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Equilibrium prices and trade under ambiguous volatility

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This article considers general equilibrium economies with a primitive uncertainty model that features ambiguity about continuous-time volatility. For the resulting non-equivalence of priors, an appropriate commodity-price space is introduced. Agents are heterogeneous in the size of captured ambiguity, endowment and preference for risk and ambiguity. Preferences are of variational type à la Maccheroni et al. (Econometrica 74(6):1447–1498, 2006). One important implication involves a problematic aspect of linear equilibrium price systems. Positive payoffs are for free on events outside the domain of the representing equilibrium pricing measure. Moreover, when aggregate risk is present and aggregate ambiguity is absent, the insurance properties of optimal allocations depend on the notion of ambiguity-free payoffs.

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  1. The indeterminacy result given by Ohtaki and Ozaki (2015) also follows from Knightian uncertainty and serves as a further hint that indeterminacy of equilibria is a frequently appearing feature that requires a careful distinction from equilibrium implications without ambiguity.

  2. Priors in \({\mathcal {P}}\) can be related in two ways. One case refers to mutual singularity and points to a disjoint support for these measures. The second possibility is a mixture: Two priors may be equivalent for a subclass of events and mutually singular for its complement.

  3. In the volatility uncertainty setting, such priors occur when the volatilities agree up to a certain point \(t>0\) and then differ (see Fig. 2 in Sect. 3). Two priors \(P_1,P_2\in int( {\varDelta }(\varOmega ))\) in the interior of the simplex are always equivalent. Two priors are singular if their supports are disjoint.

  4. In the present setting, weak convergence is shown in Dolinsky et al. (2012).

  5. \(P_0=P^1\) is not a reference measure. Its technical purpose stems from the uncertainty model. \(P_0\notin {\mathcal {P}}\) means that \(1\notin {\mathcal {D}}\). To guarantee weak compactness and convexity of \({\mathcal {P}}'\), it suffices to assume Hölder continuity of the bounds \(\underline{\sigma },\overline{\sigma }: [0,T]\rightarrow {\mathbb {R}}\); see Proposition 5 in Denis and Kervarec (2013).

  6. By Peng (2010), \({\mathbb {E}}\) is a sublinear expectation, i.e., it shows monotonicity, positive homogeneity, a constant preserving property, sub-additivity, and it constitutes the basis of the model.

  7. Note that \(\text {E}^P\vert X\vert \le \Vert X \Vert _{\mathcal {P}}\) for every \(X\in L\) implies a richer dual space: The primal norm reenters in the dual norm via \(\Vert \varPi \Vert ^*_{\mathcal {P}}= \sup \{ \vert \varPi (X)\vert : \Vert X\Vert _{\mathcal {P}}\le 1 \}\).

  8. To see this, let \(\bar{P}(Y>0)>0\) and \(Y=0\) P-a.s. for every \(P\in {\mathcal {P}}{\setminus }\{\bar{P}\}\) and let \(X\mapsto l(X)=\text {E}^{P_l}[ \psi X ]\) such that \(P_l ,\bar{P}\) are mutually singular and \(\psi >0\) \(P^l\)-a.s., hence \(l(Y)=0\).

  9. The positive cone of an infinite dimensional commodity spaces may have an empty interior. Since \(\Vert \cdot \Vert _{\mathcal {P}}\) is a maximum of integrals, this issue is still present. A properness condition is required to establish the existence of an equilibrium. A more elaborate discussion is beyond the scope of this paper. See Martins-da Rocha and Riedel (2010) for additional details.

  10. \(u_{\alpha }\) is given by \({u_{\alpha }}(e)=\max _{x\in \varLambda (e)}\sum \alpha _i u_{i}(x_i)\). Here \( L^{\infty }\) is the closure for \({\mathcal {C}}_b\) under the norm \(\Vert X\Vert _\infty =\inf \{ M\ge 0: \vert X\vert \le M q.s.\} \) [see again Denis et al. (2011)].

  11. Note that \({\mathbb {E}}_G[c]=c\) and \({\mathbb {E}}_G[C+c]={\mathbb {E}}_G[C]+ c\) for every \(C\in L\) and constant c. The second and third equalities follow from the linearity of \({\mathbb {E}}_G\) whenever one summand is an element of \(\text {M}_{\mathcal {P}}\) (see Proposition I.3.6 in Peng 2010). Moreover, we used the fact that the ambiguous Brownian motion contains no ambiguity in the weak sense: \(B_T\in \text {M}_{\mathcal {P}}\) and \(E^P[B_T]=0\) for all \(P\in {\mathcal {P}}\).


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Correspondence to Patrick Beissner.

