$\begin{aligned} A_{n-1}^n:=\{\beta ^u_n=\beta \}\quad \text {and}\quad A_{k}^n:=\{\beta ^u_n=\beta _k\}\backslash \bigcup _{i=n-1}^{k-1}A_{i}^n,\;k\ge n, \end{aligned}$$

then it follows from induction that $A_k^n\in \mathcal {G},k\ge {n-1}$. Since $\sup \left\{ \beta ,\beta _k:k\ge n \right\} =\max \left\{ \beta ,\beta _k:k\ge n \right\}$, we have that $\left( \bigcup _{k\ge n-1}A_k^n\right) ^C$ is a $\mathbb {P}$-nullset. It follows from $\mathcal {G}$-locality and the Lebesgue property of $f_{\rho _\mathcal {G}}$ that

$$\begin{aligned}&f_{\rho _\mathcal {G}}\left( f^{-1}_{\rho _\mathcal {G}}(\beta )\mathbbm {1}_{A_{n-1}^n}+\sum _{k\ge n}f^{-1}_{\rho _\mathcal {G}}(\beta _k)\mathbbm {1}_{A_k^n}\right) \\&\quad =\beta \mathbbm {1}_{A_{n-1}^n}+f_{\rho _\mathcal {G}}\left( \lim _{m\rightarrow \infty }\sum _{k=n}^m f_{\rho _\mathcal {G}}^{-1}(\beta _k)\mathbbm {1}_{A_k^n}\right) \mathbbm {1}_{\bigcup _{k\ge n}A_{k}^n}\\&\quad =\beta \mathbbm {1}_{A_{n-1}^n}+\lim _{m\rightarrow \infty }\left( \sum _{k=n}^m\beta _k\mathbbm {1}_{A_{k}^n}+f_{\rho _\mathcal {G}}\left( 0\right) \mathbbm {1}_{\bigcup _{k\ge m}A_k^n}\right) \\&\quad =\beta \mathbbm {1}_{A_{n-1}^n}+\sum _{k\ge n}\beta _k\mathbbm {1}_{A_{k}^n}=\beta ^u_n, \end{aligned}$$

which implies $\beta _n^u\in {{\mathrm{Im\,}}}f_{\rho _\mathcal {G}}$. By a similar argumentation we obtain $\beta _n^d\in {{\mathrm{Im\,}}}f_{\rho _\mathcal {G}}$. Recall that $\beta _n^u\downarrow \beta$ $\mathbb {P}$-a.s. which by antitonicity of $f_{\rho _\mathcal {G}}^{-1}$ implies that the sequence $\left( f^{-1}_{\rho _\mathcal {G}}(\beta _n^u)\right) _{n\in \mathbb {N}}$ is isotone and thus $\alpha =\lim _{n\rightarrow \infty }f_{\rho _\mathcal {G}}^{-1}(\beta _n^u)$ exists in $L^\infty (\mathcal {G})$. It follows from antitonicity and the Lebesgue property of $f_{\rho _\mathcal {G}}$ that

$$\begin{aligned} \beta =\lim _{n\rightarrow \infty }\beta _n^u=\lim _{n\rightarrow \infty }f_{\rho _\mathcal {G}}\left( f_{\rho _\mathcal {G}}^{-1}(\beta _n^u)\right) =f_{\rho _\mathcal {G}}(\alpha ), \end{aligned}$$

and hence that indeed $\alpha =f_{\rho _\mathcal {G}}^{-1}(\beta )$. Analogously, we obtain that $f_{\rho _\mathcal {G}}(\hat{\alpha })=\beta$ for $\hat{\alpha }=\lim _{n\rightarrow \infty }f_{\rho _\mathcal {G}}^{-1}(\beta _n^d)$, and thus $\hat{\alpha }=\alpha =f_{\rho _\mathcal {G}}^{-1}(\beta )$. Hence, by antitonicity of $f^{-1}_{\rho _\mathcal {G}}$

$$\begin{aligned} f_{\rho _\mathcal {G}}^{-1}(\beta )&=\lim _{n\rightarrow \infty }f_{\rho _\mathcal {G}}^{-1}(\beta ^u_n)\le \liminf _{n\rightarrow \infty }f_{\rho _\mathcal {G}}^{-1}(\beta _n)\\&\le \limsup _{n\rightarrow \infty }f_{\rho _\mathcal {G}}^{-1}(\beta _n)\le \lim _{n\rightarrow \infty }f_{\rho _\mathcal {G}}^{-1}(\beta ^d_n)= f_{\rho _\mathcal {G}}^{-1}(\beta ), \end{aligned}$$

so $\lim _{n\rightarrow \infty }f_{\rho _\mathcal {G}}^{-1}(\beta _n)= f_{\rho _\mathcal {G}}^{-1}(\beta )$, i.e. $f^{-1}_{\rho _\mathcal {G}}$ has the Lebesgue property. $\square$

An important observation is that the domain of $f^{-1}_{\rho _\mathcal {G}}$ is equal to the image of $\rho _\mathcal {G}$, i.e. $f^{-1}_{\rho _\mathcal {G}}(\rho _\mathcal {G}(X))$ is well-defined for all $X\in L_{d}^\infty (\mathcal {F})$.

Lemma A.2

For a CRM $\rho _\mathcal {G}:L_{d}^\infty (\mathcal {F})\rightarrow L_{}^\infty (\mathcal {G})$ it holds that

$$\begin{aligned} \rho _\mathcal {G}(L_{d}^\infty (\mathcal {F}))= f_{\rho _\mathcal {G}}(L_{}^\infty (\mathcal {G})). \end{aligned}$$

Proof

Clearly, $\rho _\mathcal {G}(L_{d}^\infty (\mathcal {F}))\supseteq f_{\rho _\mathcal {G}}(L_{}^\infty (\mathcal {G}))$.

For the reverse inclusion let $X\in L_{d}^\infty (\mathcal {F})$. Our aim is to show that there exists an $\alpha ^*\in L_{}^\infty (\mathcal {G})$ such that

$$\begin{aligned} \rho _\mathcal {G}(X)= f_{\rho _\mathcal {G}}(\alpha ^*). \end{aligned}$$

(A.1)

Define

$$\begin{aligned} P:=\left\{ \alpha \in L_{}^\infty (\mathcal {G})\,: f_{\rho _\mathcal {G}}(\alpha ) \ge \rho _\mathcal {G}(X) \right\} . \end{aligned}$$

As $-\Vert X\Vert _{d,\infty }\mathbf {1}_d\le X\le \Vert X\Vert _{d,\infty }\mathbf {1}_d$ we have that $-\Vert X\Vert _{d,\infty }\in P$, so $P\ne \emptyset$. Moreover, P is bounded from above by $\Vert X\Vert _{d,\infty }$ since if $A:=\{\alpha >\Vert X\Vert _{d,\infty }\}$ for $\alpha \in L_{}^\infty (\mathcal {G})$ has positive probability, then by $\mathcal {G}$-locality and strict antitonicity

$$\begin{aligned} f_{\rho _\mathcal {G}}(\alpha )\mathbbm {1}_A= f_{\rho _\mathcal {G}}(\alpha \mathbbm {1}_A)\mathbbm {1}_A\le f_{\rho _\mathcal {G}}(\Vert X\Vert _{d,\infty }\mathbbm {1}_A)\mathbbm {1}_A=f_{\rho _\mathcal {G}}(\Vert X\Vert _{d,\infty })\mathbbm {1}_A\le \rho _\mathcal {G}(X)\mathbbm {1}_A \end{aligned}$$

where the first inequality is strict with positive probability, so $\alpha \not \in P$. By $\mathcal {G}$-locality it also follows that P is upwards directed. Hence, for

$$\begin{aligned} \alpha ^{*}{:=}{{\mathrm{esssup\,}}}P \end{aligned}$$

there is a uniformly bounded sequence $(\alpha _n)_{n\in \mathbb {N}}\subset P$ such that $\alpha ^*=\lim _{n\rightarrow \infty }\alpha _n$ $\mathbb {P}$-a.s.; see Föllmer and Schied [15, Theorem A.33]. Thus it follows that $\alpha ^*\in L_{}^\infty (\mathcal {G})$ and

$$\begin{aligned} f_{\rho _\mathcal {G}}(\alpha ^*)=\lim _{n\rightarrow \infty }f_{\rho _\mathcal {G}}(\alpha _n)\ge \rho _\mathcal {G}(X), \end{aligned}$$

i.e. $\alpha ^*\in P$. Let

$$\begin{aligned} B:=\{f_{\rho _\mathcal {G}}(\alpha ^*)>\rho _\mathcal {G}(X)\} \end{aligned}$$

and note that by the Lebesgue property

$$\begin{aligned} B=\bigcup _{n\in \mathbb {N}}\{f_{\rho _\mathcal {G}}(\alpha ^*+1/n)>\rho _\mathcal {G}(X)\}\quad \mathbb {P}\text{-a.s. } \end{aligned}$$

Hence, if $\mathbb {P}(B)>0$ it follows that $\mathbb {P}(B_n)>0$ for some $B_n:=\{f_{\rho _\mathcal {G}}(\alpha ^*+1/n)>\rho _\mathcal {G}(X)\}$. Note that $B_n\in \mathcal {G}$ and that

$$\begin{aligned} \alpha ^*\mathbbm {1}_{B_n^C}+(\alpha ^*+1/n)\mathbbm {1}_{B_n}\in P \end{aligned}$$

by $\mathcal {G}$-locality of $f_{\rho _\mathcal {G}}$. But this contradicts the definition of $\alpha ^*$. Hence, $\mathbb {P}(B)=0$. $\square$

Lemma A.3

Let $\Lambda :L_{d}^\infty (\mathcal {F})\rightarrow L_{}^\infty (\mathcal {F})$ be a conditional aggregation function. Then $f_{\Lambda }$ and $f_{\Lambda }^{-1}$ are strictly isotone, $\mathcal {F}$-local, and fulfill the Lebesgue property. Moreover, $\Lambda (L_{d}^\infty (\mathcal {F}))=f_{\Lambda }(L_{}^\infty (\mathcal {F}))$ and $\Lambda (X)=\Lambda \big (f_{\Lambda }^{-1}(\Lambda (X))\mathbf {1}_d\big )$ for all $X\in L_{d}^\infty (\mathcal {F})$.

