Skip to main content
Log in

Cite this article


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, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others


  1. This concept is introduced by [26]. We refer to [13] for a detailed survey.

  2. For a first overview we refer to [25].

  3. The case of finite commodities can be treated by the same argumentation, we refer to [10].

  4. Measures on \(\Omega \times [0,T]\) which allow considerations of terminal consumption are possible. In this case the BSDE in (2) has a non-trivial terminal condition.

  5. 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 \(\sigma ,\sigma ' \in \mathbb {R}^n\).

  6. 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 measure-theoretic concept available in the finite-dimensional case”.

  7. This indicates, that the condition on the aggregate endowment in Theorem 1 is less strong than one might suspect at first glance.

  8. In principle, each \(\kappa ^k_i\) can also be a bounded \(\mathbb {F}\)-adapted process.

  9. 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.

  10. This assumption holds in our multiple-prior economies of Corollary 1 and 2. Following the implicit-function 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 \(\mathcal {E}^{\mathfrak {u}^i,\mathfrak {s}^i}_0=\alpha ^i\).

  11. 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–Backward-SDE’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) \(\mathfrak {s}_t^i\in \partial _{\sigma }f^i\left( K_i\big (t,e_t,\mathcal {E}_t, U_t\big ),\sigma ^i\right) ,\quad t\in [0,T].\)

  12. We refer to Theorem 3.1.4 in [4]

  13. \(L(e)\) denotes the order ideal \(L\). Details can be found in [2].


  1. Aliprantis, C.: Separable utility functions. J. Math. Econ. 28(4), 415–444 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  2. Aliprantis, C., Burkinshaw, O.: Locally Solid Riesz Spaces with Applications to Economics, vol. 105. American Mathematical Society, Providence (2003)

    MATH  Google Scholar 

  3. Anderson, R.M., Zame, W.R.: Genericity with infinitely many parameters. BE J. Theor. Econ. 1(1), 1–64 (2001)

    MathSciNet  Google Scholar 

  4. Barbu, V., Precupanu, T.: Convexity and Optimization in Banach Spaces, vol. 10. Springer, Dordrecht (1986)

    MATH  Google Scholar 

  5. Bewley, T.: Knightian decision theory. Part I. Decis. Econ. Financ. 25(2), 79–110 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chen, Z., Epstein, L.: Ambiguity, risk, and asset returns in continuous time. Econometrica 70(4), 1403–1443 (2002)

  7. Dana, R., Riedel, F.: Intertemporal equilibria with Knightian uncertainty. J. Econ. Theory 148(4), 1582–1605 (2013)

  8. Duffie, D.: Dynamic Asset Pricing Theory, 3rd edn. Princeton University Press, Princeton (1996)

    Google Scholar 

  9. Duffie, D., Epstein, L.: Stochastic differential utility. Econometrica 60(2), 353–394 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  10. Duffie, D., Geoffard, P., Skiadas, C.: Efficient and equilibrium allocations with stochastic differential utility. J. Math. Econ. 23(2), 133–146 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  11. Duffie, D., Zame, W.: The consumption-based capital asset pricing model. Econometrica 57(6), 1279–1297 (1989)

  12. Dumas, B., Uppal, R., Wang, T.: Efficient intertemporal allocations with recursive utility. J. Econ. Theory 93(2), 240–259 (2000)

    Article  MATH  Google Scholar 

  13. El Karoui, N., Peng, S., Quenez, M.: Backward stochastic differential equations in finance. Math. Financ. 7(1), 1–71 (1997)

    Article  MATH  Google Scholar 

  14. Epstein, L., Wang, T.: Intertemporal asset pricing under Knightian uncertainty. Econometrica 62(3), 283–322 (1994)

    Article  MATH  Google Scholar 

  15. Epstein, L.G., Miao, J.: A two-person dynamic equilibrium under ambiguity. J. Econ. Dyn. Control 27(7), 1253–1288 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Fan, K., Glicksberg, I., Hoffman, A.: Systems of inequalities involving convex functions. Proc. Am. Math. Soc. 8, 617–622 (1957)