Additional information

I thank Rose-Anne Dana, Ali Khan, Frank Riedel and Jan Werner for many fruitful discussions and an anonymous referee for useful suggestions. Financial support by the German Research Foundation (DFG) via grant RI 1128/7-1 Financial Markets under Knightian Uncertainty is gratefully acknowledged.



The following convergence results are relevant to several proofs.

Convergence properties of sublinear expectations (Denis et al. 2011):

  1. 1.

    \(\{ P_n\}\subset {\mathcal {P}}\) converges weakly to P, then \(\text {E}^{P_n}[X]\rightarrow \text {E}^P[X]\), for all \(X\in L\).

  2. 2.

    Let \(\{ X_n\} \subset L\) be such that \(X_n \searrow X\), then \({\mathbb {E}}[X_n]\searrow {\mathbb {E}}[X]\).

1.1 Details and proofs of Sect. 3

As mentioned in Sect. 3.1, two random variables \(X,Y\in L\) can be distinguished if there is a prior \(P\in {\mathcal {P}}\) such that \(P(X\ne Y)>0\). \({\mathcal {P}}\)-polar sets are elements in \({\mathcal {F}}\) with a probability of zero or one for every prior. A property holds quasi-surely (q.s.) if it holds outside a polar set. Furthermore, the commodity space L is characterized in Denis et al. (2011) via

$$\begin{aligned} L=\left\{ X\in L(\varOmega ):X\text { has a q.c. version, } \lim _{n\rightarrow \infty }{\mathbb {E}}\bigl [\vert X\vert 1_{\{\vert X\vert >n\}} \bigl ]=0\right\} . \end{aligned}$$

We say that \(X:\varOmega \rightarrow {\mathbb {R}}\) has a q.c. version if there exists a quasi-continuous function \(Y:\varOmega \rightarrow {\mathbb {R}}\) with \(X = Y\) q.s. A mapping \(X:\varOmega \rightarrow {\mathbb {R}}\) is said to be quasi-continuous (q.c.) if for all \(\epsilon > 0\) there exists an open set O with \(c(O)=\sup _{P\in {\mathcal {P}}}P(O) < \epsilon \) such that \(X|_{O^c}\) is continuous.

Proof of Proposition 1

We show \(\inf (X,Y)=X \wedge Y\in L\) for every \(X,Y\in L\) via the representation in (12). Since \(\{\vert X\vert>n\}\supset \{\vert X\wedge Y\vert >n\}\), sublinearity of \({\mathbb {E}}\) implies

$$\begin{aligned} {\mathbb {E}}\bigl [\vert X\wedge Y \vert 1_{\{\vert X\wedge Y \vert>n\}} \bigl ]\le {\mathbb {E}}\bigl [\vert X\vert 1_{\{\vert X\vert>n\}} \bigl ]+{\mathbb {E}}\bigl [\vert Y\vert 1_{\{\vert Y\vert >n\}} \bigl ] \xrightarrow [n\rightarrow \infty ]{} 0. \end{aligned}$$

Since X and Y have a q.c. version, there are \(\bar{\epsilon },\epsilon _X, \epsilon _Y>0\) such that \(\epsilon _X+ \epsilon _Y< \bar{\epsilon }\) with \(c(O_X) < \epsilon _X\), \(c(O_Y) < \epsilon _Y\) and hence \(c(O_X\cup O_Y)\le c(O_X)+c(O_Y)< \bar{\epsilon }\). Because \(X|_{(O_X\cup O_Y)^c}\) and \(Y|_{(O_X\cup O_Y)^c}\) are both continuous, the quasi-continuity of \(X \wedge Y\) follows. The order relation is indeed a lattice operation.

That L is a Banach space is shown in Denis et al. (2011). For all \(X,Y\in L\) with \(\vert X\vert \le \vert Y\vert \) q.s. imply

$$\begin{aligned} \Vert X \Vert _{\mathcal {P}}=\max _{P\in {\mathcal {P}}} \text {E}^P[\vert X\vert ]=\text {E}^{P_X}[\vert X\vert ]\le \text {E}^{P_X}[\vert Y\vert ]\le \Vert Y\Vert _{\mathcal {P}}, \end{aligned}$$

for some maximizing prior \(P_X\) at \(\vert X\vert \); hence, L is a Banach lattice. Fix a sequence of positive random variables \((X_n)\) in L such that \(X_n\searrow 0\) in L. Application of the monotone continuity of \({\mathbb {E}}\) (see the beginning of the “Appendix”) gives us

$$\begin{aligned} \lim _{n\rightarrow \infty } \Vert X_n \Vert _{\mathcal {P}}=\lim _{n\rightarrow \infty }{\mathbb {E}}[\vert X_n\vert ]={\mathbb {E}}[\vert \lim _{n\rightarrow \infty } X_n\vert ]=0. \end{aligned}$$

Hence, L is a \(\sigma \)-order-continuous Banach lattice. \(\square \)

Proof of Proposition 2

\(\tilde{L}\) is given by the space of \(\Vert \cdot \Vert _{\mathcal {P}}\)-equivalence classes of \(\overline{{\mathcal {C}}_b(\varOmega )}^{\Vert \cdot \Vert _{\mathcal {P}}}\), so that the domain is modified via the Lebesgue prolongation; see Sect. 2 of Bion-Nadal and Kervarec (2012) for this approach to function spaces. The explicit representation of the topological dual of \(\tilde{L}\) can be found in the first chapter of Kervarec (2008), Theorem I.30.