The well-definedness of $f_{\Lambda }^{-1}$ follows similarly to the well-definedness of $f_{\rho _\mathcal {G}}^{-1}$. Further the proof of Lemma A.3 is analogous to the proofs of Lemma A.1 and Lemma A.2 and therefore omitted here.

Proof of Theorem 3.9

The Proof of Theorem 3.9 is based on a result from Hoffmann et al. [17] which we in the following present in a version adapted to the framework of this paper.

Proposition B.1

Let $\rho _\mathcal {G}: L_{d}^\infty (\mathcal {F})\rightarrow L_{}^\infty (\mathcal {G})$ be a CRM and suppose that there exists a continuous realization $\rho _\mathcal {G}(\cdot ,\cdot )$ which satisfies risk-antitonicity:

$$\begin{aligned} \rho _\mathcal {G}(X(\omega ),\omega )\ge \rho _\mathcal {G}(Y(\omega ),\omega )\,\mathbb {P} \text{-a.s., } \text{ implies } \rho _\mathcal {G}(X)\ge \rho _\mathcal {G}(Y). \end{aligned}$$

Then there exists a $\mathcal {G}$-conditional aggregation function $\Lambda _\mathcal {G}:L_{d}^\infty (\mathcal {F})\rightarrow L_{}^\infty (\mathcal {F})$ and a univariate CRM $\eta _\mathcal {G}: {{\mathrm{Im\,}}}\Lambda _\mathcal {G}\rightarrow L_{}^\infty (\mathcal {G})$ such that

$$\begin{aligned} \rho _\mathcal {G}\left( X\right) =\eta _\mathcal {G}\left( \Lambda _\mathcal {G}(X)\right) \quad \text {for all }X\in L_{d}^\infty (\mathcal {F}) \end{aligned}$$

and

$$\begin{aligned} \eta _\mathcal {G}\left( \Lambda _\mathcal {G}(X)\right) =-\Lambda _\mathcal {G}(X)\quad \text{ for } \text{ all } X\in L_{d}^\infty (\mathcal {G}). \end{aligned}$$

(B.1)

This decomposition is unique.

Proof

Since $\rho _\mathcal {G}$ is antitone, ${\mathbb {R}}^d\ni x\mapsto \rho _\mathcal {G}(x)$ is antitone. It has been shown in Hoffmann et al. [17, Theorem 2.10] that this property in conjunction with the fact that $\rho _\mathcal {G}$ has a continuous realization which fulfills risk-antitonicity is sufficient for the existence and uniqueness of a function $\Lambda _\mathcal {G}:L_{d}^\infty (\mathcal {F})\rightarrow L_{}^\infty (\mathcal {F})$ which is isotone, $\mathcal {F}$-local and fulfills the Lebesgue property and a function $\eta _\mathcal {G}: {{\mathrm{Im\,}}}\Lambda _\mathcal {G}\rightarrow L_{}^\infty (\mathcal {G})$ which is antitone such that

$$\begin{aligned} \rho _\mathcal {G}=\eta _\mathcal {G}\circ \Lambda _\mathcal {G}\quad \text{ and }\quad \eta _\mathcal {G}\big (\Lambda _\mathcal {G}(x)\big )=-\Lambda _\mathcal {G}(x)\;\text{ for } \text{ all } x\in {\mathbb {R}}^d. \end{aligned}$$

(B.2)

Note that in the proof of Theorem 2.10 in Hoffmann et al. [17] $\Lambda _\mathcal {G}$ is basically constructed by setting $\Lambda _\mathcal {G}(X)(\omega )=-\rho _\mathcal {G}(X(\omega ),\omega )$, which implies that $\Lambda _\mathcal {G}$ is necessarily $\mathcal {F}$-local even though this is not directly mentioned in the paper. Indeed in Hoffmann et al. [17] we do not require or mention locality at all.

It remains to be shown that $\Lambda _\mathcal {G}$ is a $\mathcal {G}$-conditional aggregation function, $\eta _\mathcal {G}$ is a univariate CRM on ${{\mathrm{Im\,}}}\Lambda _\mathcal {G}$, and that (B.1) holds. First of all, we show that $\mathcal {F}$-locality and (B.2) imply (B.1). To this end denote by $\mathcal {S}$ the set of $\mathcal {F}$-measurable simple random vectors, i.e. $X\in \mathcal {S}$ if X is of the form $X=\sum _{i=1}^k x_i\mathbbm {1}_{A_i}$, where $k\in \mathbb {N}$, $x_i\in {\mathbb {R}}^d$ and $A_i\in \mathcal {F}$, $i=1,\ldots ,k$, are disjoint sets such that $\mathbb {P}(A_i)>0$ and $\mathbb {P}(\bigcup _{i=1}^k A_i)=1$. Now let $X\in L_{d}^\infty (\mathcal {G})$. Pick a uniformly bounded sequence $(X_n)_{n\in \mathbb {N}}=\left( \sum _{i=1}^{k_n} x^n_i\mathbbm {1}_{A^n_i}\right) _{n\in \mathbb {N}}\subset \mathcal {S}$ such that $A^n_i\in \mathcal {G}$ for all $i=1,\ldots , k_n$, $n\in \mathbb {N}$, and $X_n\rightarrow X$ $\mathbb {P}$-a.s. Then by (B.2), $\mathcal {F}$-locality and the Lebesgue property of $\Lambda _\mathcal {G}$ and $\rho _\mathcal {G}$ we infer that

$$\begin{aligned} -\Lambda _\mathcal {G}(X)= & {} -\lim _{n\rightarrow \infty }\Lambda _\mathcal {G}(X_n)\;=\;\lim _{n\rightarrow \infty }\sum _{i=1}^{k_n}-\Lambda _\mathcal {G}(x_i^n)\mathbbm {1}_{A_i^n} \\= & {} \lim _{n\rightarrow \infty }\sum _{i=1}^{k_n}\rho _\mathcal {G}(x_i^n)\mathbbm {1}_{A_i^n}\; =\; \lim _{n\rightarrow \infty } \rho _\mathcal {G}(X_n) \; =\; \rho _\mathcal {G}(X), \end{aligned}$$

which proves (B.1). Next we show that $\Lambda _\mathcal {G}$ is a $\mathcal {G}$-conditional aggregation function. The yet missing properties which need to be verified are strict antitonicity and that $\Lambda _\mathcal {G}$ is $\mathcal {G}$-conditional. The latter follows from Hoffmann et al. [17, Lemma 3.1]. As for strict antitonicity let $X,Y\in L_{d}^\infty (\mathcal {F})$ with $X\ge Y$ such that $\mathbb {P}(X>Y)>0$. Then by isotonicity of $\Lambda _\mathcal {G}$ we have that $\Lambda _\mathcal {G}(X)\ge \Lambda _\mathcal {G}(Y)$. Suppose that $\Lambda _\mathcal {G}(X)=\Lambda _\mathcal {G}(Y)$ $\mathbb {P}$-a.s., then

$$\begin{aligned} \rho _\mathcal {G}(X)=\eta _\mathcal {G}(\Lambda _\mathcal {G}(X))= \eta _\mathcal {G}(\Lambda _\mathcal {G}(Y))=\rho _\mathcal {G}(Y) \end{aligned}$$

which contradicts strict antitonicity of $\rho _\mathcal {G}$. Thus $\Lambda _\mathcal {G}$ fulfills all properties of a $\mathcal {G}$-conditional aggregation function.

As for $\eta _\mathcal {G}$, note that by Lemma A.3 for all $F\in {{\mathrm{Im\,}}}\Lambda _\mathcal {G}$ we have that

$$\begin{aligned} \eta _\mathcal {G}(F)=\eta _\mathcal {G}\big (\Lambda _\mathcal {G}\big (f^{-1}_{\Lambda _\mathcal {G}}(F)\mathbf {1}_d\big )\big )=\rho _\mathcal {G}\big (f_{\Lambda _\mathcal {G}}^{-1}(F)\mathbf {1}_d\big ). \end{aligned}$$

(B.3)

Since $\rho _\mathcal {G}$ and $f^{-1}_{\Lambda _\mathcal {G}}$ are strictly monotone, $\mathcal {G}$-local, and fulfill the Lebesgue property, so does $\eta _\mathcal {G}$, i.e. $\eta _\mathcal {G}$ is a univariate CRM on ${{\mathrm{Im\,}}}\Lambda _\mathcal {G}$. $\square$

The proof of Theorem 3.9 is now based on the following observations: $\rho _\mathcal {F}$ is necessarily risk-antitone as defined in Proposition B.1. Strong consistency in turn implies that risk-antitonicity of $\rho _\mathcal {F}$ is passed on (backwards) to $\rho _\mathcal {G}$, and hence Proposition B.1 applies.

