Abstract
This paper establishes, in the setting of Brownian information, a general equilibrium existence result in a heterogeneous agent economy. The existence is generic among income distributions. Agents differ moreover in their stochastic differential formulation of intertemporal recursive utility. The present class of utility functionals is generated by a recursive integral equation and incorporates preference for the local risk of the stochastic utility process. The setting contains models in which Knightian uncertainty is represented in terms of maxmin preferences of Chen and Epstein (Econometrica 70:1403–1443, 2002). Alternatively, Knightian decision making in terms of an inertia formulation from Bewley (Decis. Econ. Financ. 25:79–110, 2002) can be modeled as well.
This is a preview of subscription content,
to check access.Similar content being viewed by others
Notes
For a first overview we refer to [25].
The case of finite commodities can be treated by the same argumentation, we refer to [10].
Measures on \(\Omega \times [0,T]\) which allow considerations of terminal consumption are possible. In this case the BSDE in (2) has a nontrivial terminal condition.
This means \(\vert f(t,c,u,\sigma )f(t,c,u',\sigma ') \vert \le k\vert (u,\sigma )(u',\sigma ')\vert \) for all \(u,u'\in \mathbb {R}\) and \(\sigma ,\sigma ' \in \mathbb {R}^n\).
An alternative would be related to Baire’s Category theorem. A set of first category is contained in a countable union of closed sets with an empty interior. However it detects an empty interior for first category sets. This notion has little measure theoretic connection. As mentioned in [24] on page 318, a topological generic set “has to be thought of much less sharp than measuretheoretic concept available in the finitedimensional case”.
This indicates, that the condition on the aggregate endowment in Theorem 1 is less strong than one might suspect at first glance.
In principle, each \(\kappa ^k_i\) can also be a bounded \(\mathbb {F}\)adapted process.
Strict concavity is achieved by the same arguments as in the last part of the proof in Lemma A.8. of [7]. As in the present case all priors in \(\mathcal {P}_i\) are mutually equivalent.
This assumption holds in our multipleprior economies of Corollary 1 and 2. Following the implicitfunction argument in Sect. 2.5 of [10] there is a twice continuously differentiable function \(K_i\), depending on \((e_t,\mathcal {E}_t,U_t)=(e_t,\{\mathcal {E}_t^{\mathfrak {u}^i, \mathfrak {s}^i},U^i_t\}_{i=1}^m)\), such that the \(\alpha \)efficient consumption of agent \(i\) can be written as \(c^i_t=K^i(t,e_{t},\mathcal {E}_{t},U_{t})\), where \(\mathcal {E}^{\mathfrak {u}^i,\mathfrak {s}^i}_0=\alpha ^i\).
The dynamics of the efficient allocation are related to the solution \((\mathcal {E}_{t},U_t,\sigma _t)\) of a fully coupled system of Forward–BackwardSDE’s. In [12], the system is discussed in the SDU case. See also [22] for a treatment of linked recursive utility. At this stage, the Pareto weights are time dependent and correspond in the present case to the stochastic process \(\mathcal {E}^{0,\mathfrak {s}^i}_t\) as a solution of (3), where the adapted selection \(\mathfrak {s}_t^i\) satisfies (by Proposition 2 and in the notation of footnote 10) \(\mathfrak {s}_t^i\in \partial _{\sigma }f^i\left( K_i\big (t,e_t,\mathcal {E}_t, U_t\big ),\sigma ^i\right) ,\quad t\in [0,T].\)
We refer to Theorem 3.1.4 in [4]
\(L(e)\) denotes the order ideal \(L\). Details can be found in [2].