Clearly, every \(E^P[\psi \cdot ]\) is linear on L. Continuity under \(\Vert \cdot \Vert _{\mathcal {P}}\) follows from \(E^P[\psi \cdot ]\le M\cdot E^P[\psi \cdot ]\le M {\mathbb {E}}[\cdot ]\), for some constant \(M>0\). \(\square \)

Proof of Corollary 1

\(\Rightarrow \)”: Proposition 2 implies \(\varPi (X)=\text {E}^P[ \psi X]\), since \(\psi >0\) P-a.s and \(P(X>0)>0\); therefore, \(\varPi (X)>0\) follows.

\(\Leftarrow \)”: \(\varPi \in L^*\) implies, again by Proposition 2, that \(\varPi (X)=\text {E}^P[ \psi X] \). Suppose \(\psi \notin L^\infty (P)_{++}\) then \(P(\{\psi>0\} \cap \{ X>0\})=0\) for some \(P\in {\mathcal {P}}\) is possible, a contradiction. \(\square \)

Proof of Lemma 1

  1. 1.

    Monotonicity follows directly from the monotonicity of the utility index u. Let \(P_C\in M(C)\) be a minimizer at C. Semi-strict monotonicity follows from \(u'(C)>0\) on a set with a positive measure with respect to \(P_C\) and

    $$\begin{aligned} U(C+h)-U(C)\ge \text {E}^{P_C}[u(C+h)-u(C)]>\text {E}^{P_C}[u'(C+h)\cdot h]\ge 0, \end{aligned}$$

    where the strict inequality follows from the strict concavity of u and \(h\in L_\oplus \).

  2. 2.

    The mapping \(C\mapsto \text {E}^P[u(C)]+{\mathfrak {c}}(P)\) is concave for each \(P\in {\mathcal {P}}\) and the \(\inf \) operation preserves concavity. To prove strict concavity on \(\text {M}_{\mathcal {P}}\), let \(\alpha \in (0,1)\) and \(C,X\in L_+\cap \text {M}_{\mathcal {P}}\), with \(C\ne X\). This yields

    $$\begin{aligned} \alpha U(C)+ (1-\alpha ) U(X)\le & {} \min _{P\in {\mathcal {P}}} \text {E}^P[\alpha u(C) + (1-\alpha ) u(X)]+{\mathfrak {c}}(P)\\< & {} \min _{P\in {\mathcal {P}}} \text {E}^P[ u( \alpha C + (1-\alpha ) X)]+{\mathfrak {c}}(P)\\= & {} U( \alpha C+ (1-\alpha ) X), \end{aligned}$$

    where the first inequality follows from the concavity of \(C\mapsto U(C)\).

    U is not strictly concave on its whole domain, since \(C\ne X\) in L does not imply \(C\ne X\) under every \(P\in {\mathcal {P}}\); hence, one can easily pick two elements C and X which are P-a.s. equal, where \(P\in M(\lambda C+(1-\lambda ) X)\), and deduce a contradiction, when following the proof of strict concavity on \(\text {M}_{\mathcal {P}}\).

  3. 3.

    Let \((C_n)\subset L_+\) converge to C in L. To prove upper semi-continuity, every subsequence \((Y_{n_k})_{k\in {\mathbb {N}}}\) of \((C_n)\) must have in turn a subsequence \((Z_n)_{n\in {\mathbb {N}}}\) such that

    $$\begin{aligned} \limsup _{n\rightarrow \infty } U(Z_n)\le U(C). \end{aligned}$$