Proof of Theorem 3.9

In case we already know that (3.4) holds, then by antitonicity of $\eta _\mathcal {G}$ it follows that $\{\rho _\mathcal {G}, -\Lambda _\mathcal {G}\}$ is strongly consistent, and clearly $-\Lambda _\mathcal {G}:L_{d}^\infty (\mathcal {F})\rightarrow L_{}^\infty (\mathcal {F})$ is also a CRM. Thus the last assertion of Theorem 3.9 is proved.

In order to show the first part of Theorem 3.9, we recall that the only property which remains to be shown in order to apply Proposition B.1 is risk-antitonicity of $\rho _\mathcal {G}$: For this purpose we first consider simple random vectors $X,Y\in \mathcal {S}$ where $\mathcal {S}$ was defined in the proof of Proposition B.1. Note that there is no loss of generality by assuming that $X=\sum _{i=1}^n x_i\mathbbm {1}_{A_i}\in \mathcal {S}$ and $Y=\sum _{i=1}^n y_i\mathbbm {1}_{A_i}\in \mathcal {S}$, i.e. the partition $(A_i)_{i=1,\ldots ,n}$ of $\Omega$ is the same for X and Y. Suppose that $\rho _\mathcal {G}(X(\omega ),\omega )\ge \rho _\mathcal {G}(Y(\omega ),\omega )$ $\mathbb {P}$-a.s. It follows that $\rho _\mathcal {G}(x_i,\omega )\ge \rho _\mathcal {G}(y_i,\omega )$ for all $\omega \in A_i\backslash N,i=1,\ldots ,n,$ where N is a $\mathbb {P}$-nullset. Let $B_i:=\{\omega \in \Omega \mid \rho _\mathcal {G}(x_i,\omega )\ge \rho _\mathcal {G}(y_i,\omega )\}\in \mathcal {G}$. As $(A_i\setminus N)\subseteq B_i$, using antitonicity and $\mathcal {G}$-locality of $f_{\rho _\mathcal {G}}^{-1}$ we obtain

$$\begin{aligned} f^{-1}_{\rho _\mathcal {G}}\big (\rho _\mathcal {G}(x_i)\big )\mathbbm {1}_{A_i}&=f^{-1}_{\rho _\mathcal {G}}\big (\rho _\mathcal {G}(x_i)\mathbbm {1}_{B_i}\big )\mathbbm {1}_{A_i} \le f^{-1}_{\rho _\mathcal {G}}\big (\rho _\mathcal {G}(y_i)\mathbbm {1}_{B_i}\big )\mathbbm {1}_{A_i} =f^{-1}_{\rho _\mathcal {G}}\big (\rho _\mathcal {G}(y_i)\big )\mathbbm {1}_{A_i}. \end{aligned}$$

Now by strong consistency of $\{\rho _\mathcal {G},\rho _\mathcal {F}\}$, $\mathcal {F}$-locality of $\rho _\mathcal {F}$ and $f^{-1}_{\rho _\mathcal {F}}$, and by (3.3) as well as antitonicity of $\rho _\mathcal {G}$ we arrive at

$$\begin{aligned} \rho _\mathcal {G}(X)= & {} \rho _\mathcal {G}\left( f_{\rho _\mathcal {F}}^{-1}\big (\rho _\mathcal {F}(X)\big )\mathbf {1}_d\right) \quad =\quad \rho _\mathcal {G}\left( \sum _{i=1}^n f_{\rho _\mathcal {F}}^{-1}\big (\rho _\mathcal {F}(x_i)\big )\mathbbm {1}_{A_i}\mathbf {1}_d\right) \\= & {} \rho _\mathcal {G}\left( \sum _{i=1}^n f_{\rho _\mathcal {G}}^{-1}\big (\rho _\mathcal {G}(x_i)\big )\mathbbm {1}_{A_i}\mathbf {1}_d\right) \quad \ge \quad \rho _\mathcal {G}\left( \sum _{i=1}^n f_{\rho _\mathcal {G}}^{-1}\big (\rho _\mathcal {G}(y_i)\big )\mathbbm {1}_{A_i}\mathbf {1}_d\right) \\= & {} \rho _\mathcal {G}(Y), \end{aligned}$$

which proves risk-antitonicity for simple random vectors $X,Y\in \mathcal {S}$. For general $X,Y\in L_{d}^\infty (\mathcal {F})$ with $\rho _\mathcal {G}(X(\omega ),\omega )\ge \rho _\mathcal {G}(Y(\omega ),\omega )$ for $\mathbb {P}$-a.e. $\omega \in \Omega$ we can find uniformly bounded sequences $(X_n)_{n\in \mathbb {N}},(Y_n)_{n\in \mathbb {N}}\subset \mathcal {S}$ such that $X_n\nearrow X$ and $Y_n\searrow Y$ $\mathbb {P}$-a.s. for $n\rightarrow \infty$. Then by antitonicity

$$\begin{aligned} \rho _\mathcal {G}(X_n(\omega ),\omega )\ge \rho _\mathcal {G}(X(\omega ),\omega )\ge \rho _\mathcal {G}(Y(\omega ),\omega )\ge \rho _\mathcal {G}(Y_n(\omega ),\omega )\;\mathbb {P}\text {-a.s.} \end{aligned}$$

Therefore, $\rho _\mathcal {G}(X_n)\ge \rho _\mathcal {G}(Y_n)$ and the Lebegue property of $\rho _\mathcal {G}$ yields

$$\begin{aligned} \rho _\mathcal {G}(X)=\lim _{n\rightarrow \infty }\rho _\mathcal {G}(X_n)\ge \lim _{n\rightarrow \infty }\rho _\mathcal {G}(Y_n)=\rho _\mathcal {G}(Y). \end{aligned}$$

Thus $\rho _\mathcal {G}$ is risk-antitone and we apply Proposition B.1. Hence, there is a $\mathcal {G}$-conditional aggregation function $\Lambda _\mathcal {G}:L_{d}^\infty (\mathcal {F})\rightarrow L_{}^\infty (\mathcal {F})$ and a univariate CRM $\eta _\mathcal {G}: {{\mathrm{Im\,}}}\Lambda _\mathcal {G}\rightarrow L_{}^\infty (\mathcal {G})$ such that $\rho _\mathcal {G}=\eta _\mathcal {G}\circ \Lambda _\mathcal {G}$ and $\eta _\mathcal {G}\big (\Lambda _\mathcal {G}(X)\big )=-\Lambda _\mathcal {G}(X)$ for all $X\in L_{d}^\infty (\mathcal {G})$.

Using locality it follows that (3.3) indeed holds for all $\alpha \in L_{d}^\infty (\mathcal {G})\cap \mathcal {S}$ and thus by continuity $\bar{\rho }_\mathcal {F}(\alpha )=\bar{\rho }_\mathcal {G}(\alpha )\in L_{}^\infty (\mathcal {G})$ for all $\alpha \in L_{d}^\infty (\mathcal {G})$. Thus also $\Lambda _\mathcal {F}(\alpha )=-\rho _\mathcal {F}(\alpha )\in L_{}^\infty (\mathcal {G})$ for all $\alpha \in L_{d}^\infty (\mathcal {G})$. Finally by the same procedure as above, i.e. approximation via elements in $\mathcal {S}$, using locality, strong consistency, and continuity, we obtain (3.6). $\square$

Proof of Theorem 4.4

Lemma C.1

Let $\{\rho ,\rho _\mathcal {H}\}$ be strongly consistent and suppose that $\rho$ is law-invariant (and thus $\rho _\mathcal {H}$ is conditionally law-invariant by Lemma 4.3). If $(\Omega ,\mathcal {H},\mathbb {P})$ is an atomless probability space and $X\in L_{d}^\infty (\mathcal {F})$ is independent of $\mathcal {H}$, then

$$\begin{aligned} f_{\rho _\mathcal {H}}^{-1}\big (\rho _\mathcal {H}(X)\big )=f_{\rho }^{-1}\big (\rho (X)\big ). \end{aligned}$$

The proof of Lemma C.1 is adapted from Kupper and Schachermayer [20].

Proof

We distinguish three cases:

• Suppose that $f_{\rho _{\mathcal {H}}}^{-1}\big (\rho _{\mathcal {H}}(X)\big )\le f_{\rho }^{-1}\big (\rho (X)\big )$ and strictly smaller with positive probability. Then by strong consistency

  $$\begin{aligned} f_{\rho }^{-1}\big (\rho (X)\big )&=f_{\rho }^{-1}\left( \rho \left( f_{\rho _{\mathcal {H}}}^{-1}\big (\rho _{\mathcal {H}}(X)\big )\mathbf {1}_d\right) \right) \\&<f_{\rho }^{-1}\left( \rho \left( f_{\rho }^{-1}\big (\rho (X)\big )\mathbf {1}_d\right) \right) =f_{\rho }^{-1}\big (\rho (X)\big ), \end{aligned}$$

  by strict antitonicity of $\rho$ which is a contradiction.

• Analogously it follows that it is not possible that $f_{\rho _{\mathcal {H}}}^{-1}\big (\rho _{\mathcal {H}}(X)\big )\ge f_{\rho }^{-1}\big (\rho (X)\big )$ and $\mathbb {P}(f_{\rho _{\mathcal {H}}}^{-1}\big (\rho _{\mathcal {H}}(X)\big )> f_{\rho }^{-1}\big (\rho (X)\big ))>0$.