References
Aliprantis, C.: Separable utility functions. J. Math. Econ. 28(4), 415–444 (1997)
Aliprantis, C., Burkinshaw, O.: Locally Solid Riesz Spaces with Applications to Economics, vol. 105. American Mathematical Society, Providence (2003)
Anderson, R.M., Zame, W.R.: Genericity with infinitely many parameters. BE J. Theor. Econ. 1(1), 1–64 (2001)
Barbu, V., Precupanu, T.: Convexity and Optimization in Banach Spaces, vol. 10. Springer, Dordrecht (1986)
Bewley, T.: Knightian decision theory. Part I. Decis. Econ. Financ. 25(2), 79–110 (2002)
Chen, Z., Epstein, L.: Ambiguity, risk, and asset returns in continuous time. Econometrica 70(4), 1403–1443 (2002)
Dana, R., Riedel, F.: Intertemporal equilibria with Knightian uncertainty. J. Econ. Theory 148(4), 1582–1605 (2013)
Duffie, D.: Dynamic Asset Pricing Theory, 3rd edn. Princeton University Press, Princeton (1996)
Duffie, D., Epstein, L.: Stochastic differential utility. Econometrica 60(2), 353–394 (1992)
Duffie, D., Geoffard, P., Skiadas, C.: Efficient and equilibrium allocations with stochastic differential utility. J. Math. Econ. 23(2), 133–146 (1994)
Duffie, D., Zame, W.: The consumptionbased capital asset pricing model. Econometrica 57(6), 1279–1297 (1989)
Dumas, B., Uppal, R., Wang, T.: Efficient intertemporal allocations with recursive utility. J. Econ. Theory 93(2), 240–259 (2000)
El Karoui, N., Peng, S., Quenez, M.: Backward stochastic differential equations in finance. Math. Financ. 7(1), 1–71 (1997)
Epstein, L., Wang, T.: Intertemporal asset pricing under Knightian uncertainty. Econometrica 62(3), 283–322 (1994)
Epstein, L.G., Miao, J.: A twoperson dynamic equilibrium under ambiguity. J. Econ. Dyn. Control 27(7), 1253–1288 (2003)
Fan, K., Glicksberg, I., Hoffman, A.: Systems of inequalities involving convex functions. Proc. Am. Math. Soc. 8, 617–622 (1957)
Gilboa, I., Schmeidler, D.: Maxmin expected utility with nonunique prior. J. Math. Econ. 18(2), 141–153 (1989)
Giles, J.: Convex Analysis with Application in the Differentiation of Convex Functions. Pitman Pub, Boston (1982)
Lazrak, A.: Generalized stochastic differential utility and preference for information. Ann. Appl. Probab. 14(4), 2149–2175 (2004)
Lazrak, A., Quenez, M.C.: A generalized stochastic differential utility. Math. Oper. Res. 28(1), 154–180 (2003)
Le Van, C.: Complete characterization of Yannelis–Zame and Chichilnisky–Kalman–MasColell properness conditions on preferences for separable concave functions defined in \(L_+^p\) and \(L^p\). Econ. Theory 8(1), 155–166 (1996)
Levental, S., Sinha, S., Schroder, M.: Linked recursive preferences and optimality. To Appear: Math. Financ. (2014). doi:10.1111/mafi.12047
Maccheroni, F., Marinacci, M., Rustichini, A.: Ambiguity aversion, robustness, and the variational representation of preferences. Econometrica 74(6), 1447–1498 (2006)
MasColell, A.: The Theory of General Economic Equilibrium: A Differentiable Approach. Cambridge University Press, Cambridge (1990)
MasColell, A., Zame, W.: Equilibrium theory in infinite dimensional spaces. Handb. Math. Econ. 4, 1835–1898 (1991)
Pardoux, E., Peng, S.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14(1), 55–61 (1990)
Podczeck, K.: Equilibria in vector lattices without ordered preferences or uniform properness. J. Math. Econ. 25(4), 465–485 (1996)
Rigotti, L., Shannon, C.: Uncertainty and risk in financial markets. Econometrica 73(1), 203–243 (2005)
Schroder, M., Skiadas, C.: Optimal lifetime consumptionportfolio strategies under trading constraints and generalized recursive preferences. Stoch. Process. Appl. 108(2), 155–202 (2003)
Skiadas, C.: Recursive utility and preferences for information. Econ. Theory 12(2), 293–312 (1998)
Wegge, L.: Mean value theorem for convex functions. J. Math. Econ. 1(2), 207–208 (1974)
Yannelis, N., Zame, W.: Equilibria in Banach lattices without ordered preferences. J. Math. Econ. 15(2), 85–110 (1986)
Acknowledgments
I thank Frank Riedel for valuable advice, a referee for comments and suggestions, and Chiaki Hara, Frederik Herzberg and Kasper Larsen for fruitful discussions. Financial support through the German Research Foundation (DFG) and the International Graduate College “Stochastics and Real World Models” are gratefully acknowledged. This paper is based on a former working paper entitled “Existence of Arrow–Debreu Equilibrium with Generalized Stochastic Differential Utility.”