    Let \(P_C\in M(C)\) be a minimizing prior and \((Y_{n_k})\) be a subsequence of \((C_n)\). There is a subsequence \((Z_{n})\) in \((Y_{n_k})_{k\in {\mathbb {N}}}\) and some \(Z\in L_+\) satisfying \( Z_n(\omega )\rightarrow X(\omega )\) and \( 0\le Z_n(\omega )\le Z(\omega ) \) for \( P_C\)–a.s. Take \(Z=C+ \sum _{n\in {\mathbb {N}}}\vert Z_{n+1}-Z_n\vert \), with \(\Vert Z_{n+1}-Z_n\Vert _{\mathcal {P}} \le 2^{-n}\). Monotonicity of u implies \(0\le u(Z_n (\omega ))\le u(Z (\omega ))\) and \(u(Z_n (\omega ))\rightarrow u(C(\omega ))\) for \(P_C\)-a.s. \(\omega \in \varOmega \). As such, the \(\limsup \)-version of Fatou’s lemma under \(P_C\) yields

    $$\begin{aligned} \limsup _{n\rightarrow \infty } U(Z_n) \le \limsup _{n\rightarrow \infty } \text {E}^{P_C}[u(Z_n)]+{\mathfrak {c}}(P_C) \le \text {E}^{P_C}[ u(C)]+{\mathfrak {c}}(P_C) =U(C). \end{aligned}$$

    For the set of super-gradients, note that \({\mathcal {P}}\) is also \(\sigma (\tilde{L}^*, \tilde{L})\)-weakly compact which follows from the same arguments as in the proof for Proposition 2.4 of Bion-Nadal and Kervarec (2012), since it is a closed subset of the nonnegative part of the unit ball in \(\tilde{L}^*\). Minimizing priors exist, since \(P\mapsto \text {E}^P[X]\) is weakly continuous in P for every \(X\in L\), and build a convex weakly compact subset of \( {\mathcal {P}}\).

Let \(P_C\in M( C)\) and \(X\in L_+\). By the concavity and differentiability of the utility index u, this implies

$$\begin{aligned} U(X)-U( C)= & {} {\min _{P\in {\mathcal {P}}} \text {E}^P[u(X)]+{\mathfrak {c}}(P)- \min _{P\in {\mathcal {P}}} \text {E}^P[u( C)]+{\mathfrak {c}}(P)}\\\le & {} \text {E}^{P_C}[u(X)]+{\mathfrak {c}}(P_C)- \big ( \text {E}^{P_C}[u( C)]+{\mathfrak {c}}(P_C)\big )\\\le & {} \text {E}^{P_C}[ u'( C)(X- C)]. \end{aligned}$$

\(\square \)

1.2 Details and proofs for Sect. 4

The proof of Theorem 1 needs some preparation and relies on Propositions 3 to 5. Their respective proofs are collected at the end of the “Appendix.”

Proposition 3

Under Assumption 1 with \(\text {dom}({\mathfrak {c}}_i)=\{P\}\), for all \(i\in {\mathbb {I}}\), there is a P-a.s. unique equilibrium. The equilibrium allocation is a continuous function of \(E\in L_+\).

\({\mathbb {GE}}:{\mathcal {P}}\rightarrow \varDelta _I\) denotes the single-valued correspondence. It assigns to every P the unique equilibrium weight \(\alpha ^P\) of the relevant single-prior economy \({\mathcal {E}}^P=\{L_+,U_i^P, E_i\}_{i\in {\mathbb {I}}}\) with \(\text {E}^P\)-expected utility \(U^P_i\). By the first welfare theorem and Proposition 3, consider a single-valued solution selection of the concave program \((U^P_\alpha ,\varLambda (E))\), given by \(( C_i^\alpha )\in \text {arg}\max _{(C_i)\in \varLambda (E) } \sum \alpha _i U^P_i(C_i)\). This motivates the following definition.

Definition 2

The excess utility map \(\varPhi :\varDelta _I\times {\mathcal {P}}\rightarrow {\mathbb {R}}^I\) is given by

$$\begin{aligned} \varPhi _i(\alpha ,P)=\alpha _i^{-1}\text {E}^P\left[ u'_{\alpha }(E)\cdot (C_i^{\alpha }-E_i )\right] ,\quad i\in {\mathbb {I}}. \end{aligned}$$

The modification in the definition is caused by the equilibrium prior in the first-order conditions of individual maximization. As explained at the end of Sect. 2, a zero \((\alpha ,P)\in \varDelta _I\times {\mathcal {P}}\) of \(\varPhi \) is not sufficient to guarantee existence, since an arbitrary \(P\in {\mathcal {P}}\) may not lie in the set of common minimizers \( \bigcap _{i\in {\mathbb {I}}} M_i(C_i^\alpha )\).

To show that there is a common minimizing prior at \((C_1^\alpha ,\ldots , C_I^\alpha )\), consider the superdifferential of \( U_\alpha \), denoted by \(\partial U_{\alpha }\).

