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Stability of marketable payoffs with long-term assets

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Abstract

We consider a stochastic financial exchange economy with a finite date-event tree representing time and uncertainty and a financial structure with possibly long-term assets. We exhibit a sufficient condition under which the set of marketable payoffs depends continuously on the arbitrage free asset prices. This generalizes previous results of Angeloni–Cornet and Magill–Quinzii involving only short-term assets. We also show that, under the same condition, the useless portfolios do not depend on the arbitrage free asset prices. We then provide an existence result of financial equilibrium for long term nominal assets for any given state prices with assumptions only on the fundamental datas of the economy.

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Notes

  1. We use the following notations. A \(\left( \mathbb {D}\times {\fancyscript{J}}\right) \)-matrix \(A\) is an element of \(\mathbb {R}^{\mathbb {D}\times {\fancyscript{J}}}\), with entries \((a^j_\xi )_{\left( \xi \in \mathbb {D},j\in {\fancyscript{J}}\right) }\); we denote by \(A_\xi \in \mathbb {R}^{{\fancyscript{J}}}\) the \(\xi \)-th row of \(A\) and by \(A^j \in \mathbb {R}^\mathbb {D}\) the \(j\)-th column of \(A\). We recall that the transpose of \(A\) is the unique \(\left( {\fancyscript{J}}\times \mathbb {D}\right) \)-matrix \({}^{t}\mathrm A \) satisfying \(\left( A x\right) \bullet _\mathbb {D} y = x\bullet _{{\fancyscript{J}}} \left( {}^{t}\mathrm Ay \right) \) for every \(x \in \mathbb {R}^{{\fancyscript{J}}}\), \(y\in \mathbb {R}^\mathbb {D}\), where \(\bullet _\mathbb {D}\) [resp. \(\bullet _{{\fancyscript{J}}}\)] denotes the usual inner product in \(\mathbb {R}^\mathbb {D}\) [resp. \(\mathbb {R}^{{\fancyscript{J}}}\)]. We denote by \(\mathrm{rank }A \) the rank of the matrix \(A\) and by \(\hbox {Vect}\left( A\right) \) the range of \(A\), that is the linear sub-space spanned by the column vectors of \(A\). For every subset \(\tilde{\mathbb {D}}\subset \mathbb {D}\) and \(\tilde{{\fancyscript{J}}} \subset {\fancyscript{J}}\), the matrix \(A^{\tilde{{\fancyscript{J}}}}_{\tilde{\mathbb {D}}}\) is the \( (\tilde{\mathbb {D}}\times \tilde{{\fancyscript{J}}})\)-sub-matrix of \(A\) with entries \(a^j_\xi \) for every \(\left( \xi , j\right) \in (\tilde{\mathbb {D}}\times \tilde{{\fancyscript{J}}} )\). Let \(x\), \(y\) be in \(\mathbb {R}^n\); \(x\ge y\) (resp. \(x \gg y\) ) means \(x_h \ge y_h\) (resp. \(x_h > y_h\)) for every \(h= 1, \ldots ,n\) and we let \({\mathbb {R}}^n_{+}= \left\{ x\in \mathbb {R}^n : x \ge 0\right\} \), \( {\mathbb {R}}^n_{++} = \left\{ x \in \mathbb {R}^n : x \gg 0\right\} \). We also use the notation \(x >y\) if \(x \ge y\) and \(x \ne y\). The Euclidean norm in the different Euclidean spaces is denoted \(\left\| .\right\| \) and the closed ball centered at \(x\) and of radius \(r>0\) is denoted \(\bar{B}(x,r) :=\{y \in \mathbb {R}^n \mid \Vert y - x \Vert \le r\}\).

  2. See below Remark 3.2 for a short discussion about the retrading of assets.

  3. Linear subspaces \(U_1, U_2,\ldots , U_m\) (with \(U_\ell \ne \{0\}\) for all \(\ell \)) are linearly independent if for all \((u_\ell ) \in \prod _{\ell =1}^m U_\ell \), \(u_1+u_2+\ldots +u_m=0\) implies \(u_\ell =0\) for all \(\ell \). The sum of the linearly independent linear subspaces is called a direct sum of these linear subspaces and is denoted by \(\bigoplus _{\ell =1}^m U_\ell =U_1\bigoplus \ldots \bigoplus U_m\). Note that for all \(u \in \bigoplus _{\ell =1}^m U_\ell \), there exists a unique \((u_\ell ) \in \prod _{\ell =1}^m U_\ell \) such that \(u=\sum _{\ell =1}^m u_\ell \).

  4. Recall that an asset \(j_0\) is redundant for the payoff matrix \(V\) at the spot price \(p\) if the column vector \(V^{j_0}(p)\) representing its payoffs on \(\mathbb {D}\) is a linear combination of the column vectors representing the payoffs of other assets, i.e., if there exists \(\alpha =(\alpha ^j)_{j \in {\fancyscript{J}}\backslash \{j_0\}} \in \mathbb {R}^{{\fancyscript{J}}\backslash \{j_0\}}\) such that \(V^{j_0}(p)=\sum _{j \in {\fancyscript{J}}\backslash \{j_0\}}\alpha ^j V^j(p)\).

  5. For \(x = \left( x\left( \xi \right) \right) _{\xi \in \mathbb {D}}, p = \left( p\left( \xi \right) \right) _{\xi \in \mathbb {D}}\) in \( {\mathbb {R}}^{\mathbb {L}}= {\mathbb {R}}^{\mathbb {H}\times \mathbb {D}}\) (with \(x\left( \xi \right) ,~p\left( \xi \right) \) in \({\mathbb {R}}^{\mathbb {H}}\)) we let \(p\scriptscriptstyle {\Box }\textstyle x = \left( p\left( \xi \right) \bullet _{\mathbb {H}} x\left( \xi \right) \right) _{\xi \in \mathbb {D}}\in {\mathbb {R}}^{\mathbb {D}}\).

  6. we recall that \(N_{Z_i}\left( z_i\right) \) is the normal cone to \(Z_i\) at \(z_i\), which is defined as \(N_{Z_i}\left( z_i\right) =\left\{ \eta \in {\mathbb {R}}^{{\fancyscript{J}}}: \eta \bullet _{{\fancyscript{J}}} z_i \ge \eta \bullet _{{\fancyscript{J}}} z^{\prime }_i, \forall z^{\prime }_i \in Z_i\right\} \).

  7. A subset \(C \subset {\mathbb {R}}^n\) is said to be polyhedral if it is the intersection of finitely many closed half-spaces, namely \(C=\left\{ x \in {\mathbb {R}}^n : Ax \le b\right\} \), where \(A\) is a real \(\left( m \times n\right) \)-matrix, and \(b\in {\mathbb {R}}^m\). Note that polyhedral sets are always closed and convex and that the empty set and the whole space \({\mathbb {R}}^n\) are both polyhedral.