    Article  MATH  MathSciNet  Google Scholar 

  17. Gilboa, I., Schmeidler, D.: Maxmin expected utility with non-unique prior. J. Math. Econ. 18(2), 141–153 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  18. Giles, J.: Convex Analysis with Application in the Differentiation of Convex Functions. Pitman Pub, Boston (1982)

    MATH  Google Scholar 

  19. Lazrak, A.: Generalized stochastic differential utility and preference for information. Ann. Appl. Probab. 14(4), 2149–2175 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  20. Lazrak, A., Quenez, M.C.: A generalized stochastic differential utility. Math. Oper. Res. 28(1), 154–180 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  21. Le Van, C.: Complete characterization of Yannelis–Zame and Chichilnisky–Kalman–Mas-Colell properness conditions on preferences for separable concave functions defined in \(L_+^p\) and \(L^p\). Econ. Theory 8(1), 155–166 (1996)

    Article  MATH  Google Scholar 

  22. Levental, S., Sinha, S., Schroder, M.: Linked recursive preferences and optimality. To Appear: Math. Financ. (2014). doi:10.1111/mafi.12047

  23. Maccheroni, F., Marinacci, M., Rustichini, A.: Ambiguity aversion, robustness, and the variational representation of preferences. Econometrica 74(6), 1447–1498 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  24. Mas-Colell, A.: The Theory of General Economic Equilibrium: A Differentiable Approach. Cambridge University Press, Cambridge (1990)

    Google Scholar 

  25. Mas-Colell, A., Zame, W.: Equilibrium theory in infinite dimensional spaces. Handb. Math. Econ. 4, 1835–1898 (1991)

    Article  MathSciNet  Google Scholar 

  26. Pardoux, E., Peng, S.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett. 14(1), 55–61 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  27. Podczeck, K.: Equilibria in vector lattices without ordered preferences or uniform properness. J. Math. Econ. 25(4), 465–485 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  28. Rigotti, L., Shannon, C.: Uncertainty and risk in financial markets. Econometrica 73(1), 203–243 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  29. Schroder, M., Skiadas, C.: Optimal lifetime consumption-portfolio strategies under trading constraints and generalized recursive preferences. Stoch. Process. Appl. 108(2), 155–202 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  30. Skiadas, C.: Recursive utility and preferences for information. Econ. Theory 12(2), 293–312 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  31. Wegge, L.: Mean value theorem for convex functions. J. Math. Econ. 1(2), 207–208 (1974)

    Article  MATH  Google Scholar 

  32. Yannelis, N., Zame, W.: Equilibria in Banach lattices without ordered preferences. J. Math. Econ. 15(2), 85–110 (1986)

    Article  MATH  MathSciNet  Google Scholar 

Download references


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

Correspondence to Patrick Beißner.


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 super-gradient of the aggregator with respect to the corresponding component \(x\) is denoted by \(D_x f(t,\cdot ,x,\cdot )\). The partial super-differential 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 left-hand derivatives represent the superdifferential in terms of the order interval:

An application of results on Backward-SDE’s depending on parameters (see Proposition 2.4 [13]), proofs that

$$\begin{aligned} \lim _{\alpha \searrow 0} \frac{ U(c)- U(c-\alpha h) }{\alpha } =\langle \nabla ^+ U(c),h\rangle =\Big \langle \mathcal {E}^{D^+_U f,D^+_\sigma f}\cdot \partial _c f,h\Big \rangle . \end{aligned}$$

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

$$\begin{aligned} U_t(c+h)-U_t(c)\le \mathbb {E}\left[ \int _t^T \frac{\mathcal {E}_s}{\mathcal {E}_t} \partial _c f(s,c_s,U_s,\sigma _s) h_s ds|\mathcal {F}_t\right] . \end{aligned}$$

Proof of Auxillary Lemma 1

Take a \(c\) and \(h\) as stated. The related utility processes \(U\) and \(U^h\) are given by

$$\begin{aligned} d U_t= -f(t,c_t, U_t, \sigma _t ) dt + \sigma _t dB_t \textit{ and } d U^h_t= -f(t,c_t+h_t, U^h_t, \sigma ^h_t ) dt + \sigma ^h_t dB_t, \end{aligned}$$