Proposition 4

Let \((C_i^\alpha )\in \varLambda (E)\) be \(\alpha \)-efficient. The set of common risk-unadjusted priors

$$\begin{aligned} {\mathbb {P}}(\alpha ) =\text {arg}\min _{P\in {\mathcal {P}}}\left\{ \text {E}^P[u_\alpha (E)] +{\mathfrak {c}}_\alpha (P)\right\} , \end{aligned}$$

where \({\mathfrak {c}}_\alpha (P)=\sum _{i\in {\mathbb {I}}}\alpha _i{\mathfrak {c}}_i(P)\) is the \(\alpha \)-weighted sum of penalty terms, satisfies

$$\begin{aligned} {\mathbb {P}}(\alpha )=\bigcap _{i\in {\mathbb {I}}} M_i(C_i^\alpha ). \end{aligned}$$

The correspondence \({\mathbb {P}}:\varDelta _I\Rightarrow {\mathcal {P}}\) is upper hemicontinuous on \(\varDelta _I\), nonempty, weakly compact and convex valued.

The next step connects Propositions 3 and 4 with an equilibrium in the original economy \({\mathcal {E}}\). The graphs of the correspondences in Proposition 5 are \(\text {gr}({\mathbb {P}})=\{(\alpha , P)\in \varDelta _I\times {\mathcal {P}}:P\in {\mathbb {P}}(\alpha )\}\) and \(\text {gr}({\mathbb {GE}}^{-1})=\{(\alpha , P)\in \varDelta _I\times {\mathcal {P}}: \alpha \in {\mathbb {GE}}(P)\}\).

Proposition 5

The tuple \(\bigl ((C_i^{\alpha ^*}),\text {E}^{P^*}[ u'_{\alpha ^*}(E)\cdot ]\bigr )\) is an Arrow–Debreu equilibrium in the economy \({\mathcal {E}}\) if and only if \( \varPhi (\alpha ^*,P^*)=0 {\textit{ and }} P^*\in {\mathbb {P}}(\alpha ^*)\), which is equivalent to the coincidence \((\alpha ,P)\in \text {gr}({\mathbb {GE}}^{-1})\cap \text {gr}({\mathbb {P}})\) or \(P\in {\mathbb {P}}\circ {\mathbb {GE}}(P)\).

Proof of Theorem 1

Define the functional \(\rho :\varDelta _I\times {\mathcal {P}}\rightarrow {\mathbb {R}}\) by

$$\begin{aligned} \rho (\alpha ,P)=\min _{i\in {\mathbb {I}}} \varPhi _i(\alpha ,P). \end{aligned}$$

By the second part of Lemma 2, \(\varPhi _i(\alpha , \cdot )\) is linear and weakly continuous; hence, \(P\mapsto \rho (\alpha , P)\) is weakly continuous. From an application of the second part of Lemma 2, the continuity of \(\alpha \mapsto \rho (\alpha ,P)\) follows, since the pointwise infimum of a finite collection of continuous functions is again continuous. Since the maximum of \(\alpha \mapsto \rho (\alpha ,P)\) over \(\varDelta _I\) is by construction a zero (the unique equilibrium weight in \({\mathcal {E}}^P\)), the solution mapping \({\mathbb {GE}}\) is also given by

$$\begin{aligned} {\mathbb {GE}}(P)=\text {arg}\max _{\alpha \in \varDelta _I} \rho (\alpha ,P). \end{aligned}$$

Therefore, by Berge’s maximum theorem and Proposition 3, \({\mathbb {GE}}\) is a single-valued and upper hemicontinuous correspondence and hence continuous when viewed as a function. By Proposition 4, \({\mathbb {P}}\circ {\mathbb {GE}}:{\mathcal {P}}\rightarrow {\mathcal {P}}\) is a composition of upper hemicontinuous correspondences and hence again upper hemicontinuous.

\({\mathbb {P}}\) is convex and weakly compact valued and hence so is \({\mathbb {P}}\circ {\mathbb {GE}}\). The fact that the vector space of a signed measure on \((\varOmega ,{\mathcal {F}})\) equipped with the topology of weak convergence is a locally convex topological vector space allows us to apply the Kakutani–Glicksberg–Fan fixed-point theorem (Theorem 17.55 in Aliprantis and Border 2006) with respect to \({\mathbb {P}}\circ {\mathbb {GE}}\), and the result follows by Proposition 5. \(\square \)

Lemma 2

  1. 1.

    For each \(P\in {\mathcal {P}}\), the function \(\varPhi (\cdot ,P)\) is continuous in the interior of \(\varDelta _I\) and \(\Vert \varPhi (\alpha ,P)\Vert _{{\mathbb {R}}^I}\rightarrow +\infty \) whenever \(\alpha _i\rightarrow 0\) for some \(i\in {\mathbb {I}}\).

  2. 2.

    For each \(\alpha \in \varDelta _I\), the function \(\varPhi (\alpha ,\cdot )\) is weakly continuous.

Proof of Lemma 2

  1. 1.