  8. This is satisfied, in particular, when \(P_i\left( \bar{x}\right) \) is open in \(X_i\) ( for its relative topology ).

  9. Let \(Z\) a nonempty subset of \({\mathbb {R}}^{{\fancyscript{J}}}\) and let \(H\) a subspace of \({\mathbb {R}}^{{\fancyscript{J}}}\) such that \(Z \subset H\). We call relative interior of \(Z\) with respect to \(H\) denoted \(\mathrm{ri}_H (Z)\) the set \(\{z \in {\mathbb {R}}^{{\fancyscript{J}}} \mid \exists r > 0; B(z,r)\cap H\subset Z\}\).

  10. Note that we do not use the fact that the asset price is an arbitrage free price in this part of the proof.

References

  • Angeloni, L., Cornet, B.: Existence of financial equilibria in a multi-period stochastic economy. Math. Econ. 8, 1–31 (2006)

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  • Aouani, Z., Cornet, B.: Existence of financial equilibria with restricted participations. J. Math. Econ. 45, 772–786 (2009)

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  • Cornet, B., Gopalan, R.: Arbitrage and equilibrium with portfolio constraints. Econ. Theory 45, 227–252 (2010)

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  • Cornet, B., Ranjan, A.: A remark on arbitrage free prices in multi-period economy. Working paper, University of Paris 1 (2012)

  • Magill, M., Quinzii, M.: Theory of Incomplete Markets. MIT Press, Cambridge (1996)

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Acknowledgments

We would like to thank Bernard Cornet for very helpful comments, Zaier Aouani, who has detected a mistake after a careful reading of a previous version of the paper and an anonymous referee for her/his fruitful suggestion to explicitly state the marketable payoff set as a direct sum of the marketable payoff sets associated to each issuance node.

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Correspondence to Jean-Marc Bonnisseau.

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This work was supported by the French National Research Agency, through the program Investissements d’Avenir, ANR-10–LABX-93-01.

Appendix

Appendix

Proof of Proposition 3.1

Let \(p \in \mathbb {R}^{\mathbb {L}}\) be a spot price vector. Let \(\bar{q} \in \mathbb {R}^{{\fancyscript{J}}}\) an asset price and let \((q^\nu )\) a sequence of \(Q_p\), which converges to \( \bar{q} \in Q_p\) and let \(\bar{w} \in H(p,\bar{q})\). Let \(\bar{z} \in \mathbb {R}^{\fancyscript{J}}\) such that \(\bar{w} = W(p, \bar{q}) \bar{z}\). Then the sequence \((w^\nu = W(p,q^\nu ) \bar{z})\) converges to \(\bar{w}\) since the sequence of matrices \((W(p,q^\nu ))\) converges to \(W(p,\bar{q})\) and \(w^\nu \in H(p, q^\nu )\) from the very definition of \(H(p,\cdot )\). Hence the correspondance \(H(p,.)\) is l.s.c. on \(Q_p\). \(\square \)

Proof of Proposition 3.2

Let \((q^\nu )\) a sequence of \(Q_p\) converging to \(\bar{q} \in Q_p\) and let \((w^\nu )\) a sequence of marketable payoffs converging to \(\bar{w}\) such that, for each \(\nu \), \(w^\nu \in H(p,q^\nu )\). We prove that \(\bar{w} \in H(p,\bar{q})\).

For each \(\nu \) there exists \(z^\nu \in \mathbb {R}^{\fancyscript{J}}\) such that \(w^\nu =W(p,q^\nu )z^\nu \). Let \(\hat{z}^\nu \) be the orthogonal projection of \(z^\nu \) on \((\mathrm{Ker}V(p))^\bot \). Then \(w^\nu = W(p,q^\nu ) \hat{z}^\nu + W(p,q^\nu ) (z^\nu - \hat{z}^\nu )= W(p,q^\nu ) \hat{z}^\nu \) since \(z^\nu - \hat{z}^\nu \in \mathrm{Ker}V(p)= \mathrm{Ker}W(p,q^\nu )\) by assumption.

We now prove that the sequence \((\hat{z}^\nu )\) is bounded. Indeed, suppose, by contradiction, that this is not true. Then, there exists a subsequence \((\hat{z}^{\phi (\nu )})\) such that \(\left\| \hat{z}^{\phi (\nu )}\right\| \rightarrow +\infty \). For each \(\nu \), let \(\zeta ^\nu =\frac{\hat{z}^{\phi (\nu )}}{\left\| \hat{z}^{\phi (\nu )}\right\| }\)

The sequence \((\zeta ^\nu )\) belongs to the unit sphere. So there exists a subsequence \((\zeta ^{\psi (\nu )})\) of \((\zeta ^\nu )\) which converges to \(\bar{\zeta }\). Clearly \(\Vert \bar{\zeta } \Vert =1\) and \(\bar{\zeta } \in (\mathrm{Ker}V(p))^\bot \) since \(\hat{z}^\nu \in (\mathrm{Ker}V(p))^\bot \) for all \(\nu \). Thus, for each \(\nu \), we have

$$\begin{aligned} W(p,q^\nu )\frac{\hat{z}^{\phi \circ \psi (\nu )}}{\left\| \hat{z}^{\phi \circ \psi (\nu )}\right\| }=W(p,q^\nu )\zeta ^{ \psi (\nu )}\rightarrow W(p,\bar{q}) \bar{\zeta } \end{aligned}$$

and

$$\begin{aligned} W(p,q^\nu )\zeta ^{ \psi (\nu )}=W(p,q^\nu )\frac{\hat{z}^{\phi \circ \psi (\nu )}}{\left\| \hat{z}^{\phi \circ \psi (\nu )}\right\| }=\frac{w^{\phi \circ \psi (\nu )}}{\left\| \hat{z}^{\phi \circ \psi (\nu )}\right\| } \rightarrow 0 \end{aligned}$$

since \((w^\nu )\) is bounded and \(\left\| \hat{z}^{\phi (\nu )}\right\| \rightarrow +\infty \). Thus \(\bar{\zeta } \in \mathrm{Ker}W(p,\bar{q})= \mathrm{Ker}V(p)\) by assumption and \(\bar{\zeta } \in ( \mathrm{Ker}V(p))^\bot \), so \(\bar{\zeta } =0\) which contradicts \(\Vert \bar{\zeta } \Vert =1\).