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 super-gradient w.r.t. the aggregator in utility

$$\begin{aligned} \lambda _t=\exp \left( \int _0^t D_U f(s,c_s,U_s,\sigma _s) ds\right) \le \exp (k t). \end{aligned}$$

Boundedness of the super-gradient w.r.t. aggregator in the intensity component \(\sigma \) implies

$$\begin{aligned} \mathbb {E}\left[ \exp \left( \frac{1}{2}\int _0^t \vert D_\sigma f(s,c_s,U_s,\sigma _s ) \vert ^2 ds\right) \right] \le \mathbb {E}\left[ \exp \left( \frac{1}{2}\int _0^t k^2 ds\right) \right] <\infty , \end{aligned}$$

the Novikov criterion is satisfied, hence the process \(\Gamma \), given by

$$\begin{aligned} \Gamma _t=\exp \left( -\frac{1}{2}\int _0^t \vert D_\sigma f(s,c_s,U_s,\sigma _s ) \vert ^2 ds+ \int _0^t D_\sigma f(s,c_s,U_s,\sigma _s )' dB_s \right) , \end{aligned}$$

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

$$\begin{aligned} \mathbb {E}[ \Gamma _{t\wedge \tau _n}^2]&= \mathbb {E}\left[ 1+2\int _0^{t\wedge \tau _n} \Gamma _s d \Gamma _s +\frac{1}{2}\int _0^{t\wedge \tau _n} 2 d \langle \Gamma \rangle _s \right] \\&= \mathbb {E}\left[ 1+\int _0^{t\wedge \tau _n} \Gamma ^2_s D_\sigma f(s,c_s,U_s,\sigma _s )^2ds \right] \le 1 +\int _0^{t} \mathbb {E}[\Gamma ^2_{s\wedge \tau _n}]k^2 ds. \end{aligned}$$

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

$$\begin{aligned} \mathbb {E}\left[ \sup _{t} \mathcal {E}_t^2\right] \le \mathbb {E}\left[ \sup _{t} \lambda ^2_t \sup _{t} \Gamma ^2_t \right] \le e^{ 2k T} 4\mathbb {E}\left[ \Gamma _t^2\right] <\infty . \end{aligned}$$

\(\square \)

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 super-differential 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 super-gradient.

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. 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. 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

$$\begin{aligned} U^{\alpha }(\hat{c}) = \max _{c'\in \Lambda (e)} U^{\alpha }(c')=\min _{c'\in \text {L}^{2,m} : g_i(c'),g(c')\le 0 }- U^{\alpha }(c'). \end{aligned}$$

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 non-negativity 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

$$\begin{aligned} 0\le -D_{\text {L}^{2,m}} U^{\alpha }(\hat{c})+ \mu \nabla _{\text {L}^{2,m}} g(\hat{c}) \text { and }\,\, \mu g(\hat{c})=0. \end{aligned}$$

Taking the feasible transfers \(h\in H(\hat{c})\) into account, we have

$$\begin{aligned} 0&\le \langle -D_{\text {L}^{2,m}} U^{\alpha }(\hat{c}) ,h\rangle _{\text {L}^{2,m}}+ \langle \mu \cdot \nabla _{\text {L}^{2,m}} g (\hat{c}),h\rangle _{\text {L}^{2,m}} \\&= -\sum \alpha _i D U^i(\hat{c}^i) h_i+ \mu \sum h_i. \end{aligned}$$

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 Backward-SDE’s:

$$\begin{aligned} c^j\mapsto (U,\sigma ),\, c^j- \lambda ^j 1_{A^j}\mapsto (U^A,\sigma ^A)\,and\,\, c^j -\lambda ^j 1_{A^j}+\lambda ^k 1_H \mapsto (U^{AH},\sigma ^{AH}), \end{aligned}$$