    This follows from Proposition 3 and the continuous differentiability of each \(u_i\). Since \(P\in {\mathcal {P}}\) is fixed, the limit behavior follows from the same argument as in the standard single-prior case (see Dana 1993).

  2. 2.

    Let \(\{P_n\}_{n\in {\mathbb {N}}}\) be a sequence in \({\mathcal {P}}\) which converges weakly to some prior P. According to the first result at the beginning of “Appendix,”

    $$\begin{aligned} \lim _{n\rightarrow \infty } \text {E}^{P_n}[u'_{\alpha }(E)\cdot (C_i^{\alpha }-E_i) ]= \text {E}^{P}[u'_{\alpha }(E)\cdot (C_i^{\alpha }-E_i ) ] \end{aligned}$$

    which proves continuity in the weak topology.

\(\square \)

Proof of Corollary 2

Theorem 1 guarantees the existence of a Pareto optimal allocation, by the application of the first welfare theorem.

To prove the existence of a continuously differentiable selection, consider that the well-defined mapping \(C^\cdot (\cdot ):\varDelta _I\times {\mathbb {R}}_{+}\rightarrow {\mathbb {R}}^I_+\) is the unique solution of the pointwise problem

$$\begin{aligned} C^\alpha (e)=\text {arg}\max _{x_i\ge 0,\sum x_i\le e}\sum \alpha _i u_i(x_i),\quad (\alpha ,e)\in \varDelta _I\times {\mathbb {R}}_{+}, \end{aligned}$$

which is continuously differentiable on \(int(\varDelta _I)\times {\mathbb {R}}_{++}\), the interior of \(\text {dom}(C)\). For every \(\alpha \in \varDelta _I\), there is a \(P\in {\mathcal {P}}\) such that the modified economy \({\mathcal {E}}^P\) with \(\text {dom}({\mathfrak {\tilde{c}}}_i)=\{P\}\), \(i\in {\mathbb {I}}\), satisfies the same first-order condition as in the original economy \({\mathcal {E}}\)

$$\begin{aligned} \mu \in L^*,\quad \text {d}\mu =u'_\alpha (E)\text {d}P=\alpha _iu'_i\left( C_i^\alpha (E)\right) \text {d}P, \end{aligned}$$

for every \(i\in {\mathbb {I}}\) such that \(\alpha _i\ne 0\). This implies the \(\alpha \)-efficiency of \((C_i^\alpha (E))_{i\in {\mathbb {I}}}\) in the original and \({\mathfrak {\tilde{c}}}_i\)-modified economy. Feasibility holds by construction. Hence, \(C\in {\mathbb {C}}\) is a continuously differentiable selection in \(\alpha \). \(\square \)

Proof of Proposition 3

Since only P-a.s. uniqueness is required, the proof follows the lines of Dana (1993), where the present commodity-price duality is given by \(\langle L,L^\infty (P) \rangle \). \(\square \)

Proof of Proposition 4

The allocation \((C_1^\alpha ,\ldots , C_i^\alpha )\in \varLambda (E)\) can be related to an \(\alpha \)-weighted program \((U^{\alpha },\varLambda (E))\). The FOCs \(\alpha _i {u_i}'(C_i^\alpha ) \text {d}P_i=\alpha _k {u_k}'(C_k^\alpha ) \text {d}P_k \), \(i,k\in {\mathbb {I}}\), require the same support of the representing measures. Since \(E>0\) q.s. and \(E_i \in L_\oplus \), we have \(\alpha _i{u_i}'(C_i^\alpha )>0\) q.s., by the strict monotonicity of each \(u_i\). By the FOC, a singularity of \(P_i\) and \(P_k\) on some \(A\in {\mathcal {F}}\) with \(\max (P_i(A),P_k(A))>0\) can be ruled out. This implies \(P^*=P_i\) for all i, since no two priors in \({\mathcal {P}}\) are mutually equivalent.

Lemma 1 yields \(\partial U_i(C_i^\alpha )\ne \emptyset \), for each \(i\in {\mathbb {I}}\). That \(\bigcap _{i\in {\mathbb {I}}}\partial \alpha _i U_i(C_i^\alpha )=\partial U_{\alpha }(E)\ne \emptyset \) is weakly compact and convex follows from Proposition 6.6.4 in Laurent (1972). The intersection of compact sets is again compact. Let \(\bar{P}\in {\mathbb {P}}(\alpha )\) and we derive

$$\begin{aligned} U^\alpha (E)=\max _{(X_i)\in \varLambda (E)}\sum \alpha _i U_i(X_i)= & {} \sum \alpha _i \min _{P\in {\mathcal {P}}} \left\{ \text {E}^P[u_i(C_i^\alpha )]+{\mathfrak {c}}_i(P)\right\} \\\le & {} \text {E}^{\bar{P}}\left[ \sum \alpha _i u_i(C_i^\alpha )\right] + \sum \alpha _ i {\mathfrak {c}}_i(\bar{P})\\= & {} \min _{P\in {\mathcal {P}}} \text {E}^P\left[ u_\alpha (E)\right] +{\mathfrak {c}}_\alpha (P)\\\le & {} \text {E}^{ P^*}\left[ \sum \alpha _i u_i(C_i^\alpha )\right] + {\mathfrak {c}}_\alpha (P^*) \\= & {} U^\alpha (E) \end{aligned}$$

where the pointwise definition of \(u_\alpha \) can be found in the footnote in Assumption 1.