Author information
Authors and Affiliations
Corresponding author
Appendices
Appendix A: Proofs
1.1 A.1 Section 2
Proof of proposition 1
The first two assertions can be found in [13], the third is a modification of Proposition 1 in [9]. \(\square \)
The partial supergradient of the aggregator with respect to the corresponding component \(x\) is denoted by \(D_x f(t,\cdot ,x,\cdot )\). The partial superdifferential in \(u\) and \(\sigma \), namely \(\partial _{u,\sigma }f \) at \((t,c,u,\sigma )\), consists of all pointwise supergradients \((D_{U} f(t,c,u,\sigma ),D_{\sigma } f(t,c,u,\sigma ))=(\mathfrak {u},\mathfrak {s})\in \mathbb {R}\times \mathbb {R}^n\) such that \(f(t,c+x,u+y_1,\sigma +y_2)\le f(t,c,u,\sigma )+\partial _c f(t,c,u,\sigma )x+ \mathfrak {u} y_1+ \mathfrak {s} y_2\). For \(k=U,\sigma \), the stochastic process \(( D_{k} f(t,c_t,U_t,\sigma _t))_{t\in [0,T]}\) is denoted by \(D_k f\).
Proof of proposition 2
The density representation of the supergradient follows from the auxillary Lemma 1, with \(t=0\). Following the proof of Theorem 4.3 in [1] and applying the concave alternative of [16], we can show that the right and lefthand derivatives represent the superdifferential in terms of the order interval:
An application of results on BackwardSDE’s depending on parameters (see Proposition 2.4 [13]), proofs that
In this case, the closed formula of the adjoint process is given by \(\mathcal {E}^{D^+_U f,D^+_\sigma f}\). The superdifferential can be written as a specific order interval in \(\text {L}^2\), i.e. \(\partial U(c)=\big [\mathcal {E}^{D^+_U f,D^+_\sigma f}\cdot \partial _c f, \mathcal {E}^{D^_U f,D^_\sigma f}\cdot \partial _c f\big ]\) and the assertion follows. \(\square \)
Auxillary Lemma 1
Fix \(t\in [0,T]\) and suppose the conditions of Proposition 2 hold, then for any direction \(h \in \text {L}^2\) such that \(c+h \in \text {L}^2_{++}\) we have
Proof of Auxillary Lemma 1
Take a \(c\) and \(h\) as stated. The related utility processes \(U\) and \(U^h\) are given by
with terminal conditions \(U_T=0=U^h_T\). Set \(\mathcal {E}_t:=\mathcal {E}_t^{D_U f,D_\sigma f}\).
Claim: We have \(\mathbb {E}[\sup _{t\in [0,T]} \mathcal {E}_t^2]<\infty \).
proof The process \(\mathcal {E}\) admits a decomposition \(\mathcal {E}_t= \lambda _t \cdot \Gamma _t\) and hence by the boundedness of the supergradient w.r.t. the aggregator in utility
Boundedness of the supergradient w.r.t. aggregator in the intensity component \(\sigma \) implies
the Novikov criterion is satisfied, hence the process \(\Gamma \), given by
is indeed a martingale. With regard to the local martingale \(\int _0^{\cdot } \Gamma _s d \Gamma _s\), we take a localizing sequence of stopping times \((\tau _n)_{n\in \mathbb {N}}\subseteq [0,T]\) such that \(\tau _n \mathop {\longrightarrow }\limits ^{n\rightarrow \infty } T\, \mathbb {P}\)a.s., and we see that for each \(n,\, \Big ( \int _0^{t\wedge \tau _n} \Gamma _s d \Gamma _s \Big )_{t\in [0,T]}\) is a martingale.