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

$$\begin{aligned}&U^j(c^j - \lambda ^j 1_{A^j}+\lambda ^k 1_H)-U^j(c^j)\\&\ge \! e^{k^jT} \mathbb {E}\left[ \int _0^T {\underline{\delta }}_{f^j} (2 h) \lambda ^k 1_H(t) \!- k^j \left( \vert \sigma ^{AH}_t-\sigma _t^A\vert \!+\!\vert \sigma _t- \sigma _t^A\vert \right) \!+\overline{\delta }_{f^j}\left( \frac{a^j}{2}\right) \lambda ^j 1_{A^j}(t) dt\!\!\right] . \end{aligned}$$

The inequality employed the estimates in Lemma 3 and Lemma 4. Next, we compute appropriate estimates for the \(\sigma \) parts. By the Cauchy-Schwartz inequality and the a priori estimates in [13], with \(\lambda ^2=2 k,\, \mu =1\) and \(\beta \ge 2k(1+ k)+1\), we derive:

$$\begin{aligned}&\mathbb {E}\left[ \int _0^T \vert \sigma _s- \sigma ^A_s\vert ds \right] \\&\quad \le \left( \frac{T\lambda ^2}{\mu ^2 (\lambda ^2-k)} \mathbb {E}\left[ \int _0^T e^{\beta s} \vert f^j(s,c^j_s,U_s,\sigma _s)- f^j(s,c^j_s -\lambda ^j 1_{A^j},U_s,\sigma _s)\vert ^2 ds\right] \right) ^{\frac{1}{2}}\\&\quad \le (2 T {e^{\beta T}})^{1/2}\mathbb {E}\left[ \int _0^T \overline{\delta }_{f^j}(a^j/2) {\lambda ^j} 1_{A^j}(s) ds\right] \end{aligned}$$

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

$$\begin{aligned}&\mathbb {E}\left[ \int _0^T \vert \sigma ^{AH}_s- \sigma ^A_s\vert ds \right] \le (2 T{e^{\beta T}})^{1/2} \mathbb {E}\left[ \int _0^T \overline{\delta }_{f^j}\left( \frac{a^j}{2}\right) {\lambda ^k} 1_{H}(s) ds\right] . \end{aligned}$$

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 \(\hat{e}^{\circ j}= e^{\circ k_jT }\cdot (2 T{e^{\beta T}})^{1/2},\, \circ \in \{+,-\}\):

$$\begin{aligned}&U^j(c^j - \lambda ^j 1_{A^j}+\lambda ^k 1_H)-U^j(c^j)\\&\ge b^je^{-j} {\underline{\delta }}_{f^j} ( 2 h)\mathbb {E}\left[ \int _0^T \lambda ^k 1_H(t) dt \right] - e^{+j}\overline{\delta }_{f^j}\left( \frac{a^j}{2}\right) \mathbb {E}\left[ \int _0^T \lambda ^j 1_{A^j}(t) dt\right] \\&-\, e^{-j }\mathbb {E}\left[ \int _0^T k_j\vert \sigma ^{AH}_t- \sigma _t^A\vert dt \right] - e^{+j }\mathbb {E}\left[ \int _0^T k_j\vert \sigma _t- \sigma _t^A\vert dt\right] \\&\ge \lambda ^k\left( e^{-j } {\underline{\delta }}_{f^j} ( 2 h) \nu (H) - \hat{e}^{+j }\overline{\delta }_{f^j}\left( \frac{a^j}{2}\right) \nu (A^j) \right) \\&- \lambda ^j \left( e^{+j}\overline{\delta }_{f^j}\left( \frac{a^j}{2}\right) \nu (A^j)+ \hat{e}^{-j }\overline{\delta }_{f^j}\left( \frac{a^j}{2}\right) \nu (H) \right) . \end{aligned}$$