The upper hemicontinuity of \({\mathbb {P}}\) follows from Berge’s maximum theorem with respect to the \(\alpha \)-parametrized program \(\min _{P\in {\mathcal {P}}} \text {E}^P\left[ u_\alpha (E)\right] +{\mathfrak {c}}_\alpha (P)\). Here the assumed continuity of each \({\mathfrak {c}}_i\) is applied. \({\mathbb {P}}\) is weakly compact and convex valued. \(\square \)

Parts of the proof of Proposition 5 follow the ideas of Sect. 3 in Dana (2004).

Proof of Proposition 5

\(\Leftarrow \)”: Each prior \(P\in {\mathbb {P}}(\alpha )\) is associated with the super-gradients, for each \(i\in {\mathbb {I}}\), \(D U_i(C_i)(X)=\text {E}^P[u'_i(C_i)X ] \) simultaneously. A possible prior \(P\notin {\mathbb {P}}(\alpha )\) with a zero in the excess demand is not related to at least one relevant agent k’s first-order condition of \(\alpha \)-efficiency. If \(\varPhi \ne 0\) then only an equilibrium with transfer payment results.

\(\Rightarrow \)”: By Proposition 3, \(\varPhi (\alpha ^*,P^*)=0\) and \(P^*\in {\mathbb {P}}(\alpha ^*)\) implies the existence of an equilibrium \(( \{C^{P^*,\alpha ^*}_i\}_{i\in {\mathbb {I}}},u_{\alpha ^*}'(E))\) under \({\mathcal {E}}^{P^*}\). Equilibrium consumption \(C^{P^*,\alpha ^*}_i\) corresponds to the \(\alpha ^*\)-efficient consumption of agent i. We get

$$\begin{aligned} \text {E}^{P^*}[u_{\alpha ^*}'(E) (C-E_i)]\le 0 \Longrightarrow \text {E}^{P^*}\left[ u_i(C)\right] \le \text {E}^{P^*}[u_i(C^{P^*,\alpha ^*}_i)]. \end{aligned}$$

This implies \(U_i(C)\le U_i(C^{P^*,\alpha ^*}_i)\), due to \(U_i(C)\le \text {E}^{P^*}[u_i(C)]-{\mathfrak {c}}_i (P^*)\). Hence, the \(I+1\) tuple \((\{C^{P^*,\alpha ^*}_i\}_{i\in {\mathbb {I}}},\text {E}^{P^*}[ u_{\alpha ^*}'(E)\cdot ])\) is also an equilibrium of the original economy \({\mathcal {E}}\).

The last statement of the proposition can be seen as follows:

$$\begin{aligned} (\alpha ,P)\in \text {gr}({\mathbb {GE}}^{-1})\cap \text {gr}({\mathbb {P}}) \Leftrightarrow&P\in {\mathbb {P}}(\alpha )\text { and } {P}\in {\mathbb {GE}}^{-1}(\alpha ) \\ \Leftrightarrow&P\in {\mathbb {P}}(\alpha )\text { and } \alpha \in {\mathbb {GE}}(P) \\ \Leftrightarrow&P\in {\mathbb {P}}\circ {\mathbb {GE}}(P) \end{aligned}$$

\(\square \)

1.2.1 Details of Examples 4 and 5 and Corollary 3

G-martingale representation theorem (Soner et al. (2011)): Every \(X\in L\) has the unique representation \(X={\mathbb {E}}_G[X]+\int _0^t \eta _s\text {d}{B}_s-K_T\), where K is a continuous, increasing process with \(K_0=0, K_T\in L\). A direct consequence yields an alternative representation of all weakly ambiguity-free claim in L:

$$\begin{aligned} \text {M}_{\mathcal {P}}=\left\{ X\in L: X= {\mathbb {E}}_G[X]+\int _0^T \eta _t\text {d} {B}_t\right\} , \end{aligned}$$

where \(\eta \) is an adapted process with \(\big \Vert \int _0^T (\eta _t)^2 \text {d} t\big \Vert _{\mathcal {P}}<\infty \).

