Brownian equilibria under Knightian uncertainty

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.

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 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

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

(7)

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

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

(8)

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. 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

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

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. $y\notin P_i(y)$ 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. $L(e)$ 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 $L^{\star }$.