By It\(\hat{O}\)’s formula, the quadratic variation of \(\Gamma \), the boundedness of the supergradient in the intensity component \(\sigma \) and Fubini’s theorem, we get
Applying the Gronwall lemma with \(g(s)= \mathbb {E}[\Gamma ^2_{s\wedge \tau _n}]\), we conclude that \(g(T) \le \exp (T k^2)<\infty \) and by the dominated convergence, \(\mathbb {E}[\Gamma _{T}^2] \le \exp (T k^2)\). Since \(\Gamma \) is a martingale, \(\Gamma ^2\) is a submartingale. By virtue of Doob’s maximal inequality, (7) and (8), we deduce
\(\square \)
To see that \(\mathcal {E}\partial _c f\in \text {L}^2\), we argue that there is a constant \(C>0\) with \(c>C\, \mathbb {P}\otimes dt\)a.e. and, by Assumption 1, the process \(t\mapsto \partial _c f(t,c_t,U_t,\sigma _t)\) takes values in \([0,K]\) \(\mathbb {P}\otimes dt\)a.e., where \(K=\sup _{(t,u,\sigma )}\partial _c f(t,C,u,\sigma )\). Since \(c\) is bounded away from zero, we have \(\partial _c f\in \text {L}^{\infty }( \mathbb {P}\otimes dt)\) and \(\mathcal {E}\partial _c f\in \text {L}^2\) follows by the previous claim.
The remaining part follows from Lemma A.5 in [29]. \(\square \)
1.2 A.2 Section 3
We begin with the first order conditions of optimality for concave and not necessarily Gateaux differentiable functionals. Define the set of feasible directions at \(c^i\) given by \(F(c^i)= \left\{ h \in \text {L}^2 : \exists \mu >0 \quad c^i+\mu h \in \text {L}_+^2 \right\} \) and the set of feasible transfers \(H(c)=\left\{ h\in \text {L}^{2,m}: \sum h^i=0, h^i\in F(c^i), \quad 1\le i\le m\right\} \).
By \(\partial _{\text {L}^{2,m}}U\) we denote the superdifferential of a functional \(U\) on \(\text {L}^{2,m}\). We write \(\langle D U(c),h \rangle \) for \(DU(c)(h)\), where \(D U(c)\in \partial U(c)\) is a supergradient.
Proof of proposition 3
The properties of the aggregator imply the norm continuity and concavity of the utility functionals. Alaouglu’s theorem implies the weak compactness of \(\Lambda (e)\). Under concavity and upper semicontinuity, weak upper semicontinuity of the utility functionals follows. \(\alpha \)efficient allocation exists by an abstract Weierstrass argument. The equivalence between \(\alpha \)efficiency and Pareto optimality is standard in economic theory.

1.
Let \((h^1,\ldots ,h^m)=h\in H(\hat{c})\). By assuming there is a \(DU\in \bigcap _{i=1}^m \partial \alpha _i U^i(\hat{c}^i)\), with Riesz representation \(\pi \). This means for each \(i\), there is a \(D \alpha _i U^i(\hat{c}^i) \in \partial \alpha _i U^i(\hat{c}^i)\) such that \( D \alpha _i U^i(\hat{c}^i)=\langle \pi ,\cdot \rangle \) and therefore
$$\begin{aligned} \sum \langle D \alpha _i U^i(\hat{c}^i), h^i \rangle = \sum \langle \pi , h^i \rangle =\Big \langle \pi ,\sum h^i \Big \rangle =\langle \pi ,0\rangle =0. \end{aligned}$$Since each \(U^i\) satisfies the conditions of Proposition 4, \((\hat{c}_1,\ldots ,\hat{c}_m)\) is an \(\alpha \)efficient allocation.

2.