We have repeatedly used the following version of Itô’s Lemma for one-dimensional G-Brownian motion.

Itô-formula, Li and Peng (2011): Let \(\varPhi : {\mathbb {R}}\rightarrow {\mathbb {R}}\) be twice differentiable and \(\text {d} X_t = V_t \text {d} {B}_t\), then \(\varPhi (X_t)=\varPhi (X_0)+\int _0^t \varPhi _x(X_r)V_r \text {d}{B}_r+ \frac{1}{2}\int _0^t {\varPhi }_{xx}(X_r)V^2_r \text {d}\langle {B}\rangle _r\).

Proof of Corollary 3

We are in the setting of Theorem 4.1 in Riedel and Beissner (2014). Implementability via a Bachelier asset market is characterized by the requirement \(\xi _i^\psi \in \text {M}_{\mathcal {P}} \) for each \(i\in {\mathbb {I}}\).

Define \(x\mapsto H_i(x):= u_\alpha '(x)(C^\alpha _i(x) - \varphi _i(x)) \); by the same arguments as in Corollary 2, each function \(H_i:{\mathbb {R}}_{++}\rightarrow {\mathbb {R}}\) is twice continuously differentiable. Since \(E=E_T \in \text {M}_{\mathcal {P}} \) and by the same argument as in Example 4, we have

$$\begin{aligned} \xi ^\psi _i=H_i(E)=H_i ({\mathbb {E}}[E])+ \int _0^T H_i' (E_t) E_t\sigma ^E_t \text {d} B_t+\frac{1}{2} \int _0^T H''_i (E_t) (E_t\sigma ^E_t)^2\text {d}\langle {B}\rangle _t,\nonumber \\ \end{aligned}$$

where \(H_i ({\mathbb {E}}[E])={\mathbb {E}}[ H_i (E)]\) holds by assumption. \(H_i(E)\in \text {M}_{\mathcal {P}}\) holds if and only if the third term on the right-hand side in (14) is zero. This follows from the alternative characterization of weakly ambiguity-free payoffs in (13).

In view of 14, this is equivalent to \(H''_i (E_t) =0\), since \(E_t \sigma ^E_t>0\). Differentiating each \(H_i\) twice gives (11), since \({u_\alpha }'(x)=\alpha _i {u_i}'(C_i^\alpha (x))\). \(\square \)

Finally, I give the postponed derivation of the equation (9) in Example 5. Suppose each optimal consumption has the complete representation property, we can write

$$\begin{aligned} F^\alpha _2(B_T)= G\left[ F_\alpha ^2(B_T)\right] +\int _0^T \theta _t^ 2\text {d}B_t- \int _0^T G\left( \eta ^2_t\right) \text {d}t+\frac{1}{2}\int _0^T \eta ^2_t \text {d}\langle B\rangle _t, \end{aligned}$$

where \(\theta _t^2=f^2_{x}(t,B_t)\), \(\eta _t^2=f^2_{xx}(t,B_t)\) and \(f^2(T,B_T)=F^\alpha _2(B_T)=C_2^\alpha \). A sufficient condition for the decomposition of K is the boundedness of \(\partial _x F^\alpha _i(x)\) on \({\mathbb {R}}_+\) (see Theorem 2.2 in Chapter 4 of Peng (2010)).

As illustrated in Example 5, \(F^2_\alpha \) is convex, and by Sect. 1 in Chapter II of Peng (2010), it follows that \(f^2(t,\cdot ):{\mathbb {R}}\rightarrow {\mathbb {R}}_+ \) is convex for each \(t\in [0,T]\). Hence \(f^2_{xx}\ge 0\) and we deduce that the last two terms of (15) can be written as

$$\begin{aligned} -K_T^2= & {} -\int _0^T G\left( f^2_{xx}(t,B_t)\right) \text {d}t+\frac{1}{2}\int _0^T f^2_{xx}(t,B_t) \text {d}\langle B\rangle _t\\= & {} -\frac{1}{2} \int _0^T \sup _{\sigma \in [\underline{\sigma }, \overline{\sigma }]} \sigma f^2_{xx}(t,B_t) +\hat{a}_t f^2_{xx}(t,B_t) \text {d}t\\= & {} \frac{1}{2}\int _0^T \left( \hat{a}_t -\overline{\sigma } \right) f^2_{xx}(t,B_t) \text {d}t. \end{aligned}$$

We have \(-K^2_t\equiv 0\) \(P^{\overline{\sigma }}\)-a.s. and \(-K^2_t< 0\) under every other prior in \({\mathcal {P}}{\setminus }\{P^{\overline{\sigma }}\}\).

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Beissner, P. Equilibrium prices and trade under ambiguous volatility. Econ Theory 64, 213–238 (2017).

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