• There exist $A,B\in \mathcal {H}$ such that $\mathbb {P}(A)=\mathbb {P}(B)>0$ and

  $$\begin{aligned} f_{\rho _{\mathcal {H}}}^{-1}\big (\rho _{\mathcal {H}}(X)\big )>f_{\rho }^{-1}\big (\rho (X)\big )\text { on }A\text { and }f_{\rho _{\mathcal {H}}}^{-1}\big (\rho _{\mathcal {H}}(X)\big )<f_{\rho }^{-1}\big (\rho (X)\big )\text { on }B. \end{aligned}$$

  Then we have for an arbitrary $m=a\mathbf {1}_d$ where $a\in {\mathbb {R}}$ that

  $$\begin{aligned} \rho (X\mathbbm {1}_A+m\mathbbm {1}_{A^C})&=\rho \left( f_{\rho _{\mathcal {H}}}^{-1}\big (\rho _{\mathcal {H}}(X\mathbbm {1}_A+m\mathbbm {1}_{A^C})\big )\mathbf {1}_d\right) \nonumber \\&=\rho \left( f_{\rho _{\mathcal {H}}}^{-1}\big (\rho _{\mathcal {H}}(X)\big )\mathbbm {1}_A\mathbf {1}_d+m\mathbbm {1}_{A^C}\right) \nonumber \\&<\rho \left( f_{\rho }^{-1}\big (\rho (X)\big )\mathbbm {1}_A\mathbf {1}_d+m\mathbbm {1}_{A^C}\right) \end{aligned}$$

  (C.1)

  and similarly

  $$\begin{aligned} \rho (X\mathbbm {1}_B+m\mathbbm {1}_{B^C})>\rho \left( f_{\rho }^{-1}\big (\rho (X)\big )\mathbbm {1}_B\mathbf {1}_d+m\mathbbm {1}_{B^C}\right) . \end{aligned}$$

  (C.2)

  However, as X is independent of $\mathcal {H}$ the random vector $X\mathbbm {1}_A+m\mathbbm {1}_{A^C}$ has the same distribution under $\mathbb {P}$ as $X\mathbbm {1}_B+m\mathbbm {1}_{B^C}$. Note that also $f_{\rho }^{-1}\big (\rho (X)\big )\mathbbm {1}_A+a\mathbbm {1}_{A^C}$ and $f_{\rho }^{-1}\big (\rho (X)\big )\mathbbm {1}_B+a\mathbbm {1}_{B^C}$ share the same distribution under $\mathbb {P}$. Hence, as $\rho$ is law-invariant, (C.1) and (C.2) yield a contradiction.

$\square$

Proof of Theorem 4.4

For the last assertion of the theorem note that since u is a deterministic function, we have for $\alpha \in L_{}^\infty (\mathcal {H})$ that

$$\begin{aligned} f_{\rho _\mathcal {H}}(\alpha )&=\rho _\mathcal {H}(\alpha \mathbf {1}_d)=g_{\mathcal {H}}\left( f_{u}^{-1}\big (\mathbb {E}_{\mathbb {P}}\left[ \left. u(\alpha \mathbf {1}_d)\,\right| \,\mathcal {H}\right] \big )\right) \\&=g_{\mathcal {H}}\left( f_{u}^{-1}\big (f_{u}(\alpha )\big )\right) =g_{\mathcal {H}}(\alpha ) \end{aligned}$$

and analogously we obtain $f_\rho \equiv g$.

Next we prove sufficiency in the first statement of the theorem: Let $\rho _\mathcal {H}$ and $\rho$ be as in (4.2) and (4.1). It is easily verified that $\rho _\mathcal {H}$ and $\rho$ are (conditionally) law-invariant CRMs. Furthermore, since $f_{u}^{-1}$ is strictly increasing and $g_{\mathcal {H}}$ is strictly antitone and $\mathcal {H}$-local, we have for each $X,Y\in L_{d}^\infty (\mathcal {F})$ with $\rho _\mathcal {H}(X)\ge \rho _\mathcal {H}(Y)$ that

$$\begin{aligned} \mathbb {E}_{\mathbb {P}}\left[ \left. u(X)\,\right| \,\mathcal {H}\right] \le \mathbb {E}_{\mathbb {P}}\left[ \left. u(Y)\,\right| \,\mathcal {H}\right] . \end{aligned}$$

But this implies that also $\mathbb {E}_{\mathbb {P}}\left[ u(X)\right] \le \mathbb {E}_{\mathbb {P}}\left[ u(Y)\right]$ and thus that $\rho (X)\ge \rho (Y)$, i.e. $\{\rho ,\rho _\mathcal {H}\}$ is strongly consistent.

Now we prove necessity in the first statement of the theorem: We assume in the following that $\rho$ and $\rho _\mathcal {H}$ are normalized on constants and follow the approach of Föllmer [12, Theorem 3.4]. The idea is to introduce a preference order $\prec$ on multivariate distributions $\mu ,\nu$ on $({\mathbb {R}}^d,\mathcal{B}({\mathbb {R}}^d))$ with bounded support given by

$$\begin{aligned} \mu \prec \nu \quad \Longleftrightarrow \quad \rho (X)>\rho (Y),\quad \text{ with } X\sim \mu \text{ and } Y\sim \nu . \end{aligned}$$

Here $\mathcal{B}({\mathbb {R}}^d)$ denotes the Borel-$\sigma$-algebra on ${\mathbb {R}}^d$ and $X\sim \mu$ means that the distribution of $X\in L_{d}^\infty (\mathcal {F})$ under $\mathbb {P}$ is $\mu$. It is well-known that if this preference order fulfills a set of conditions, then there exists a von Neumann-Morgenstern representation, that is

$$\begin{aligned} \mu \prec \nu \quad \Longleftrightarrow \quad \int u(x)\,\mu (dx)<\int u(x)\,\nu (dx), \end{aligned}$$

(C.3)

where $u:{\mathbb {R}}^d\rightarrow {\mathbb {R}}$ is a continuous function. Sufficient conditions to guarantee (C.3) are that $\prec$ is continuous and fulfills the independence axiom; cf. Föllmer and Schied [15, Corollary 2.28]. We refer to Föllmer and Schied [15] for a definition and comprehensive discussion of preference orders and the mentioned properties. Suppose for the moment that we have already proved (C.3). Note that strict antitonicity of $\rho$ implies that $\delta _x\succ \delta _y$ whenever $x,y\in {\mathbb {R}}^d$ satisfy $x\ge y$ and $x\ne y$. Hence $u(x)=\int u(s)\,\delta _x(ds)>\int u(s)\,\delta _y(ds)=u(y)$, and we conclude that u is necessarily strictly increasing as claimed.

Now we prove (C.3): The proof of continuity of $\prec$ is completely analogous to the corresponding proof in Föllmer [12, Theorem 3.4], so we omit it here. The crucial property is the independence axiom, which states that for any three distributions $\mu ,\nu ,\vartheta$ such that $\mu \preceq \nu$ and for all $\lambda \in (0,1]$, we have

$$\begin{aligned} \lambda \mu +(1-\lambda )\vartheta \preceq \lambda \nu +(1-\lambda )\vartheta . \end{aligned}$$

Since $(\Omega ,\mathcal {F},\mathbb {P})$ is conditionally atomless given $\mathcal {H}$, we can find $X,Y,Z\in L_{d}^\infty (\mathcal {F})$ which are independent of $\mathcal {H}$ such that $X\sim \mu ,Y\sim \nu$ and $Z\sim \vartheta$. Furthermore, since $(\Omega ,\mathcal {H},\mathbb {P})$ is atomless, we can find an $A\in \mathcal {H}$ with $\mathbb {P}(A)=\lambda$. It can be easily seen that $X\mathbbm {1}_A+Z\mathbbm {1}_{A^C}\sim \lambda \mu +(1-\lambda )\vartheta$ and $Y\mathbbm {1}_A+Z\mathbbm {1}_{A^C}\sim \lambda \nu +(1-\lambda )\vartheta$. Moreover, since $\mu \preceq \nu$, we have that $\rho (X)\ge \rho (Y)$. As $\{\rho ,\rho _\mathcal {H}\}$ is strongly consistent and as $\rho$ is law-invariant, we know from Lemma 4.3 that $\rho _\mathcal {H}$ is conditionally law-invariant. This ensures that we can apply Lemma C.1 to the random vectors X and Y which are independent of $\mathcal {H}$. Therefore, by $\mathcal {H}$-locality of $\rho _\mathcal {H}$ and recalling Remark 3.4

$$\begin{aligned} \rho \left( X\mathbbm {1}_A+Z\mathbbm {1}_{A^C}\right)&=\rho \left( -\rho _\mathcal {H}\left( X\mathbbm {1}_A+Z\mathbbm {1}_{A^C}\right) \mathbf {1}_d\right) \\&= \rho \left( -\rho _\mathcal {H}(X)\mathbbm {1}_A\mathbf {1}_d-\rho _\mathcal {H}(Z)\mathbbm {1}_{A^C}\mathbf {1}_d\right) \\&=\rho \left( -\rho (X)\mathbbm {1}_A\mathbf {1}_d-\rho _\mathcal {H}(Z)\mathbbm {1}_{A^C}\mathbf {1}_d\right) \\&\ge \rho \left( -\rho (Y)\mathbbm {1}_A\mathbf {1}_d-\rho _\mathcal {H}(Z)\mathbbm {1}_{A^C}\mathbf {1}_d\right) \;=\;\rho \left( Y\mathbbm {1}_A+Z\mathbbm {1}_{A^C}\right) , \end{aligned}$$

which is equivalent to $\lambda \mu +(1-\lambda )\vartheta \preceq \lambda \nu +(1-\lambda )\vartheta$. Thus there exists a von Neumann-Morgenstern representation (C.3) with a continuous and strictly increasing utility function $u:{\mathbb {R}}^d\rightarrow {\mathbb {R}}$.