Since the sequence \((\hat{z}^\nu )\) is bounded, there exists a converging subsequence \((\hat{z}^{\varphi (\nu )})\) which converges to \(\bar{z} \in (\mathrm{Ker}V)^\bot \) and we easily checks that

$$\begin{aligned} \bar{w}=\lim _{\nu \rightarrow +\infty } w^{\varphi (\nu )}=\lim _{\nu \rightarrow +\infty } W(p,q^{\varphi (\nu )})z^{\varphi (\nu )}= W(p,\bar{q})\bar{z}\in H(p,\bar{q}). \end{aligned}$$

\(\square \)

Proof of Lemma 3.1

Let \({\mathbb {D}}^e\) (resp. \({\mathbb {D}}^{\prime e}\)) be the set of issuance nodes for the financial structure \({\fancyscript{F}}\) (reap. \({\fancyscript{F}}^\prime \)). Clearly, \({\mathbb {D}}^{\prime e}\subset {\mathbb {D}}^e\). By Assumption R, for all \(\xi \in {\mathbb {D}}^{\prime e}\),

$$\begin{aligned} \hbox {Vect}\left( V_{{\mathbb {D}}^+(\xi )}^{{{\fancyscript{J}}}(\mathbb {D}^-(\xi )}\right) \bigcap \hbox {Vect}\left( V_{{\mathbb {D}}^+(\xi )}^{{{\fancyscript{J}}}(\xi )}\right) = \left\{ 0\right\} . \end{aligned}$$

Since \({\fancyscript{J}}^\prime \subset {\fancyscript{J}}\),

$$\begin{aligned} \hbox {Vect}\left( V_{{\mathbb {D}}^+(\xi )}^{{{\fancyscript{J}}}^\prime ((\mathbb {D}^-(\xi ))}\right) \subset \hbox {Vect}\left( V_{{\mathbb {D}}^+\left( \xi \right) }^{{{\fancyscript{J}}} ((\mathbb {D}^-(\xi ))}\right) \hbox { and } \hbox {Vect}\left( V_{{\mathbb {D}}^+(\xi )}^{{{\fancyscript{J}}}^\prime (\xi )}\right) \subset \hbox {Vect}\left( V_{{\mathbb {D}}^+(\xi )}^{{{\fancyscript{J}}} (\xi )}\right) . \end{aligned}$$

So,

$$\begin{aligned} \hbox {Vect}\left( V_{{\mathbb {D}}^+(\xi )}^{{{\fancyscript{J}}}^\prime (\xi ^-)}\right) \cap \hbox {Vect}\left( V_{{\mathbb {D}}^+(\xi )}^{{{\fancyscript{J}}}^\prime (\xi )}\right) \subset \hbox {Vect}\left( V_{{\mathbb {D}}^+(\xi )}^{{{\fancyscript{J}}}(\mathbb {D}^-(\xi )}\right) \bigcap \hbox {Vect}\left( V_{{\mathbb {D}}^+(\xi )}^{{{\fancyscript{J}}}(\xi )}\right) = \left\{ 0\right\} . \end{aligned}$$

hence the financial structure \({\fancyscript{F}}^\prime \) satisfies Assumption R. \(\square \)

Proof of Corollary 3.1

First, we remark that \(V_{\xi ^+}^{{{\fancyscript{J}}}\left( \xi \right) }(p) \) is a sub-matrix of \(V^{{{\fancyscript{J}}}\left( \xi \right) }(p) \), so \(n(\xi )= \mathrm{rank }V_{\xi ^+}^{{{\fancyscript{J}}}\left( \xi \right) } (p)\le \mathrm{rank }V^{{{\fancyscript{J}}}\left( \xi \right) }(p)\). On the other hand, \(\mathrm{rank }V^{{{\fancyscript{J}}}\left( \xi \right) }(p) \le n(\xi )\) since the number of column of \(V^{{{\fancyscript{J}}}\left( \xi \right) }(p)\) is \(n(\xi )\). Hence \(n(\xi )=\mathrm{rank }V^{{{\fancyscript{J}}}\left( \xi \right) }(p)\).

We now prove that Assumption R is satisfied. Let \(\kappa \in \left\{ 2,\ldots ,k\right\} \), \( \xi \in {\mathbb {D}}^e_{\tau _\kappa } \) and \(y \in {{\mathbb {R}}^{{\mathbb {D}}^+\left( \xi \right) } \setminus \left\{ 0\right\} }\) such that

$$\begin{aligned} y \in \hbox {Vect}\left( V_{{\mathbb {D}}^+\left( \xi \right) }^{{{\fancyscript{J}}} \left( \mathbb {D}^-(\xi )\right) }(p)\right) \cap \hbox {Vect}\left( V_{{\mathbb {D}}^+\left( \xi \right) }^{{{\fancyscript{J}}} \left( \xi \right) }(p)\right) \end{aligned}$$

Then, there exists \((a_j) \in \mathbb {R}^{{\fancyscript{J}}(\xi )}\) such that \(y= \sum _{j \in {\fancyscript{J}}(\xi )} a_j V^j_{\mathbb {D}^+(\xi )}(p)\) and there exists \((b_j) \in \mathbb {R}^{{\fancyscript{J}}(\mathbb {D}^-(\xi ))}\) such that \(y= \sum _{j \in {\fancyscript{J}}(\mathbb {D}^- (\xi ))} b_j V^j_{\mathbb {D}^+(\xi )}(p)\). Restricting the above equality to the coordinates in \(\xi ^+\), one gets \(y_{\xi ^+} = \sum _{j \in {\fancyscript{J}}(\xi )} a_j V^j_{\xi ^+}(p)= \sum _{j \in {\fancyscript{J}}( \mathbb {D}^- (\xi ))} b_j V^j_{\xi ^+}(p)\). From our second assumption, this implies that \(y_{\xi ^+}=0\). From the first assumption, since the vectors \((V^j_{\xi ^+}(p))_{j \in {\fancyscript{J}}(\xi )}\) are of maximal rank hence linearly independent, this implies that \(a_j=0\) for all \(j \in {\fancyscript{J}}(\xi )\). So, \(y=0\), which proves that \( \hbox {Vect}\left( V_{{\mathbb {D}}^+\left( \xi \right) }^{{{\fancyscript{J}}} \left( \mathbb {D}^-(\xi )\right) }(p)\right) \bigcap \hbox {Vect}\left( V_{{\mathbb {D}}^+\left( \xi \right) }^{{{\fancyscript{J}}} \left( \xi \right) }(p)\right) =\{0\}\). Consequently Assumption R is satisfied. \(\square \)

Proof of Proposition 3.4

Let \(q\) be an arbitrage free price and let \(\lambda =\left( \lambda _\xi \right) \in {\mathbb {R}}^{\mathbb {D}}_{++}\) such that \({}^tW(p,q)\lambda =0\). From Proposition 3.5, \(\mathrm{rank }V(p) = \mathrm{rank }W(p,q)\) and this implies that \(\dim \mathrm{Ker}V(p)= \dim \mathrm{Ker}W(p,q)\). So, to get the equality of the kernels, it remains to show \(\mathrm{Ker}V(p) \subset \mathrm{Ker}W(p,q)\).