For each \(i\), the consumption process \(\hat{c}^i\) is bounded away from zero. This implies \(\text {L}^\infty ( \mathbb {P}\otimes dt) \subseteq F(\hat{c}^i)\). Suppose the converse, there are two agents \(i\) and \(j\) such that \(\partial \alpha _i U^i(\hat{c}^i) \cap \partial \alpha _j U^j(\hat{c}^j)=\emptyset \). Then there is an \(h_i\in F(\hat{c}^i)\setminus \left\{ 0 \right\} \), an \(h_j\in F(\hat{c}^j)\setminus \left\{ 0 \right\} \) and an \(h\in H(\hat{c})\) with \(h^k=0\) if \(k\notin \left\{ i,j\right\} \) such that, for all \(D \alpha _i U^i (c^i)\in \partial \alpha _i U^i(c^i)\) and \(D\alpha _j U^j(\hat{c}^j)\in \partial \alpha _j U^j(\hat{c}^j)\), we have
$$\begin{aligned} 0&< \mathbb {E}\left[ \int _0^T h^i_t \pi ^i(\hat{c}^i)_t  h^i_t \pi ^j(\hat{c}^j)_t dt\right] \\&= \mathbb {E}\left[ \int _0^T h^i_t \pi ^i(\hat{c}^i)_t + h^j_t \pi ^j(\hat{c}^j)_t dt\right] = \sum \langle D \alpha _i U^i( \hat{c}^i), h^i\rangle _{\text {L}^{2}}, \end{aligned}$$where \(\pi ^j(\hat{c}^j)\) is the Riesz representation of \(D\alpha _j U^j(\hat{c}^j)\), a contradiction to Proposition 4.
\(\square \)
For the proof of Proposition 3 we applied the following result.
Proposition 4
Assume that for each \(i\), the utility functional \(U^i\) is upper semicontinuous, strictly increasing, concave and let the aggregate endowment \(e\) be bounded away from zero.
Then \(\alpha \)efficiency of \(\hat{c}\in \Lambda (e)\) is equivalent to the existence of a \(D U^i ( \hat{c}^i) \in \partial U^i(\hat{c}^i)\), for each \(i\), such that \(0 \ge \sum \langle D \alpha _i U^i( \hat{c}^i), h^i\rangle ,\, h\in H( \hat{c})\).
Proof of proposition 4
Let \(g(c^1,\ldots ,c^m)=\sum c^i e\) and \(g_i(c^1,\ldots ,c^m)=c^i\). Then \(\alpha \)efficiency for \(\hat{c}=(\hat{c}^1,\ldots ,\hat{c}^m)\) can be written as
Since \(e\) is bounded away from zero, the Slater condition holds. We apply the Kuhn–Tucker theorem (see Theorem 3.1.4 in [4]), to \(U^{\alpha }\). Hence, \(\hat{c}\) is \(\alpha \)efficient if and only if there are constants \(\mu _i,\mu \ge 0\) such that \(0\in (\partial _{\text {L}^{2,m}} U^{\alpha })(\hat{c})+ \mu \nabla _{\text {L}^{2,m}} g(\hat{c}) +\sum \mu _i \nabla _{\text {L}^{2,m}} g_i(\hat{c})\) and \(\mu g(\hat{c})=0,\, \mu _i g_i(\hat{c})=0,\, i=1,\ldots ,m\). Taking the nonnegativity constraints into account and the existence of a \(D_{\text {L}^{2,m}}U^{\alpha }(\hat{c})\in (\partial _{\text {L}^{2,m}}  U^{\alpha })(\hat{c})\), this is equivalent to
Taking the feasible transfers \(h\in H(\hat{c})\) into account, we have
Since the \(U^i\)’s are strictly increasing, \(g(\hat{c})=0\) follows.
Proof of Lemma 1
Let \(\nu =\mathbb {P} \otimes dt\) and take a \(c\in \text {L}_{++}^{2,m}\). For every \(i\) we have \(U^i(c^i)>U^i(0)\) since each \(U^i\) is strictly increasing.
Suppose some \(c^j\) is not bounded away from zero. Then for every \(h>0\) there is an \(\hat{H}=\hat{H}(h)\in \mathcal {O}\) such that \(\nu (\hat{H})>0\) and \(c^j\le h\) on \(\hat{H}\). Since \(e\) is bounded away from zero, we have \(e>C \nu \)a.e. for some constant \(C>0\). This gives us, if \(C\) is small enough, that there is an agent \(k\) such that \(c^k\ge \frac{C}{m} \) on \(H'\subset \hat{H}\). We choose \(H=\{c^j<h \}\cap \{ \frac{C}{m}\le c^k\le C^k \}\) which has a positive measure.