In the next step we define $f_{u}:{\mathbb {R}}\rightarrow {\mathbb {R}};x\mapsto u(x\mathbf {1}_d)$. Then $f_{u}$ is strictly increasing and continuous and thus $f_{u}^{-1}$ exists. Let $\mu$ be an arbitrary distribution on $({\mathbb {R}}^d,\mathcal{B}({\mathbb {R}}^d))$ with bounded support and $X\sim \mu$. Then

$$\begin{aligned} \rho \big (\Vert X\Vert _{d,\infty }\mathbf {1}_d\big )\le \rho (X)\le \rho \big (-\Vert X\Vert _{d,\infty }\mathbf {1}_d\big ) \end{aligned}$$

and hence

$$\begin{aligned} f_{u}(-\Vert X\Vert _{d,\infty })&=\int u(x)\;\delta _{-\Vert X\Vert _{d,\infty }\mathbf {1}_d}(dx) \le \int u(x)\;\mu (dx)\\&\le \int u(x)\;\delta _{\Vert X\Vert _{d,\infty }\mathbf {1}_d}(dx)= f_{u}(\Vert X\Vert _{d,\infty }). \end{aligned}$$

The intermediate value theorem now implies the existence of a constant $c(\mu )\in {\mathbb {R}}$ such that

$$\begin{aligned} f_{u}\big (c(\mu )\big )=\int u(x)\;\mu (dx)\quad \Longleftrightarrow \quad c(\mu )= f_{u}^{-1}\left( \int u(x)\;\mu (dx)\right) . \end{aligned}$$

Finally, since $\delta _{c(\mu )\mathbf {1}_d}\approx \mu$, we have

$$\begin{aligned} \rho (X)&=\rho \big (c(\mu )\mathbf {1}_d\big )=-c(\mu )=- f_{u}^{-1}\left( \int u(x)\;\mu (dx)\right) =- f_{u}^{-1}\big (\mathbb {E}_{\mathbb {P}}\left[ u(X)\right] \big ). \end{aligned}$$

Hence, we have proved (4.1) (with $g\equiv -{\text {id}}$). Define

$$\begin{aligned} \psi _{\mathcal {H}}(X):=- f_{u}^{-1}\big (\mathbb {E}_{\mathbb {P}}\left[ \left. u(X)\,\right| \,\mathcal {H}\right] \big ), \quad X\in L_{d}^\infty (\mathcal {F}), \end{aligned}$$

then we have seen in the first part of the proof that $\psi _{\mathcal {H}}$ is a CRM which is strongly consistent with $\rho$. Moreover, $\psi _\mathcal {H}$ is normalized on constants. Thus it follows by Lemma 3.5 that $\rho _\mathcal {H}=\psi _{\mathcal {H}}$. If $\rho$ and/or $\rho _\mathcal {H}$ are not normalized on constants, then considering the normalized CRMs $-f_{\rho }^{-1}\circ \rho$ and $-f_{\rho _\mathcal {H}}^{-1}\circ \rho _\mathcal {H}$, the result follows from $\rho =f_{\rho }\circ \big (-(-f_{\rho }^{-1}\circ \rho )\big )$ and $\rho _\mathcal {H}=f_{\rho _\mathcal {H}}\circ \big (-(-f_{\rho _\mathcal {H}}^{-1}\circ \rho _\mathcal {H})\big )$, i.e. $g=f_\rho$ and $g_\mathcal {H}=f_{\rho _\mathcal {H}}$. $\square$

Positive affine transformations of stochastic utilities

Proposition D.1

Let $U_\mathcal {H}$ be the stochastic utility from Theorem 4.6 and let $\widetilde{U}_\mathcal {H}:{{\mathrm{Im\,}}}\Lambda _\mathcal {H}\rightarrow L_{}^\infty (\mathcal {F})$ be another function which is strictly isotone, $\mathcal {F}$-local, fulfills the Lebesgue property and $\widetilde{U}_\mathcal {H}({{\mathrm{Im\,}}}\Lambda _\mathcal {H}\cap L_{}^\infty (\mathcal {H}))\subseteq L_{}^\infty (\mathcal {H})$, such that

$$\begin{aligned} \widetilde{U}^{-1}_\mathcal {H}\left( \mathbb {E}_{\mathbb {P}}\left[ \left. \widetilde{U}_\mathcal {H}(F)\,\right| \,\mathcal {H}\right] \right) =U^{-1}_\mathcal {H}\left( \mathbb {E}_{\mathbb {P}}\left[ \left. U_\mathcal {H}(F)\,\right| \,\mathcal {H}\right] \right) ,\quad \text {for all }F\in {{\mathrm{Im\,}}}\Lambda _\mathcal {H}. \end{aligned}$$

(D.1)

Then $\widetilde{U}_\mathcal {H}$ is an $\mathcal {H}$-measurable positive affine transformation of $U_\mathcal {H}$, i.e. there exist $\alpha ,\beta \in L_{}^\infty (\mathcal {H})$ with $\mathbb {P}(\alpha >0)=1$ such that $\widetilde{U}_\mathcal {H}(F)=\alpha U_\mathcal {H}(F)+\beta$ for all $F\in {{\mathrm{Im\,}}}\Lambda _\mathcal {H}$.

Proof

We have seen in Theorem 4.6 that $U_\mathcal {H}\circ \Lambda _\mathcal {H}=u$, where u is strictly increasing and continuous. Thus

$$\begin{aligned} \mathcal {X}:={{\mathrm{Im\,}}}U_\mathcal {H}=u(L_{d}^\infty (\mathcal {F}))\subseteq L_{}^\infty (\mathcal {F}) \end{aligned}$$

and it follows that for all $F\in \mathcal {X}$ there exists a sequence of $\mathcal {F}$-simple random variables $(F_n)_{n\in \mathbb {N}}\subseteq \mathcal {X}$ such that $F_n\rightarrow F\,\mathbb {P}$-a.s. Moreover, by the intermediate value theorem we can find for each $X,Y\in L_{d}^\infty (\mathcal {F})$ and $\lambda \in L_{}^\infty (\mathcal {F})$ with $0\le \lambda \le 1$ a random variable Z such that $\min \{-\Vert X\Vert _{d,\infty },-\Vert Y\Vert _{d,\infty }\}\le Z\le \max \{\Vert X\Vert _{d,\infty },\Vert Y\Vert _{d,\infty }\}$ and for all $\mathbb {P}$-almost all $\omega \in \Omega$

$$\begin{aligned} \lambda (\omega ) u\big (X(\omega )\big )+(1-\lambda ) u\big (Y(\omega )\big )=u\big (Z(\omega )\mathbf {1}_d\big ) \end{aligned}$$

where $X(\cdot ),Y(\cdot )$ and $\lambda (\cdot )$ are arbitrary representatives of X, Y and $\lambda$. Indeed, it can be shown by a measurable selection argument that Z can be chosen to be $\mathcal {F}$-measurable and hence $\mathcal {X}$ is $\mathcal {F}$-conditionally convex in the sense that $\lambda F+(1-\lambda )G\in \mathcal {X}$ for all $F,G\in \mathcal {X}$ and $\lambda \in L_{}^\infty (\mathcal {F})$ with $0\le \lambda \le 1$.

Next define the strictly isotone and $\mathcal {F}$-local function

$$\begin{aligned} V_\mathcal {H}:\mathcal {X}\rightarrow L_{}^\infty (\mathcal {F});\;X\mapsto \widetilde{U}_\mathcal {H}\left( U_\mathcal {H}^{-1}(F)\right) , \end{aligned}$$

that is $\widetilde{U}_\mathcal {H}=V_\mathcal {H}\circ U_\mathcal {H}$. Moreover, it easily follows that $V_\mathcal {H}$ fulfills the Lebesgue property and $V_\mathcal {H}(\mathcal {X}\cap L_{}^\infty (\mathcal {H}))\subseteq L_{}^\infty (\mathcal {H})$. We show that $V_\mathcal {H}$ is an affine function, that is $V_\mathcal {H}(F)=\alpha F+\beta$ for all $F\in \mathcal {X}$, where $\alpha ,\beta \in L_{}^\infty (\mathcal {F})$. Note that affinity can be equivalently expressed via $V_\mathcal {H}(\lambda F+(1-\lambda )G)=\lambda V_\mathcal {H}(F)+(1-\lambda ) V_\mathcal {H}(G)$ for all $F,G\in \mathcal {X}$ and $\lambda \in L_{}^\infty (\mathcal {F})$ with $0\le \lambda \le 1$.