Let \(z=\left( z^j\right) _{j \in {\fancyscript{J}}} \in {\mathbb {R}}^{{\fancyscript{J}}}\) be an element of the kernel of the payoff matrix \(V(p)\) so \(\sum _{j \in {\fancyscript{J}}} z^j V^j(p)=0\). From Proposition 3.5, \(\mathrm{Im}V(p)=\bigoplus _{\xi \in \mathbb {D}^e} \mathrm{Im}V^{{\fancyscript{J}}(\xi )}(p)\), hence, for all \(\xi \in \mathbb {D}^e\), \(\sum _{j\in {\fancyscript{J}}(\xi )} z^j V^j(p)=0.\)

For all \(\xi \in \mathbb {D}^e\), for all \(j \in {\fancyscript{J}}(\xi )\) and for all \(\eta \in {\mathbb {D}\setminus \left\{ \xi \right\} }\), \(V^j_\eta (p)= W^j_\eta (p,q)\). At the node \(\xi \), \(\sum _{j \in {\fancyscript{J}}(\xi )}z^j W^j_\xi (p,q) = -\sum _{j \in {\fancyscript{J}}(\xi )}z^j q_j\). But

$$\begin{aligned} q_j=(1/\lambda _{\xi }) \sum _{\xi ^\prime \in \mathbb {D}^+(\xi )}\lambda _{\xi ^\prime } V^j_{\xi ^\prime }(p). \end{aligned}$$

Hence,

$$\begin{aligned} \sum \limits _{j \in {{\fancyscript{J}}}(\xi ) }z^j q_j&= (1/\lambda _{\xi }) \sum \limits _{j \in {{\fancyscript{J}}}(\xi ) }z^j \left[ \sum \limits _{\xi ^\prime \in \mathbb {D}^+(\xi )}\lambda _{\xi ^\prime }V^j_{\xi ^\prime }(p)\right] \\&= (1/\lambda _{\xi }) \sum \limits _{\xi ^\prime \in \mathbb {D}^+(\xi )}\lambda _{\xi ^\prime }\left[ \sum \limits _{j \in {{\fancyscript{J}}}(\xi ) }z^jV^j_{\xi ^\prime }(p)\right] \\&= 0\\ \end{aligned}$$

So, we have proved that \(\sum _{j \in {\fancyscript{J}}(\xi )}z^j W^j (p,q) =0\), and since it holds true for all \(\xi \in \mathbb {D}^e\), \(\sum _{j \in {\fancyscript{J}}}z^j W^j(p,q)=0\) that is \(z \in \mathrm{Ker}W(p,q)\). \(\square \)

Proof of Proposition 3.5

For all \( \xi \in \mathbb {D}^e\), we denote by \(n(\xi )\) the number of assets issued at this node and by \(\mathrm{rk}(\xi )\) the rank of \(V_\mathbb {D}^{{{\fancyscript{J}}}\left( \xi \right) }(p)\). We also simplify the notation by defining \(V^{{{\fancyscript{J}}}\left( \xi \right) }(p) := V_\mathbb {D}^{{{\fancyscript{J}}}\left( \xi \right) }(p)\) and \(W^{{{\fancyscript{J}}}\left( \xi \right) } (p,q):= W_\mathbb {D}^{{{\fancyscript{J}}}\left( \xi \right) }(p,q)\).

Step 1: For all \( \xi \in \mathbb {D}^e\), \(\mathrm{rank }W^{ {{\fancyscript{J}}} \left( \xi \right) }(p,q) = \mathrm{rk}\left( \xi \right) \).

Let us consider \(\lambda =\left( \lambda _\xi \right) _{\xi \in \mathbb {D}} \in {\mathbb {R}}^{\mathbb {D}}_{++}\) such that \({}^tW\left( p,q\right) \lambda = 0\). Such \( \lambda \) exists since \(q\) is an arbitrage free price.

For all \(\xi \in \mathbb {D}^e\), let \({\fancyscript{J}}^\prime (\xi ) \subset {\fancyscript{J}}(\xi )\) such that \(\# {\fancyscript{J}}^\prime (\xi )=\mathrm{rk}(\xi )\) and \((V^j_\xi (p))_{j \in {\fancyscript{J}}^\prime (\xi ) }\) is a linearly independent family. Since \(W^{{\fancyscript{J}}^\prime (\xi )}(p,q)\) is obtained from \(V^{{\fancyscript{J}}^\prime (\xi )}(p)\) by replacing a zero row by the row of the opposite of asset prices issued at \(\xi \), the regular \(\mathrm{rk}(\xi )\) square sub-matrix of \(V^{{\fancyscript{J}}^\prime (\xi )}(p)\) is also a sub-matrix of \(W^{{\fancyscript{J}}^\prime (\xi )}(p,q)\), hence the rank of \(W^{{\fancyscript{J}}^\prime (\xi )}(p,q)\) is higher or equal to \(\mathrm{rk}(\xi )\) Footnote 10.

Let us now prove that the rank of \( W^{{{\fancyscript{J}}}\left( \xi \right) }(p,q)\) is not strictly larger than \(\mathrm{rk}(\xi )\). It suffices to prove that for all \(j_0 \notin {\fancyscript{J}}^\prime (\xi )\), \(W^{j_0} (p,q) \in \hbox {Vect}((W^{j} (p,q))_{j \in {\fancyscript{J}}^\prime (\xi )})\).

\(V^{j_0}(p)\) is a linear combination of \((V^{j}(p))_{j \in {\fancyscript{J}}^\prime (\xi )}\) since the rank of \(V^{{\fancyscript{J}}(\xi )}(p)\) is \(\mathrm{rk}(\xi )\). Hence there exists \((\alpha _j)_{j \in {\fancyscript{J}}^\prime (\xi )}\) such that \(\sum _{j \in {\fancyscript{J}}^\prime (\xi )} \alpha _j V^{j}(p)= V^{j_0}(p)\). Since \({}^tW\left( p,q\right) \lambda = 0\), \(\lambda _\xi q_{j_0}= \sum _{\xi ^\prime \in \mathbb {D}^+(\xi )} \lambda _{ \xi ^\prime } V^{j_0}_{\xi ^\prime }(p)\). Hence \(\lambda _\xi q_{j_0}\) is equal to

$$\begin{aligned} \sum \limits _{\xi ^\prime \in \mathbb {D}^+(\xi )}\left[ \lambda _{\xi ^\prime } \sum \limits _{j \in {\fancyscript{J}}^\prime (\xi )}\alpha _j V^{j}_{\xi ^\prime }(p) \right]&= \sum \limits _{j \in {\fancyscript{J}}^\prime (\xi )} \left[ \alpha _j\sum \limits _{\xi ^\prime \in \mathbb {D}^+(\xi )} \lambda _{\xi ^\prime } V^{j}_{\xi ^\prime }(p) \right] \\&= \sum \limits _{j \in {\fancyscript{J}}^\prime (\xi )} \left[ \alpha _j \lambda _\xi q_{j}\right] = \lambda _\xi \sum \limits _{j \in {\fancyscript{J}}^\prime (\xi )}\alpha _j q_{j} \end{aligned}$$