On the other hand, since \(c=(c_1,\ldots ,c_m)\) is in the quasi interior of \(\text {L}_{+}^{2,m}\), for every \(i\), there is a set \(A^i\in \mathcal {O}\) with \(\nu (A^i)>0\) and a number \(a^i>0\) such that \(c^i\ge a^i\) on \(A^i\).
We show a Pareto improvement when multiples of \(H\) and \(A^j\) are traded between agent \(j\) and \(k\). Let \(\lambda ^k\in (0,h)\) and \(\lambda ^j \in (0,\frac{a_j}{2})\). Define the following BackwardSDE’s:
where \(U_0= U^j(c^j),\, U^A_0=U^j(c^j \lambda ^j 1_{A^j})\) and \(U^{AH}_0=U^j(c^j  \lambda ^j 1_{A^j}+\lambda ^k 1_H)\) are the corresponding evaluated utility functionals. We derive
The inequality employed the estimates in Lemma 3 and Lemma 4. Next, we compute appropriate estimates for the \(\sigma \) parts. By the CauchySchwartz inequality and the a priori estimates in [13], with \(\lambda ^2=2 k,\, \mu =1\) and \(\beta \ge 2k(1+ k)+1\), we derive:
The second inequality is a pointwise application of the mean value theorem, the usage of \(\lambda ^j<\frac{a^j}{2}\) and \(c^j\ge a_j\) on \(A^j\) and because \(\partial _c f^j\) is decreasing. Analogous arguments yield
Since \(h\) can be taken to be arbitrarily small, \({\underline{\delta }}_f ( 2 h)\) becomes arbitrarily large and by the last two derivations with \(e^{\circ j}=e^{\circ k_jT}\) and \(\hat{e}^{\circ j}= e^{\circ k_jT }\cdot (2 T{e^{\beta T}})^{1/2},\, \circ \in \{+,\}\):
A utility improvement of agent \(j\) is related to the strict positivity of the last term. An analogous derivation and a modification of Lemma 4 and Lemma 5 yield the corresponding inequality for agent \(k\). Hence, in order to achieve a Pareto improvement
must hold. If we take a sufficiently small \(h\), then, by the Inada style condition, \({\underline{\delta }}_f (2 h)\) becomes arbitrarily large. Consequently \(\nu (\hat{H})\) and hence \(\nu (H)\) become arbitrary small. We may choose \(A^j\) such that \(\nu (H)=\nu (A^j)>0\), this gives us
by choosing appropriate multiples \(\lambda ^k\in (0,h)\) and \(\lambda ^j \in (0,\frac{a_j}{2})\) we finally get:
This yields a Pareto improvement, contradicting that \((c^1,\ldots ,c^m)\) is a Pareto optimal allocation. Therefore, each \(c^j\) of the efficient allocation is bounded away from zero. \(\square \)
Proof of Lemma 2
By a modification to Lemma 1, each \(c^i\) is bounded away from zero. The assumption of a quasi interior allocation may be substituted by individual rationality.
Fix \(v\equiv 1\) as the properness vector. According to Proposition 2, a super gradient density \(\pi (D_{U,\sigma } f^i)\in \text {L}^{2}_{++}\) at \(c^i\) is given by
The parametrization is related to the superdifferential \(\partial _{U,\sigma } f^i\) of the aggregator \(f^i\). For later use define
where \(V(D_{U,\sigma } f^i)=\left\{ z\in \text {L}^{2}:\langle \pi (D_{U,\sigma } f^i), (1z)\rangle _{\text {L}^2} >0 \right\} \). We show that \(V\) is a neighborhood of \(0\) in \(\text {L}^{2}\). For each \(D_{U,\sigma } f^i\) there exists an open ball around zero which is contained in \(V(D_{U,\sigma } f^i)\). Choose an arbitrary
The positivity of \(\pi \) implies \(\langle \pi (D_{U,\sigma } f^i),z\rangle _{\text {L}^2}<\Vert \pi \Vert _{\text {L}^{1}} \langle \pi (D_{U,\sigma } f^i),1\rangle _{\text {L}^2}\). Hence, there is an open ball which is contained in \(V\).