We suppose that $V_\mathcal {H}$ is not affine, i.e. there are $F,G\in \mathcal {X}$ and $\lambda \in L_{}^\infty (\mathcal {F})$ with $0\le \lambda \le 1$ such that

$$\begin{aligned} \mathbb {P}\left( V_\mathcal {H}(\lambda F+(1-\lambda )G)\ne \lambda V_\mathcal {H}(F)+(1-\lambda ) V_\mathcal {H}(G)\right) >0. \end{aligned}$$

(D.2)

First note that it suffices to assume that (D.2) holds for deterministic F, G and $\lambda$. To see this suppose that $V_\mathcal {H}$ is affine on deterministic values, but not on the whole of $\mathcal {X}$, i.e. (D.2) holds for some $F,G\in \mathcal {X}$ and $\lambda \in L_{}^\infty (\mathcal {F})$ with $0\le \lambda \le 1$. We know that there exist sequences of $\mathcal {F}$-simple functions $(F_n)_{n\in \mathbb {N}},(G_n)_{n\in \mathbb {N}}\subset \mathcal {X}\cap \mathcal {S}$ and $(\lambda _n)_{n\in \mathbb {N}}\subset L_{}^\infty (\mathcal {F})\cap \mathcal {S}$ with $0\le \lambda _n\le 1$ for all $n\in \mathbb {N}$ such that $F_n\rightarrow F,G_n\rightarrow G,\lambda _n\rightarrow \lambda$ $\mathbb {P}$-a.s., where $\mathcal {S}$ was defined in the proof of Proposition B.1. Without loss of generality we might assume that $F_n=\sum _{i=1}^{k_n}F_i^n\mathbbm {1}_{A_i^n},G_n=\sum _{i=1}^{k_n}G_i^n\mathbbm {1}_{A_i^n}$ and $\lambda _n=\sum _{i=1}^{k_n}\lambda _i^n\mathbbm {1}_{A_i^n}$ have the same disjoint $\mathcal {F}$-partition $(A_i^n)_{i=1,\ldots ,k_n}$. By the $\mathcal {F}$-locality and Lebesgue property and since $F_i^n,G_i^n,\lambda _i^n\in {\mathbb {R}}$ for all $i=1,\ldots ,k_n$ and $n\in \mathbb {N}$ we have

$$\begin{aligned} V_\mathcal {H}(\lambda F+(1-\lambda )G)&=\lim _{n\rightarrow \infty } V_\mathcal {H}(\lambda _n F_n+(1-\lambda _n)G_n)\\&=\lim _{n\rightarrow \infty } V_\mathcal {H}\left( \sum _{i=1}^{k_n}(\lambda _i^n F^n_i+(1-\lambda ^n_i)G^n_i)\mathbbm {1}_{A_i^n}\right) \\&=\lim _{n\rightarrow \infty } \sum _{i=1}^{k_n} V_\mathcal {H}\big (\lambda _i^n F^n_i+(1-\lambda ^n_i)G^n_i\big )\mathbbm {1}_{A_i^n}\\&=\lim _{n\rightarrow \infty } \sum _{i=1}^{k_n}\Big (\lambda ^n_i V_\mathcal {H}(F^n_i)+(1-\lambda _i^n) V_\mathcal {H}(G_i^n)\Big )\mathbbm {1}_{A_i^n}\\&=\lim _{n\rightarrow \infty }\lambda _n V_\mathcal {H}(F_n)+(1-\lambda _n) V_\mathcal {H}(G_n)\\&=\lambda V_\mathcal {H}(F)+(1-\lambda ) V_\mathcal {H}(G), \end{aligned}$$

which contradicts (D.2). Moreover we assume that $0<\lambda <1$ since otherwise this would also contradict (D.2). Finally, we assume w.l.o.g. that

$$\begin{aligned} A:=\{V_\mathcal {H}(\lambda F+(1-\lambda )G)<\lambda V_\mathcal {H}(F)+(1-\lambda ) V_\mathcal {H}(G)\}\in \mathcal {H}\end{aligned}$$

has positive probability. Next define $H_1:=F\mathbbm {1}_A+G\mathbbm {1}_{A^C}$ and $H_2:=G$, then $H_i\in \mathcal {X}\cap L_{}^\infty (\mathcal {H}), i=1,2$ and by $\mathcal {F}$-locality of $V_\mathcal {H}$

$$\begin{aligned} V_\mathcal {H}(\lambda H_1+(1-\lambda )H_2)\le \lambda V_\mathcal {H}(H_1)+(1-\lambda ) V_\mathcal {H}(H_2) \end{aligned}$$

and the inequality is strict with positive probability.

Since $(\Omega ,\mathbb {P},\mathcal {F})$ is conditionally atomless given $\mathcal {H}$ there exists a $B\in \mathcal {F}$ with $\mathbb {P}(B)=\lambda$ and which is independent of $\mathcal {H}$. Since $H_1,H_2\in \mathcal {X}$ and $\mathcal {X}$ is $\mathcal {F}$-conditionally convex

$$\begin{aligned} H:=H_1\mathbbm {1}_B+H_2\mathbbm {1}_{B^C}\in \mathcal {X}. \end{aligned}$$

Now by $\mathcal {F}$-locality of $V_\mathcal {H}$, $V_\mathcal {H}(\mathcal {X}\cap L_{}^\infty (\mathcal {H}))\subseteq L_{}^\infty (\mathcal {H})$ and $B\perp \perp \mathcal {H}$ we get

$$\begin{aligned} \mathbb {E}_{\mathbb {P}}\left[ \left. V_\mathcal {H}\left( H\right) \,\right| \,\mathcal {H}\right]&=\mathbb {E}_{\mathbb {P}}\left[ \left. V_\mathcal {H}\left( H_1\mathbbm {1}_B+H_2\mathbbm {1}_{B^C}\right) \,\right| \,\mathcal {H}\right] \\&=V_\mathcal {H}(H_1)\mathbb {E}_{\mathbb {P}}\left[ \left. \mathbbm {1}_B \,\right| \,\mathcal {H}\right] +V_\mathcal {H}(H_2)\mathbb {E}_{\mathbb {P}}\left[ \left. \mathbbm {1}_{B^C} \,\right| \,\mathcal {H}\right] \\&= V_\mathcal {H}(H_1)\mathbb {E}_{\mathbb {P}}\left[ \mathbbm {1}_B \right] +V_\mathcal {H}(H_2)\mathbb {E}_{\mathbb {P}}\left[ \mathbbm {1}_{B^C} \right] \\&=\lambda V_\mathcal {H}(H_1)+(1-\lambda )V_\mathcal {H}(H_2)\\&\ge V_\mathcal {H}(\lambda H_1+(1-\lambda )H_2)\\&=V_\mathcal {H}\left( \mathbb {E}_{\mathbb {P}}\left[ \left. H_1\mathbbm {1}_B+H_2\mathbbm {1}_{B^C}\,\right| \,\mathcal {H}\right] \right) \\&=V_\mathcal {H}\left( \mathbb {E}_{\mathbb {P}}\left[ \left. H\,\right| \,\mathcal {H}\right] \right) , \end{aligned}$$

and the inequality is strict with positive probability. Moreover $\mathcal {X}={{\mathrm{Im\,}}}U_\mathcal {H}$ implies the existence of a $\widetilde{H}\in {{\mathrm{Im\,}}}\Lambda _\mathcal {H}$ such that $H=U_\mathcal {H}(\widetilde{H}).$ Finally we get

$$\begin{aligned} \widetilde{U}_\mathcal {H}^{-1}\left( \mathbb {E}_{\mathbb {P}}\left[ \left. \widetilde{U}_\mathcal {H}(\widetilde{H})\,\right| \,\mathcal {H}\right] \right)&=U_\mathcal {H}^{-1}\left( V_\mathcal {H}^{-1}\left( \mathbb {E}_{\mathbb {P}}\left[ \left. V_\mathcal {H}\left( U_\mathcal {H}(\widetilde{H})\right) \,\right| \,\mathcal {H}\right] \right) \right) \\&=U_\mathcal {H}^{-1}\left( V_\mathcal {H}^{-1}\left( \mathbb {E}_{\mathbb {P}}\left[ \left. V_\mathcal {H}\left( H\right) \,\right| \,\mathcal {H}\right] \right) \right) \\&\ge U_\mathcal {H}^{-1}\left( V_\mathcal {H}^{-1}\left( V_\mathcal {H}\left( \mathbb {E}_{\mathbb {P}}\left[ \left. H\,\right| \,\mathcal {H}\right] \right) \right) \right) \\&=U_\mathcal {H}^{-1}\left( \mathbb {E}_{\mathbb {P}}\left[ \left. H\,\right| \,\mathcal {H}\right] \right) \\&=U_\mathcal {H}^{-1}\left( \mathbb {E}_{\mathbb {P}}\left[ \left. U_\mathcal {H}(\widetilde{H})\,\right| \,\mathcal {H}\right] \right) , \end{aligned}$$

and the inequality is strict with positive probability, since $\widetilde{U}^{-1}_\mathcal {H}$ and $U_\mathcal {H}^{-1}$ are strictly isotone (c.f. Lemma A.1). Thus we have the desired contradiction of (D.1) and hence $V_\mathcal {H}$ is affine, i.e. $V_\mathcal {H}(F)=\alpha F+\beta$ for all $F\in \mathcal {X}$, where $\alpha ,\beta \in L_{}^\infty (\mathcal {F})$. Moreover, since we know that $V_\mathcal {H}(x)\in L_{}^\infty (\mathcal {H})$ for all $x\in {\mathbb {R}}\cap \mathcal {X}$, we obtain that $\alpha ,\beta$ are actually $\mathcal {H}$-measurable. That $\alpha >0$ follows immediately from the fact that $\widetilde{U}_\mathcal {H},U^{-1}_\mathcal {H}$ are strictly isotone. $\square$

Proof of Proposition 5.9

Lemma E.1

Let $u:{\mathbb {R}}^d\rightarrow {\mathbb {R}}$ be a deterministic utility, i.e. u is strictly increasing and continuous, and let $\mathcal {G}$ and $\mathcal {H}$ be a sub-$\sigma$-algebras of $\mathcal {F}$ such that $\mathcal {G}\subseteq \mathcal {H}$. Then

$$\begin{aligned} \mathbb {E}_{\mathbb {P}}\left[ \left. u(L_{d}^\infty (\mathcal {H}))\,\right| \,\mathcal {G}\right] =u(L_{d}^\infty (\mathcal {G})). \end{aligned}$$