Hence \(q_{j_0}= \sum _{j \in {\fancyscript{J}}^\prime (\xi )}\alpha _j q_{j} \). Since \(\sum _{j \in {\fancyscript{J}}^\prime (\xi )} \alpha _j V^{j}(p)= V^{j_0}(p)\), we obtain for all \(\eta \ne \xi \),

$$\begin{aligned} \sum _{j \in {\fancyscript{J}}^\prime (\xi )} \alpha _j W^{j}_\eta (p,q)=\sum _{j \in {\fancyscript{J}}^\prime (\xi )} \alpha _j V^{j}_\eta (p)=V^{j_0}_\eta (p)= W^{j_0}_\eta (p,q) \end{aligned}$$

and

$$\begin{aligned} \sum _{j \in {\fancyscript{J}}^\prime (\xi )} \alpha _j W^{j}_\xi (p,q)=-\sum _{j \in {\fancyscript{J}}^\prime (\xi )} \alpha _j q_j=-q_{j_0}= W^{j_0}_\xi (p,q) \end{aligned}$$

So,

$$\begin{aligned} \sum _{j \in {\fancyscript{J}}^\prime (\xi )} \alpha _j W^{j}(p,q)= W^{j_0}(p,q) \end{aligned}$$

and \(W^{j_0} (p,q)\) belongs to \(\hbox {Vect}((W^{j} (p,q))_{j \in {\fancyscript{J}}^\prime (\xi )})\).

Step 2: \( \mathrm{rank }V(p) = \sum _{\xi \in \mathbb {D}^e} \mathrm{rk}(\xi )= \mathrm{rank }W\left( p,q\right) \), \(\mathrm{Im}V(p)=\bigoplus _{\xi \in \mathbb {D}^e} \mathrm{Im}V^{{\fancyscript{J}}(\xi )}(p)\) and \(\mathrm{Im}W (p,q)=\bigoplus _{\xi \in \mathbb {D}^e} \mathrm{Im}W^{{\fancyscript{J}}(\xi )}(p,q)\).

Since \(\hbox {Vect}\left( V(p)\right) \) is the sum of the subspaces \(\left( \hbox {Vect}\left( V^{{{\fancyscript{J}}(\xi )}}(p)\right) \right) _{\xi \in \mathbb {D}^e}\), \(\mathrm{rank }V(p)\) \(\le \sum _{\xi \in \mathbb {D}^e} \mathrm{rank }V^{{{\fancyscript{J}}(\xi )}}(p) = \sum _{\xi \mathbb {D}^e} \mathrm{rk}(\xi )\).

Like in the proof of Step 1, for all \(\xi \in \mathbb {D}^e\), let \({\fancyscript{J}}^\prime (\xi ) \subset {\fancyscript{J}}(\xi )\) such that \(\# {\fancyscript{J}}^\prime (\xi )=\mathrm{rk}(\xi )\) and the family \((V^j_\xi (p))_{j \in {\fancyscript{J}}^\prime (\xi ) }\) is linearly independent. Let \({\fancyscript{J}}^\prime = \cup _{\xi \in \mathbb {D}^e} {\fancyscript{J}}^\prime (\xi )\) and for all \(\kappa = 1, \ldots , k\), \({\fancyscript{J}}^\prime _\kappa = \cup _{\xi \in \mathbb {D}^e_{\tau _\kappa }} {\fancyscript{J}}^\prime (\xi )\). We now prove that the family \((V^j(p))_{j \in {\fancyscript{J}}^\prime }\) is linearly independent. Since \((V^j(p))_{j \in {\fancyscript{J}}^\prime (\xi )}\) is a basis of \(\mathrm{Im}V^{{\fancyscript{J}}(\xi )}(p)\) for all \(\xi \), note that this implies \( \mathrm{rank }V(p) = \sum _{\xi \in \mathbb {D}^e} \mathrm{rk}(\xi )\) and \(\mathrm{Im}V(p)=\bigoplus _{\xi \in \mathbb {D}^e} \mathrm{Im}V^{{\fancyscript{J}}(\xi )}(p)\).

Let \(\left( \alpha _j\right) \in {\mathbb {R}}^{{{\fancyscript{J}}}^\prime }\) such that \(\sum _{j\in {{\fancyscript{J}}}^\prime } \alpha _j V^j(p) = 0\). We work by backward induction on \(\kappa \) from \(k\) to \(1\).

For all \(\xi \in {\mathbb {D}}^e_{\tau _k}\), \(\sum _{j\in {{\fancyscript{J}}}^\prime } \alpha _j V^j_{{\mathbb {D}}^+\left( \xi \right) }(p) = 0\). Since \(\tau _\kappa < \tau _k\) for all \(\kappa =1, \ldots , k-1\), for all \(j\) such that \(\xi (j) \notin \mathbb {D}^-(\xi ) \cup \{\xi \}\), \(V^j_{{\mathbb {D}}^+\left( \xi \right) }(p)=0\). So, one gets

$$\begin{aligned} \sum _{j\in {{\fancyscript{J}}}^\prime \left( \xi \right) } \alpha _j V^j_{{\mathbb {D}}^+\left( \xi \right) }(p) +\sum _{\xi ^\prime \in \mathbb {D}^-(\xi )} \sum _{j\in {{\fancyscript{J}}}^\prime (\xi ^\prime )} \alpha _j V^j_{{\mathbb {D}}^+\left( \xi \right) }(p)= 0 \end{aligned}$$

From Assumption R,

$$\begin{aligned} \hbox {Vect}\left( V_{{\mathbb {D}}^+\left( \xi \right) }^{{{\fancyscript{J}}}\left( \mathbb {D}^-(\xi )\right) } (p)\right) \bigcap \hbox {Vect}\left( V_{{\mathbb {D}}^+\left( \xi \right) }^{{{\fancyscript{J}}} \left( \xi \right) }(p)\right) = \left\{ 0\right\} . \end{aligned}$$

From the above equality,

$$\begin{aligned} \sum _{j\in {{\fancyscript{J}}}^\prime \left( \xi \right) } \alpha _j V^j_{{\mathbb {D}}^+\left( \xi \right) }(p) \in \hbox {Vect}\left( V_{{\mathbb {D}}^+\left( \xi \right) }^{{{\fancyscript{J}}}\left( \mathbb {D}^-(\xi )\right) }(p) \right) \bigcap \hbox {Vect}\left( V_{{\mathbb {D}}^+\left( \xi \right) }^{{{\fancyscript{J}}} \left( \xi \right) }(p)\right) \end{aligned}$$

hence \(\sum _{j\in {{\fancyscript{J}}}^\prime \left( \xi \right) } \alpha _j V^j_{{\mathbb {D}}^+\left( \xi \right) }(p) =0 \).