Let \(c^i+\lambda (1z)\in \text {L}^{2}_{+}\), where \(z\in V\) is arbitrary and \(\lambda >0\) is sufficiently small. When \(\lambda >0\) tends to zero, the term \(\lambda ^{1}U^i(c^i+\lambda (1z) ) U^i(c^i)\) increases, due to the concavity of \(U^i\). Fix some \(z\in V\), whenever \(\lambda \searrow 0\), the limit of the quotient exists by Theorem 1 (p. 117) [18] and we have
The first inequality holds by Theorem 3 (p. 122) in [18]. The second inequality is valid since \(z\in V\supset B_{\varepsilon }(0)\). Now, consider a sufficiently small \(\lambda \) with \(U^i(c^i+\lambda (1z))>U^i(c^i)\). In other words, \(U^i\) is Fproper at \(c^i\). \(\square \)
The following two results are used in the proof of Lemma 1 and 2. The approach goes back to [11]. The aggregator is not differentiable in \(u\) and \(\sigma \) (but concave) and hence we need a mean value theorem for convex functions, see [31]. Lemma 3 and Lemma 4 are formulated so that an application to the contradiction argument in Lemma 1 fits the agent \(j\).
Lemma 3
Assume that \(U\) is a generalized stochastic differential utility generated by an aggregator \(f\) that satisfies Assumption 1. Let \(A\in \mathcal {O}\) and \(a>0\) be arbitrary. If \(y,x\in \text {L}^2_+\) with \(y\ge a\) on \(A,\, x=0\) on \(A^c\) and \(x\le \frac{a}{2}\), then
Proof of Lemma 3
Let \((U_t,\sigma _t)_{t\in [0,T]}=(U,\sigma )\) be the solution of the utility process related to \(y\) and \((\bar{U},\bar{\sigma })\) the solution of the utility process related to \(yx\) where \(x\) is chosen as above. By assumption, \(f\) is differentiable in \(c\). We apply the classical mean value theorem to the consumption component. Since \(f\) is uniformly Lipschitz continuous in \(u\) and \(\sigma \), upper semicontinuity follows, we apply the mean value theorem for convex functions of [31] to \(f(t,c,\cdot ,\cdot )\). Hence, there is an \(\mathbb {R}\times \mathbb {R}\times \mathbb {R}^n\) valued process \((\xi ^c,\xi ^U ,\xi ^\sigma )\) such that
Observe \(U_t  \bar{U}_t\ge 0\), for all \(t\in [0,T]\), by Proposition 1 since \(x\ge 0\) and \(f\) is increasing in consumption. Combined with the boundedness of the supergradients, we derive:
The last inequality holds because \(x\mapsto \partial _c f(s,x,v,\sigma )\) is decreasing and using the estimate \(\overline{\delta }_f(\frac{a}{2})\), since \(y_s(\omega )+\xi _s^c(\omega )\ge \frac{a}{2}\) on \(A\). Finally, the first Stochastic Gronwall inequality (see Corollary B in the Appendix of [9]), evaluated at time zero yields
\(\square \)
Lemma 4
Assume that \(U\) is a generalized stochastic differential utility generated by an aggregator \(f\) that satisfies Assumption 1. Let \(H\in \mathcal {O},\, h>0\) and \(y\in \text {L}^2_+\) with \(y\le h\) on \(H\). Then for every \(\lambda \in [0,h]\)
Proof of Lemma 4
Let \((U_t, \sigma _t)_{t\in [0,T]}=(U,\sigma )\) be the solution of the utility process of the process \(y\) and \((\bar{U},\bar{\sigma })\) the solution of the utility process of \(y+\lambda 1_H\). \(f\) is differentiable in consumption and concave in the other components. Applying the mean value theorem for \(c\), there is a \(\mathbb {R}^{2+n}\) valued process \((\xi ^c,\xi ^V,\xi ^\sigma )\) and we have
The inequality follows from the application of the estimates \({\underline{\delta }}_f(2h)\) (since \(y_s(\omega )+\xi _s^c(\omega )\le 2 h\) on \(H\)) and arguments similar to Lemma 4. We have \(U_s \bar{U}_s\ge 0\) since \(\lambda 1_H\ge 0\) and \(f\) is increasing. Finally, the second Stochastic Gronwall inequality (see again Corollary B in the Appendix of [9]), evaluated at time zero gives us
Appendix B: Quasiequilibrium in normed lattices
Let \((L,\tau )\) be the commodity space, a vector lattice with a Hausdorff, locally convex topology \(\tau \). We fix a pure exchange economy with \(m\in \mathbb {N}\) agents \(\mathtt {E}=\left\{ L_+,P_i,e^i\right\} _{1\le i\le m}\) in \(L\) such that \(P_i:L_+\rightarrow 2^{L_+}\) are the preference relations on the consumption set \(L_+\) and \(e^i\in L_+\) is the initial endowment of each agent.