Proof

”$\supseteq$”: Obvious. ”$\subseteq$”: Define the CRM $\rho _\mathcal {G}:L_{d}^\infty (\mathcal {H})\rightarrow L_{}^\infty (\mathcal {G});X\mapsto -\mathbb {E}_{\mathbb {P}}\left[ \left. u(X)\,\right| \,\mathcal {G}\right]$. By Lemma A.2 it follows that

$$\begin{aligned} \mathbb {E}_{\mathbb {P}}\left[ \left. u(L_{d}^\infty (\mathcal {H}))\,\right| \,\mathcal {G}\right]&=-\rho _\mathcal {G}(L_{d}^\infty (\mathcal {H}))=-f_{\rho _\mathcal {G}}(L_{}^\infty (\mathcal {G}))=\mathbb {E}_{\mathbb {P}}\left[ \left. u(L_{}^\infty (\mathcal {G})\mathbf {1}_d)\,\right| \,\mathcal {G}\right] \\&\subseteq \mathbb {E}_{\mathbb {P}}\left[ \left. u(L_{d}^\infty (\mathcal {G}))\,\right| \,\mathcal {G}\right] =u(L_{d}^\infty (\mathcal {G})). \end{aligned}$$

$\square$

Lemma E.2

For an arbitrary $\mathcal {T}\in \mathcal {I}$ let $u_\mathcal {T}:{\mathbb {R}}^d\rightarrow {\mathbb {R}}$ be a deterministic utility and define $\mathcal {X}_\mathcal {H}:=u_\mathcal {T}(L_{d}^\infty (\mathcal {H}))$ for all $\mathcal {H}\in \mathcal {E}(\mathcal {T})$. Moreover, let $p_\mathcal {H}:\mathcal {X}_\mathcal {H}\rightarrow L_{}^\infty (\mathcal {H})$ be functions such that $p_\mathcal {H}$ is $\mathcal {H}$-local, strictly isotone and fulfills the Lebesgue-property. If for all $\mathcal {G},\mathcal {H}\in \mathcal {E}(\mathcal {T})$ with $\mathcal {G}\subseteq \mathcal {H}$ and $\mathcal {H}$ atomless it holds that

$$\begin{aligned} p_\mathcal {G}\left( \mathbb {E}_{\mathbb {P}}\left[ \left. F\,\right| \,\mathcal {G}\right] \right) =\mathbb {E}_{\mathbb {P}}\left[ \left. p_\mathcal {H}(F)\,\right| \,\mathcal {G}\right] \text { for all }F\in \mathcal {X}_\mathcal {H},\end{aligned}$$

(E.1)

then

$$\begin{aligned} p_\mathcal {H}(F)=a F+\beta _\mathcal {H}, \end{aligned}$$

where $a\in {\mathbb {R}}^+\backslash \{0\}$ and $\beta _\mathcal {H}\in L_{}^\infty (\mathcal {H})$ such that $\mathbb {E}_{\mathbb {P}}\left[ \left. \beta _\mathcal {H}\,\right| \,\mathcal {G}\right] =\beta _\mathcal {G}$.

Note that (E.1) is well-defined by Lemma E.1.

Proof

Firstly, we consider the case where $\mathcal {G}$ is the trivial $\sigma$-algebra. We write $p:=p_{\{\Omega ,\emptyset \} }$. Note that, since p is a deterministic function, $p\left( \mathbb {E}_{\mathbb {P}}\left[ F\right] \right)$ is law-invariant and thus by (E.1) also $\mathbb {E}_{\mathbb {P}}\left[ p_\mathcal {H}(F)\right]$.

Now suppose that there exist $x,y\in \mathcal {X}:=\mathcal {X}_{\{\Omega ,\emptyset \}}$ with $p_\mathcal {H}(x)-p_\mathcal {H}(y)\not \in {\mathbb {R}}$, i.e. there exists a $c\in {\mathbb {R}}$ such that $\mathbb {P}(p_\mathcal {H}(x)\le p_\mathcal {H}(y)+c)\in (0,1)$. Since $\mathcal {H}$ is an atomless space we can choose $A_1,A_2,A_3\in \mathcal {H}$ with

$$\begin{aligned} \mathbb {P}(A_1)=\mathbb {P}(A_2):=q>0 \end{aligned}$$

such that

$$\begin{aligned} A_1\subseteq \{p_\mathcal {H}(x)\le p_\mathcal {H}(y)+c\},A_2\subseteq \{p_\mathcal {H}(x)> p_\mathcal {H}(y)+c\},A_3:=(A_1\cup A_2)^C. \end{aligned}$$

Moreover, we define

$$\begin{aligned} F_1:=x\mathbbm {1}_{A_1}+y\mathbbm {1}_{A_2}+x\mathbbm {1}_{A_3}\quad \text {and}\quad F_2:=y\mathbbm {1}_{A_1}+x\mathbbm {1}_{A_2}+x\mathbbm {1}_{A_3}. \end{aligned}$$

Obviously $F_1,F_2\sim q\delta _y+(1-q)\delta _x$, that is $F_1{\mathop {=}\limits ^{\text {d}}}F_2$. However, since $p_\mathcal {H}$ is $\mathcal {H}$-local, we have

$$\begin{aligned} \mathbb {E}_{\mathbb {P}}\left[ p_\mathcal {H}(F_1)\right] +cq&=\mathbb {E}_{\mathbb {P}}\left[ p_\mathcal {H}(x)\mathbbm {1}_{A_1}\right] +\mathbb {E}_{\mathbb {P}}\left[ (p_\mathcal {H}(y)+c)\mathbbm {1}_{A_2}\right] +\mathbb {E}_{\mathbb {P}}\left[ p_\mathcal {H}(x)\mathbbm {1}_{A_3}\right] \\&<\mathbb {E}_{\mathbb {P}}\left[ (p_\mathcal {H}(y)+c)\mathbbm {1}_{A_1}\right] +\mathbb {E}_{\mathbb {P}}\left[ p_\mathcal {H}(x)\mathbbm {1}_{A_2}\right] +\mathbb {E}_{\mathbb {P}}\left[ p_\mathcal {H}(x)\mathbbm {1}_{A_3}\right] \\&=\mathbb {E}_{\mathbb {P}}\left[ p_\mathcal {H}(F_2)\right] +cq, \end{aligned}$$

which contradicts the law-invariance of $F\mapsto \mathbb {E}_{\mathbb {P}}\left[ p_\mathcal {H}(F)\right]$.

Hence we have that $p_\mathcal {H}(x)-p_\mathcal {H}(y)\in {\mathbb {R}}$ for all $x,y\in \mathcal {X}$. Choose an arbitrary $\widetilde{x}\in \mathcal {X}$, and let

$$\begin{aligned} a(x):=p_\mathcal {H}(x)-p_\mathcal {H}(\widetilde{x}),\quad x\in \mathcal {X}, \end{aligned}$$

so $a:\mathcal {X}\rightarrow {\mathbb {R}}$. Define ${\widetilde{\beta }}_\mathcal {H}:=p_\mathcal {H}(\widetilde{x})\in L_{}^\infty (\mathcal {H})$, then $p_\mathcal {H}(x)=a(x)+{\widetilde{\beta }}_\mathcal {H}$. The function a is continuous, since otherwise there would exist a sequence $(x_n)_{n\in \mathbb {N}}\subset \mathcal {X}$ with $x_n\rightarrow x\in \mathcal {X}$, but $a(x_n)\not \rightarrow a(x)$ and the Lebesgue-property would imply the contradiction

$$\begin{aligned} p_\mathcal {H}(x)=\lim _{n\rightarrow \infty }p_\mathcal {H}(x_n)=\lim _{n\rightarrow \infty }a(x_n)+\widetilde{\beta }_\mathcal {H}\ne a(x)+{\widetilde{\beta }}_\mathcal {H}=p_\mathcal {H}(x). \end{aligned}$$

Let $F\in \mathcal {X}_\mathcal {H}$. Since the $\mathcal {H}$-measurable simple random vectors are dense in $L_{d}^\infty (\mathcal {H})$ and by the definition of $\mathcal {X}_\mathcal {H}$ there exists a sequence of $\mathcal {H}$-measurable simple random variables $(F_n)_{n\in \mathbb {N}}\subset \mathcal {X}_\mathcal {H}\cap \mathcal {S}$ with $F_n=\sum _{i=1}^{k_n}x_i^n1_{A_i^n}\rightarrow F$ $\mathbb {P}$-a.s. Thus

$$\begin{aligned} p_\mathcal {H}(F)&=\lim _{n\rightarrow \infty }p_\mathcal {H}(F_n)=\lim _{n\rightarrow \infty }\sum _{i=1}^{k_n} p_\mathcal {H}(x_i^n)\mathbbm {1}_{A_i^n}=\lim _{n\rightarrow \infty }\sum _{i=1}^{k_n} a(x_i^n)\mathbbm {1}_{A_i^n}+{\widetilde{\beta }}_\mathcal {H}\\&=\lim _{n\rightarrow \infty } a\left( \sum _{i=1}^{k_n} x_i^n\mathbbm {1}_{A_i^n}\right) +{\widetilde{\beta }}_\mathcal {H}=\lim _{n\rightarrow \infty } a(F_n)+{\widetilde{\beta }}_\mathcal {H}= a(F)+{\widetilde{\beta }}_\mathcal {H}. \end{aligned}$$