By construction, the family \((V^j (p))_{j\in {{\fancyscript{J}}}^\prime (\xi )} \) is linearly independent and for all \(\xi ^\prime \notin {\mathbb {D}}^+\left( \xi \right) \), \(V^j_{\xi ^\prime }(p)=0\), so the family \((V^j_{{\mathbb {D}}^+\left( \xi \right) }(p))_{j\in {{\fancyscript{J}}}^\prime \left( \xi \right) } \) is linearly independent. Hence, from above, one deduces that \(\alpha _j=0\) for all \(j \in {{\fancyscript{J}}}^\prime \left( \xi \right) \). Since this is true for all \(\xi \in \mathbb {D}^e_{\tau _k}\), one gets \(\alpha _j=0\) for all \(j \in {{\fancyscript{J}}}^\prime _k\).

If \(k=1\), we are done. If \(k\ge 2\), we do again the same argument as above. Indeed, since we have proved that for all \(j \in {{\fancyscript{J}}}^\prime _k\), \(\alpha _j=0\), for all \(\xi \in \mathbb {D}^e_{\tau _{k-1}}\), \(\sum _{j\in {{\fancyscript{J}}}^\prime } \alpha _j V^j_{{\mathbb {D}}^+\left( \xi \right) } (p)= 0\) implies \(\sum _{j \in {\fancyscript{J}}^\prime \setminus {\fancyscript{J}}^\prime _k} V^j_{{\mathbb {D}}^+\left( \xi \right) } (p)=0\), hence

$$\begin{aligned} \sum _{j\in {{\fancyscript{J}}}^\prime \left( \xi \right) } \alpha _j V^j_{{\mathbb {D}}^+\left( \xi \right) }(p) +\sum _{\xi ^\prime \in \mathbb {D}^-(\xi )} \sum _{j\in {{\fancyscript{J}}}^\prime (\xi ^\prime )} \alpha _j V^j_{{\mathbb {D}}^+\left( \xi \right) }(p)= 0. \end{aligned}$$

Using again Assumption R and the linear independence of \((V^j_{\mathbb {D}^+(\xi )}(p))_{j \in {\fancyscript{J}}^\prime (\xi )}\), one then deduces that for all \(j \in {{\fancyscript{J}}}^\prime _{k-1}\), \(\alpha _j=0\).

Consequently, after a finite number of steps, we deduce that all \(\alpha _j\) are equal to \(0\), which implies that the family \((V^j(p))_{j \in {\fancyscript{J}}^\prime }\) is linearly independent.

For the remaining result about the matrix \(W(p,q)\), we prove that \((W^j(p,q))_{j \in {\fancyscript{J}}^\prime }\) is a linearly independent family. We use a similar argument noticing that for all \(j \in {\fancyscript{J}}^\prime (\xi ) \cup \left( \cup _{\xi ^\prime \in \mathbb {D}^- (\xi )} {\fancyscript{J}}^\prime (\xi ^\prime )\right) \), \(V^j_{\mathbb {D}^+ (\xi )}(p)= W^j_{\mathbb {D}^+ (\xi )} (p,q)\), which implies that for all \(\xi \in \mathbb {D}^e\), the family \((W^j(p,q))_{j \in {\fancyscript{J}}^\prime (\xi )}\) is linearly independent. \(\square \)

Proof of Corollary 3.2

We provide the proof of Part A and the same reasoning applies to Part B thanks to Assumption R. Obviously, \((ii)\) implies \((i)\). Let us show the other implication.

According to the rank Theorem and thanks to Proposition 3.5, we have

$$\begin{aligned} \dim \mathrm{Ker}V(p)&= \# {\fancyscript{J}}- \dim \mathrm{Im}V(p)\\&= \sum \nolimits _{\xi \in \mathbb {D}^e}\#{\fancyscript{J}}(\xi ) - \sum \nolimits _{\xi \in \mathbb {D}^e}\dim \mathrm{Im}V^{{\fancyscript{J}}(\xi )}(p)\\&= \sum \nolimits _{\xi \in \mathbb {D}^e}[\#{\fancyscript{J}}(\xi ) - \dim \mathrm{Im}V^{{\fancyscript{J}}(\xi )}(p)]\\&= \sum \nolimits _{\xi \in \mathbb {D}^e}\dim \mathrm{Ker}V^{{\fancyscript{J}}(\xi )}(p). \end{aligned}$$

If \(\left( V^j(p)\right) _{j \in {\fancyscript{J}}(\xi )}\) is a linearly independent family for all \(\xi \in \mathbb {D}^e\), then \(\dim \mathrm{Ker}V^{{\fancyscript{J}}(\xi )}\) \((p)=0\). From the previous equality, \(\dim \mathrm{Ker}V(p)=0\), hence, the family \(\{V^j (p)\}_{j \in {\fancyscript{J}}}\) is linearly independent. So \((i)\) implies \((ii)\). \(\square \)

Proof of Proposition 3.6

1) The proof is just an adaptation of the proof of Proposition 3.5. In the first step, since the price \(q\) is not supposed to be an arbitrage free price, we get \(\mathrm{rank }W^{{\fancyscript{J}}(\xi )} (p,q) \ge \mathrm{rk}(\xi )\) instead of an equality. For the second step, the proofs never uses the fact that \(q\) is an arbitrage free price, so we can replicate them to obtain \(\mathrm{rank }W(p,q) \ge \sum _{\xi \in \mathbb {D}^e} \mathrm{rk}(\xi ) = \mathrm{rank }V(p)\).

2) If \(\mathrm{rk}(\xi )= n(\xi )\) for all \(\xi \), then \(\sum _{\xi \in \mathbb {D}^e} \mathrm{rk}(\xi ) \) is the cardinal of \({\fancyscript{J}}\), which is the number of column of the matrix \(W(p,q)\). So \(\mathrm{rank }W(p,q) \le \sum _{\xi \in \mathbb {D}^e} \mathrm{rk}(\xi ) = \mathrm{rank }V(p)\).\(\square \)

Proof of lemma 3.2

We first show that the equality of kernels implies the equality of dimensions of the images. Let \(G\) be a linear subspace of \(E\) and let \(\varphi _G\) (resp. \(\psi _G\)) be the restriction of \(\varphi \) (resp. \(\psi \)) at \(G\). We have \(\varphi (G)= \mathrm{Im}\varphi _G\) and \(\dim \mathrm{Im}\varphi _G= \dim G - \dim \left( \mathrm{Ker}\varphi _G\right) \). As \( \mathrm{Ker}\varphi = \mathrm{Ker}\psi \), we have \(\mathrm{Ker}\varphi _G= \left( \mathrm{Ker}\varphi \right) \cap G=\left( \mathrm{Ker}\psi \right) \cap G=\mathrm{Ker}\psi _G\) hence \(\dim \varphi (G)= \dim \psi (G)\).