An allocation \((x^1,\ldots ,x^m)\) is individually rational if \(e^i\notin P_i(x^i)\) for every \(i\).
A quasiequilibrium for \(\mathtt {E}\) consists of a feasible allocation \((x^1,\ldots ,x^m)\in L^m_+\), i.e. \(\sum x^i=e\), and a linear functional \(\pi :L\rightarrow \mathbb {R}\) with \(\pi \ne 0\) such that, for all \(i\, \pi (x^i)\le \pi (e^i)\) and for any \(i,\,\, y\in K_+\) with \(y \in P (x^i)\) implies \(\pi (y) \ge \pi (x^i)\). The quasiequilibrium is an equilibrium if \(y \in P (x^i)\) implies \(\pi (y) > \pi (x^i)\). Forward properness is a modification of a cone condition (see [32]).
Definition 1
A preference relation \(P:L_+\rightarrow 2^{L_+}\) is Fproper at \(x \in L_+\) if: There is a \(v\in L_+\), some constant \(\rho >0\) and a \(\tau \)neighborhood \(U\) satisfying, with \(\lambda \in (0,\rho ):\)
If \(z\in U\), then \(x+\lambda v z \in L_+\) implies \(x+\lambda v  \lambda z \in P(x)\)
The following assumption is needed to establish the existence of a quasiequilibrium.
Assumption 2
The economy satisfies the following conditions:

1.
\(y\notin P_i(y)\) and \(P_i(y)\) is for all \(y\in L_+\) and every \(i\in \{1,\ldots ,m\}\)

2.
There is a Hausdorff topology \(\eta \) on \(L\) such that \([0,e]\) is \(\eta \)compact and for every \(i\) the graph \(gr(P_i)=\left\{ (x,y)\in L \times L: x\in L_+, y\in P_i(x )\right\} \) is a relatively open subset of \(L_+\times L_+\) in the product topology \(\eta \tau \).

3.
\(P_i(y)\cap L(e)\ne \emptyset \) for all \(y\in [0,e]\) and every \(i\).

4.
\(L(e)\) ^{Footnote 13} is \(\tau \)dense in \(L\) and if \((x_1,\ldots ,x_m)\in L^m_+\) is an individually rational and Paretooptimal allocation, then, for every \(i,\, P_i\) is Fproper at \(x_i\).
Theorem 2
Suppose the economy E satisfies Assumption 2. Then there is an \(x\in L^m_+\) and a \(p\in L^*\) such that \((x,p)\) is a nontrivial quasiequilibrium.
This result is proved in [27]. If preferences are strictly monotone and continuous and the total endowment is strictly positive, the notions of equilibrium and quasiequilibrium coincide, see Corollary 8.37 in [2], where it is requested that \(L^*\) is a sublattice of the order dual \(L^{\star }\).
Rights and permissions
About this article
Cite this article
Beißner, P. Brownian equilibria under Knightian uncertainty. Math Finan Econ 9, 39–56 (2015). https://doi.org/10.1007/s1157901401331
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s1157901401331
Keywords
 Generalized stochastic differential utility
 Supergradients
 Properness
 General equilibrium
 Knightian uncertainty
 Generic existence
 Asset pricing