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

$$\begin{aligned} 1&> \frac{ e^{+j }\overline{\delta }_{f^j}\Big (\frac{a^j}{2}\Big )\nu (A^j)+ \hat{e}^{-j } \overline{\delta }_{f^j}\Big ( \frac{a^j}{2}\Big ) \nu (H)}{ e^{-j } {\underline{\delta }}_{f^j} ( 2 h) \nu (H) - \hat{e}^{+j }\overline{\delta }_{f^j}\Big ( \frac{a^j}{2}\Big )\nu (A^j) } \cdot \,\frac{ e^{+k }\overline{\delta }_{f^k}\Big (\frac{C}{2m}\Big )\nu (A^j)+ \hat{e}^{-k }\overline{\delta }_{f^k}\Big ( \frac{a^k}{2}\Big ) \nu (H) }{ e^{-k }{\underline{\delta }}_{f^k} ( 2 C^k) \nu (H) - \hat{e}^{+k }\overline{\delta }_{f^k}( \frac{a^j}{2})\nu (A^j) } \end{aligned}$$

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

$$\begin{aligned} 1&> \frac{ e^{+j }\overline{\delta }_{f^j}\Big (\frac{a^j}{2}\Big )+ \hat{e}^{-j } \overline{\delta }_{f^j}\Big ( \frac{a^j}{2}\Big ) }{ e^{-j } {\underline{\delta }}_{f^j} ( 2 h) - \hat{e}^{+j }\overline{\delta }_{f^j}\Big ( \frac{a^j}{2}\Big ) } \cdot \,\frac{ e^{+k }\overline{\delta }_{f^k}\Big (\frac{C}{2m}\Big )+ \hat{e}^{-k }\overline{\delta }_{f^k}\Big ( \frac{a^k}{2}\Big ) }{ e^{-k }{\underline{\delta }}_{f^k} ( 2 C^k) - \hat{e}^{+k }\overline{\delta }_{f^k}\Big ( \frac{a^j}{2}\Big ) } \end{aligned}$$

by choosing appropriate multiples \(\lambda ^k\in (0,h)\) and \(\lambda ^j \in (0,\frac{a_j}{2})\) we finally get:

$$\begin{aligned} U^j(c^j - \lambda ^j 1_{A^j}+\lambda ^k 1_H)> U^j(c^j)\,\, and \,\, U^k(c^k + \lambda ^j 1_{A^j}-\lambda ^k 1_H)>U^k(c^k). \end{aligned}$$

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

$$\begin{aligned} \pi (D_{U,\sigma } f^i)_t= \mathcal {E}^{D_U f^i,D_\sigma f^i}_t \cdot \partial _c f^i(t,c^i_t,U^i_t,\sigma ^i_t) . \end{aligned}$$

The parametrization is related to the super-differential \(\partial _{U,\sigma } f^i\) of the aggregator \(f^i\). For later use define

$$\begin{aligned} V:=\bigcap _{D_{U,\sigma } f^i\in \partial _{U,\sigma } f^i} V(D_{U,\sigma } f^i). \end{aligned}$$

where \(V(D_{U,\sigma } f^i)=\left\{ z\in \text {L}^{2}:\langle \pi (D_{U,\sigma } f^i), (1-z)\rangle _{\text {L}^2} >0 \right\} \). 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

$$\begin{aligned} z\in \left\{ y\in \text {L}^{2}: \Vert y \Vert _{\text {L}^{2}}< \frac{\Vert \pi (D_{U,\sigma } f^i) \Vert _{\text {L}^{1}}}{\Vert \pi (D_{U,\sigma } f^i) \Vert _{\text {L}^{2}}} \right\} . \end{aligned}$$

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 \(c^i+\lambda (1-z)\in \text {L}^{2}_{+}\), where \(z\in V\) is arbitrary and \(\lambda >0\) is sufficiently small. When \(\lambda >0\) tends to zero, the term \(\lambda ^{-1}U^i(c^i+\lambda (1-z) ) -U^i(c^i)\) 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

$$\begin{aligned} \lim _{\lambda \searrow 0} \frac{U^i(c^i+\lambda (1-z) ) -U^i(c^i)}{\lambda } \ge \langle \pi (D_{U,\sigma } f^i)_t (1-z) \rangle _{\text {L}^2}>0. \end{aligned}$$

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 \(U^i(c^i+\lambda (1-z))>U^i(c^i)\). In other words, \(U^i\) is F-proper 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

$$\begin{aligned} U(y)-U(y-x)\le e^{ k T} \mathbb {E}\left[ \int _0^T \overline{\delta }_f \Big ( \frac{a}{2}\Big ) x_t +k \vert \sigma _s-\bar{\sigma }_s \vert dt\right] . \end{aligned}$$