Let us show the converse implication. If \( \mathrm{Ker}\varphi \ne \mathrm{Ker}\psi \), then there exists \( u \in \mathrm{Ker}\varphi \) such that \(u \notin \mathrm{Ker}\psi \) or there exists \( u \in \mathrm{Ker}\psi \) such that \( u \notin \mathrm{Ker}\varphi \). In the first case, with \(G= \mathrm{Ker}\varphi \), we have \(\varphi (G)= \{0\} \ne \psi (G)\), hence \(\dim \varphi (G)= 0 <\dim \psi (G)\). In the second case, we obtain the same inequality with \(G= \mathrm{Ker}\psi \). So the equality of the dimension of \(\varphi (G)\) and \(\psi (G)\) for all linear subspace \(G\) implies the equality of kernels. \(\square \)

Proof of Proposition 4.3

Let \(\lambda \in \mathbb {R}^\mathbb {D}_{++}\) and \(q\) be the unique asset price such that \({^t}W(q) \lambda =0\). Since \({\fancyscript{Z}}_{\fancyscript{F}}\cap \mathrm{Ker}V = \{0\}\), Assumption R and Proposition 3.4 imply that \({\fancyscript{Z}}_{\fancyscript{F}}\cap \mathrm{Ker}W(q) = \{0\}\). So Proposition 4.4 implies that there exists \(\delta >0\) such that \(B^\delta (\lambda )\) is bounded. Hence, all assumptions of Theorem 3.1 of Angeloni and Cornet (2006) are satisfied but the fact that \(0 \in \mathrm{ri}_{{\fancyscript{Z}}_{\fancyscript{F}}} (Z_{i_0})\) instead of \(0 \in \mathrm{int}Z_{i_0}\) and \(B^\delta (\lambda )\) is bounded instead of \(B^1(\lambda )\). To complete the proof, we now show how to adapt the proof of Angeloni-Cornet to these slightly more general conditions.

In the preliminary definitions, \(\eta \) is chosen in \({\fancyscript{Z}}_{\fancyscript{F}}\) instead of \(\mathbb {R}^{\fancyscript{J}}\). Then the set \(B\) is replaced by

$$\begin{aligned} B^\delta =\{(p, \eta ) \in \mathbb {R}^\mathbb {L}\times {\fancyscript{Z}}_{\fancyscript{F}}\mid \Vert p \Vert \le 1, \Vert \eta \Vert \le \delta \} \end{aligned}$$

and the function \(\rho \) is defined by \(\rho (p, \eta ) = \max \{0, 1- \Vert p\Vert -(1/\delta ) \Vert \eta \Vert \}\). This choice of the set \(B^\delta \) allows us to conclude in Sub-sub-section 4.1.3 that \((\bar{x}, \bar{z}, \bar{p}, \bar{q})\) is an equilibrium and furthermore \((\bar{x}, \bar{z}) \) belongs to \(B^\delta (\lambda )\), which is used in the proof of Proposition 4.2 on page 22. In Step 2 of the proof of Claim 4.1, if \(\eta \ne 0\), we obtain \(0 < \max \{\eta \bullet _{\fancyscript{J}}z_{i_0} \mid z_{i_0} \in Z_{i_0}\}\) since \(Z_{i_0}\) is included in \({\fancyscript{Z}}_{\fancyscript{F}}\), \(\eta \in {\fancyscript{Z}}_{\fancyscript{F}}\) and \(0 \in \mathrm{ri}_{{\fancyscript{Z}}_{\fancyscript{F}}} (Z_{i_0})\), so \(r \eta \in Z_{i_0}\) for \(r>0\) small enough. In Claim 4.3 of Sub-sub-section 4.1.3, the argument holds true since \(\bar{z}_i \in {\fancyscript{Z}}_{\fancyscript{F}}\) for all \(i\) and so, \((\delta /\Vert \sum _{i \in {\fancyscript{I}}} \bar{z}_i\Vert ) \sum _{i \in {\fancyscript{I}}} \bar{z}_i\) belongs to \({\fancyscript{Z}}_{\fancyscript{F}}\). The equality \(\sum _{i \in I} (\bar{x}_i - e_i)=0\) is obtained by the same argument. Indeed, since \(\lambda _\xi >0\) for all \(\xi \), \(p \rightarrow (\lambda \Box p) \bullet _\mathbb {L}\sum _{i \in I} (\bar{x}_i - e_i) \) is a non zero linear mapping if \(\sum _{i \in I} (\bar{x}_i - e_i)\ne 0\) so its maximum on the ball is positive and reached on the boundary of \(\bar{B}_\mathbb {L}(0,1)\), which implies that \(\Vert \bar{p} \Vert = 1\) and \(\rho (\bar{p}, \bar{\eta })=0\).

In Sub-sub-section 4.2.2, to show that \(0 \in \mathrm{ri}_{{\fancyscript{Z}}_{\fancyscript{F}}} (Z_{i_0 r})\) in the truncated economy, we remark that there exists \(r^\prime >0\) such that \(B_{\fancyscript{J}}(0,r^\prime ) \cap {\fancyscript{Z}}_{\fancyscript{F}}\subset Z_{i_0}\), hence, \(B_{\fancyscript{J}}(0,\min \{r,r^\prime \}) \cap {\fancyscript{Z}}_{\fancyscript{F}}\subset Z_{i_0 r}\), which means that \(0\) belongs to the relative interior of \(Z_{i_0 r}\) with respect to \({\fancyscript{Z}}_{\fancyscript{F}}\). \(\square \)

Proof of Proposition 4.4

For every \(\delta >0\), for every \(i \in {\fancyscript{I}}, \lambda \in {\mathbb {R}}^{\mathbb {D}}_{++}\), we let \(\hat{X}^\delta _i(\lambda )\) and \(\hat{Z}_i^\delta (\lambda )\) be the projections of \(B^\delta (\lambda )\) on \(X_i\) and \(Z_i\), that is respectively:

$$\begin{aligned} \hat{X}^\delta _i(\lambda )&:= \left\{ x_i \in X_i \mid \exists \left( x_k\right) _{k\ne i} \in \prod _{k\ne i} X_k, \exists z \in \prod _{k \in {\fancyscript{I}}} Z_k, \left( x, z\right) \in B^\delta (\lambda )\right\} \\ \hat{Z}^\delta _i(\lambda )&:= \left\{ z_i \in Z_i \mid \exists \left( z_k\right) _{k\ne i} \in \prod _{k\ne i} Z_k,~\exists x \in \prod _{ k \in {\fancyscript{I}}} X_k, \left( x, z\right) \in B^\delta (\lambda )\right\} . \end{aligned}$$

Step 1: for all \(\delta \ge 0\), and for all \(i \in {\fancyscript{I}}\), \(\hat{X}^\delta _i(\lambda )\) is bounded.