The function $\mathcal {X}_\mathcal {H}\ni F\mapsto \mathbb {E}_{\mathbb {P}}\left[ F\right]$ induces a preference relation on $\mathcal {M}:=\{\mu : \exists F\in \mathcal {X}_\mathcal {H}$ such that $F\sim \mu \}$ via

$$\begin{aligned} \mu \succcurlyeq \nu \quad \Longleftrightarrow \quad \mathbb {E}_{\mathbb {P}}\left[ F\right] \ge \mathbb {E}_{\mathbb {P}}\left[ G\right] ,F\sim \mu ,G\sim \nu . \end{aligned}$$

Moreover the function $x\mapsto p^{-1}(x+{\mathbb {E}}[{\widetilde{\beta }}_\mathcal {H}])$ is strictly increasing and by (E.1)

$$\begin{aligned} \mathbb {E}_{\mathbb {P}}\left[ F\right] =p^{-1}\left( \mathbb {E}_{\mathbb {P}}\left[ p_\mathcal {H}(F)\right] \right) =p^{-1}\left( \mathbb {E}_{\mathbb {P}}\left[ a(F)\right] +{\mathbb {E}}\big [{\widetilde{\beta }}_\mathcal {H}\big ]\right) . \end{aligned}$$

Thus $\mathbb {E}_{\mathbb {P}}\left[ a(F)\right]$ is another affine numerical representation of $\succcurlyeq$. It is well-known that the affine numerical representation of $\succcurlyeq$ is unique up to a positive affine transformation (see e.g. Föllmer and Schied [15, Theorem 2.21]), i.e. there exist ${\tilde{a}},b\in {\mathbb {R}}, {\tilde{a}}>0$ such that $\mathbb {E}_{\mathbb {P}}\left[ a(F)\right] ={\tilde{a}} \mathbb {E}_{\mathbb {P}}\left[ F\right] +b$ for all $F\in \mathcal {X}_\mathcal {H}$. In particular this implies that for all $x\in \mathcal {X}$

$$\begin{aligned} a(x)=\mathbb {E}_{\mathbb {P}}\left[ a(x)\right] =a\mathbb {E}_{\mathbb {P}}\left[ x\right] +b=\tilde{a}x+b. \end{aligned}$$

By setting $b+{\widetilde{\beta }}_\mathcal {H}=:\beta _\mathcal {H}\in L_{}^\infty (\mathcal {H})$ we get for all $F\in \mathcal {X}_\mathcal {H}$ that

$$\begin{aligned} p_\mathcal {H}(F)= a(F)+{\widetilde{\beta }}_\mathcal {H}={\tilde{a}} F+b+{\widetilde{\beta }}_\mathcal {H}={\tilde{a}} F+\beta _\mathcal {H}. \end{aligned}$$

Finally we obtain by (E.1) that for every $\mathcal {G}\subseteq \mathcal {H}$ and for all $F\in \mathcal {X}_\mathcal {G}$

$$\begin{aligned} p_\mathcal {G}(F)=p_\mathcal {G}\left( \mathbb {E}_{\mathbb {P}}\left[ \left. F\,\right| \,\mathcal {G}\right] \right) =\mathbb {E}_{\mathbb {P}}\left[ \left. p_\mathcal {H}(F)\,\right| \,\mathcal {G}\right] =a F+\mathbb {E}_{\mathbb {P}}\left[ \left. \beta _\mathcal {H}\,\right| \,\mathcal {G}\right] , \end{aligned}$$

which proves the martingale property of $(\beta _\mathcal {G})_{\mathcal {G}\subseteq \mathcal {H}}$.

Proof of Proposition 5.9

Let $(\rho _{\mathcal {H},\mathcal {T}})_{(\mathcal {H},\mathcal {T})\in \mathcal {E}}$ be a strongly consistent family such that (5.2) holds for all $(\mathcal {H},\mathcal {T})\in \mathcal {E}$, i.e.

$$\begin{aligned} \rho _{\mathcal {H},\mathcal {T}}(X)=f_{\rho _{\mathcal {H},\mathcal {T}}}\left( f_{u_{\mathcal {T}}}^{-1}\big (\mathbb {E}_{\mathbb {P}}\left[ \left. u_{\mathcal {T}}(X)\,\right| \,\mathcal {H}\right] \big )\right) ,\quad \text {for all } X\in L_{d}^\infty (\mathcal {T}), \end{aligned}$$

We define the functions

$$\begin{aligned} h_{\mathcal {H},\mathcal {T}}:u_{\mathcal {T}}(L_{d}^\infty (\mathcal {H}))\rightarrow L_{}^\infty (\mathcal {H});F\mapsto f_{\rho _{\mathcal {H},\mathcal {T}}}\circ f_{u_{\mathcal {T}}}^{-1}(F) \end{aligned}$$

and

$$\begin{aligned} p_{\mathcal {H},\mathcal {T}_1,\mathcal {T}_2}: u_{\mathcal {T}_1}(L_{d}^\infty (\mathcal {H}))\rightarrow L_{}^\infty (\mathcal {H}); F\mapsto h^{-1}_{\mathcal {H},\mathcal {T}_2}\circ h_{\mathcal {H},\mathcal {T}_1}(F). \end{aligned}$$

By strong consistency, we obtain for $\mathcal {G}\subseteq \mathcal {H}\subseteq \mathcal {T}_1\cap \mathcal {T}_2$, $X\in L_{d}^\infty (\mathcal {T}_1)$ and $F:=\mathbb {E}_{\mathbb {P}}\left[ \left. u_{\mathcal {T}_1}(X)\,\right| \,\mathcal {H}\right]$ that

$$\begin{aligned} p_{\mathcal {G},\mathcal {T}_1,\mathcal {T}_2}\left( \mathbb {E}_{\mathbb {P}}\left[ \left. F\,\right| \,\mathcal {G}\right] \right)&=h^{-1}_{\mathcal {G},\mathcal {T}_2}\left( h_{\mathcal {G},\mathcal {T}_1}\big (\mathbb {E}_{\mathbb {P}}\left[ \left. \mathbb {E}_{\mathbb {P}}\left[ \left. u_{\mathcal {T}_1}(X)\,\right| \,\mathcal {H}\right] \,\right| \,\mathcal {G}\right] \big )\right) \nonumber \\&=h^{-1}_{\mathcal {G},\mathcal {T}_2}\left( \rho _{\mathcal {G},\mathcal {T}_1}(X)\right) \nonumber \\&=h^{-1}_{\mathcal {G},\mathcal {T}_2}\left( \rho _{\mathcal {G},\mathcal {T}_2}\left( f^{-1}_{\rho _{\mathcal {H},\mathcal {T}_2}}\big (\rho _{\mathcal {H},\mathcal {T}_1}(X)\big )\mathbf {1}_d\right) \right) \nonumber \\&=\mathbb {E}_{\mathbb {P}}\left[ \left. h^{-1}_{\mathcal {H},\mathcal {T}_2}\left( h_{\mathcal {H},\mathcal {T}_1}\big (\mathbb {E}_{\mathbb {P}}\left[ \left. u_{\mathcal {T}_1}(X)\,\right| \,\mathcal {H}\right] \big )\right) \,\right| \,\mathcal {G}\right] \nonumber \\&=\mathbb {E}_{\mathbb {P}}\left[ \left. p_{\mathcal {H},\mathcal {T}_1,\mathcal {T}_2}(F)\,\right| \,\mathcal {G}\right] . \end{aligned}$$

(E.2)

By Lemma E.2 (E.2) is fulfilled, if and only if

$$\begin{aligned} p_{\mathcal {H},\mathcal {T}_1,\mathcal {T}_2}(F)=a_{\mathcal {T}_1,\mathcal {T}_2}F+b_{\mathcal {H},\mathcal {T}_1,\mathcal {T}_2},\quad \text {for all } F\in u_{\mathcal {T}_1}(L_{d}^\infty (\mathcal {H})), \end{aligned}$$

where $a_{\mathcal {T}_1,\mathcal {T}_2}\in {\mathbb {R}}^+\backslash \{0\}$, $b_{\mathcal {H},\mathcal {T}_1,\mathcal {T}_2}\in L_{}^\infty (\mathcal {H})$ and $\mathbb {E}_{\mathbb {P}}\left[ \left. b_{\mathcal {H},\mathcal {T}_1,\mathcal {T}_2}\,\right| \,\mathcal {G}\right] =b_{\mathcal {G},\mathcal {T}_1,\mathcal {T}_2}$ for all $\mathcal {G}\in \mathcal {I}$ with $\mathcal {G}\subseteq \mathcal {H}$. Thus

$$\begin{aligned} h_{\mathcal {H},\mathcal {T}_1}(F)=h_{\mathcal {H},\mathcal {T}_2}(a_{\mathcal {T}_1,\mathcal {T}_2}F+b_{\mathcal {H},\mathcal {T}_1,\mathcal {T}_2}),\quad F\in u_{\mathcal {T}_1}(L_{d}^\infty (\mathcal {H})), \end{aligned}$$

which implies that

$$\begin{aligned} \rho _{\mathcal {H},\mathcal {T}_1}(X)=f_{\rho _{\mathcal {H},\mathcal {T}_2}}\left( f_{u_{\mathcal {T}_2}}^{-1}\big (a_{\mathcal {T}_1,\mathcal {T}_2}\mathbb {E}_{\mathbb {P}}\left[ \left. u_{\mathcal {T}_1}(X)\,\right| \,\mathcal {H}\right] +b_{\mathcal {H},\mathcal {T}_1,\mathcal {T}_2}\big )\right) . \end{aligned}$$