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 \(y-x\) 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

$$\begin{aligned} U_t - \bar{U}_t&= \mathbb {E}\left[ \int _t^T f(s,y_s,U_s,\sigma _s) -f(s,y_s-x_s,\bar{U}_s, \bar{\sigma }_s)ds | \mathcal {F}_t\right] \\&= \mathbb {E}\Big [\int _t^T \partial _c f(s,y_s+\xi _s^c,U_s+\xi _s^U,\sigma _s+\xi _s^\sigma )x_s \qquad \qquad \qquad \\&+\,D_U f(s,y_s+\xi _s^c,U_s+\xi _s^U,\sigma _s+\xi _s^Z ) (U_s-\bar{U}_s) \\&+\, \langle D_\sigma f(s,y_s+\xi _s^c,U_s+\xi _s^U ,\sigma _s+\xi _s^\sigma ) ,(\sigma _s-\bar{\sigma }_s)\rangle ds | \mathcal {F}_t\Big ]. \end{aligned}$$

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 super-gradients, we derive:

$$\begin{aligned} U_t - \bar{U}_t&\le \mathbb {E}\left[ \int _t^T \partial _c f(s,y_s+\xi _s^c,U_s+\xi _s^U,\sigma _s+\xi _s^\sigma )x_s + k (U_s-\bar{U}_s)\right. \\&\left. + \,\langle D_\sigma f(s,y_s+\xi _s^c,U_s+\xi _s^U, \sigma _s+\xi _s^\sigma ) ,(\sigma _s-\bar{\sigma }_s)\rangle ds | \mathcal {F}_t\right] \\&\le \mathbb {E}\left[ \int _t^T \overline{\delta }_f(\frac{a}{2}) x_s + k ( U_s - \bar{U}_s)+k\vert \sigma _s-\bar{\sigma }_s \vert ds | \mathcal {F}_t\right] . \end{aligned}$$

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

$$\begin{aligned} U(y)-U(y-x) =U_0-\bar{U}_0 \le e^{ k T}\mathbb {E}\left[ \int _0^T \overline{\delta }_f\Big (\frac{a}{2}\Big ) x_s + k \vert \sigma _s-\bar{\sigma }_s \vert ds \right] . \end{aligned}$$

\(\square \)

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 \(\lambda \in [0,h]\)

$$\begin{aligned} U(y+\lambda 1_H)-U(y)\ge e^{- k T}\mathbb {E}\left[ \int _0^T {\underline{\delta }}_f ( 2 h)\lambda 1_H(t)-k \vert \bar{\sigma }_s -\sigma _s \vert dt\right] . \end{aligned}$$

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

$$\begin{aligned} \bar{U}_t - U_t \ge \mathbb {E}\left[ \int _t^T {\underline{\delta }}_f(2h) \lambda 1_H(t) - k ( \bar{U}_s - U_s)-k \vert \bar{\sigma }_s - \sigma _s \vert ds | \mathcal {F}_t\right] . \end{aligned}$$

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

$$\begin{aligned} U(y)-U(y-x)=U_0-\bar{U}_0\ge e^{ -k T} \mathbb {E}\left[ \int _0^T {\underline{\delta }}_f(2h) \lambda 1_H(t) -k \vert \bar{\sigma }_s - \sigma _s \vert ds \right] . \end{aligned}$$

Appendix B: Quasi-equilibrium 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 quasi-equilibrium 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 quasi-equilibrium 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 F-proper 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 quasi-equilibrium.

Assumption 2

The economy satisfies the following conditions:

  1. 1.

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

  2. 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. 3.

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

  4. 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 Pareto-optimal allocation, then, for every \(i,\, P_i\) is F-proper 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 non-trivial quasi-equilibrium.

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 quasi-equilibrium 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

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Beißner, P. Brownian equilibria under Knightian uncertainty. Math Finan Econ 9, 39–56 (2015).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


JEL Classifications