Indeed, let \(\delta \ge 0\), and \(i \in {\fancyscript{I}}\), since the sets \(X_i\) are bounded below, there exists \(\underline{x}_i \in {\mathbb {R}}^{\mathbb {L}}\) such that \(X_i \subset \{ \underline{x}_i \} + {\mathbb {R}}^{\mathbb {L}}_+\). If \(x_i \in \hat{X}_i\left( \lambda \right) \), there exists \(x_k \in X_k\), for every \(k\ne i\), such that \(x_i + \sum _{k \ne i} x_k = \sum _{\kappa \in {\fancyscript{I}}} e_\kappa \). Consequently,

$$\begin{aligned} \underline{x}_i \le x_i = -\sum _{k\ne i} x_k + \sum _{\kappa \in {\fancyscript{I}}} e_\kappa \le -\sum _{k\ne i} \underline{x}_k + \sum _{\kappa \in {\fancyscript{I}}} e_\kappa \end{aligned}$$

and so \(\hat{X}^\delta _i\left( \lambda \right) \) is bounded.

Step 2: for all \(i \in {\fancyscript{I}}\), \(\hat{Z}^0_i(\lambda )\) is bounded.

For all \(i \in {\fancyscript{I}}\) and for every \(z_i \in \hat{Z}^0_i(\lambda )\), there exists \(\left( z_k\right) _{k\ne i} \in \prod _{k\ne i} Z_k\), \(x \in \prod _{ \kappa \in {\fancyscript{I}}} X_\kappa \) and \(p \in \bar{B}_{\mathbb {L}} \left( 0,1\right) \), such that \(z_i + \sum _{k \ne i} z_k = 0\) and \(\left( x_\kappa ,z_\kappa \right) \in B^\kappa _{\fancyscript{F}} \left( p,q\right) \) for every \(\kappa \in {\fancyscript{I}}\). As \((x_\kappa ,z_\kappa )\in B^\kappa _{\fancyscript{F}}(p,q)\) and \(\left( x_\kappa ,p\right) \in \hat{X}^0_j(\lambda ) \times \bar{B}_{\mathbb {L}} (0,1)\), a compact set, there exists \(\alpha _j \in {\mathbb {R}}^{\mathbb {D}}\) such that

$$\begin{aligned} \alpha _j \le p\scriptscriptstyle {\Box }\textstyle \left( x_\kappa - e_\kappa \right) \le W(q)z_\kappa . \end{aligned}$$

Using the fact that \(\sum _{\kappa \in {\fancyscript{I}}} z_\kappa =0\), we have

$$\begin{aligned} \alpha _i\le W(q)z_i = W(q)\left( -\sum _{k\ne i} z_k\right) \le -\sum _{k\ne i} \alpha _k, \end{aligned}$$

hence there exists \(r >0\) such that \(W(q)z_i \in \bar{B}_{\mathbb {D}}\left( 0,r\right) \).

By assumption, \({\fancyscript{Z}}_{\fancyscript{F}}\cap \mathrm{Ker}W(q)=\{0\}\). So, the linear mapping \(W(q)_{\mid {\fancyscript{Z}}_{\fancyscript{F}}}\) from \({\fancyscript{Z}}_{\fancyscript{F}}\) to \(W(q) {\fancyscript{Z}}_{\fancyscript{F}}\) is an isomorphism. Since we have proved that for every \(z_i \in \hat{Z}^0_i(\lambda )\), \(W(q)z_i \in \bar{B}_{\mathbb {D}}\left( 0,r\right) \) and \(Z_i \subset {\fancyscript{Z}}_{\fancyscript{F}}\), we can conclude that

$$\begin{aligned} \hat{Z}^0_i(\lambda ) \subset [W(q)_{\mid {\fancyscript{Z}}_{\fancyscript{F}}}]^{-1} ( \bar{B}_{\mathbb {D}}\left( 0,r\right) \cap W(q){\fancyscript{Z}}_{\fancyscript{F}}) \end{aligned}$$

a bounded subset, so \(\hat{Z}^0_i(\lambda ) \) is a bounded subset of \({\fancyscript{Z}}_{\fancyscript{F}}\).

Let \( M \in \mathbb {R}^*_+\) such that for all \((x,z) \in B^0(\lambda )\), \(\left\| z\right\| < M.\)

Step 3: there exists \(\delta >0\) such that \(B^\delta (\lambda )\) is bounded.

By contradiction. Suppose that for all \(\delta >0\), \(B^\delta (\lambda )\) is not bounded. This implies that for all \(\nu \in \mathbb {N}^*\), \(B^{1/\nu }(\lambda )\) is not bounded. We build a sequence \((x^\nu ,z^\nu )_{\nu \in \mathbb {N}^*}\) in \( \prod _{i \in {\fancyscript{I}}}X_i \times \prod _{i \in {\fancyscript{I}}}Z_i\) by induction in the following way: \((x^1,z^1) \in B^1(\lambda )\) such that \(\left\| z^1\right\| > M+1\) and for all \(\nu \in \mathbb {N}^*\), \((x^{\nu +1},z^{\nu +1}) \in B^{\frac{1}{\nu +1}}(\lambda )\) and \(\left\| z^{\nu +1}\right\| >\left\| z^\nu \right\| +1\). So \(\left\| z^\nu \right\| \) converges to \(+\infty \).

Since for all \(\nu \in \mathbb {N}^*, (x^\nu ,z^\nu ) \in B^{1/\nu }(\lambda )\), there exists a sequence \((p^\nu ,q^\nu )_{\nu \in \mathbb {N}^*}\) such that for all \(\nu \in \mathbb {N}^*,\) \(\left\| p^\nu \right\| \le 1\), \(p^\nu \scriptscriptstyle {\Box }\textstyle (x^\nu _i-e_i)\le W(q^\nu )z^\nu _i\) and \(0\le \left\| {}^tW(q^\nu )\lambda \right\| \le \frac{1}{\nu }\). We remark that for all \(\nu \in \mathbb {N}^*, B^{\frac{1}{\nu +1}}(\lambda )\subset B^{\frac{1}{\nu }}(\lambda )\) so the sequence \((x^\nu ,z^\nu ) \subset B^1(\lambda )\). By Step 1, the sequence \((x^\nu )\) is bounded. For each \(\nu \in \mathbb {N}^*\), let \(\zeta ^\nu =M\frac{z^\nu }{\left\| z^\nu \right\| }\). \({}^tW(q^\nu )\lambda \in \bar{B}_{\fancyscript{J}}(0,\frac{1}{\nu })\) implies that for all \(\nu \in \mathbb {N}^*\) and for all \(j \in {\fancyscript{J}}\) there exists \(\eta ^\nu \in \bar{B}_{\fancyscript{J}}(0,\frac{1}{\nu })\) such that \(\lambda _{\xi (j)}q^{\nu j}=\sum _{\xi \in \mathbb {D}^+(\xi (j))}\lambda _\xi V^j_\xi + \eta ^{\nu j}\). Hence the sequence \((q^{\nu j})\) is bounded for all \(j\). Consequently the sequence \((x^\nu ,\zeta ^\nu ,p^\nu ,q^\nu )\) is bounded so it has a subsequence \((x^{\phi (\nu )},\zeta ^{ \phi (\nu )},p^{ \phi (\nu )}, q^{\phi (\nu )})\), which converges to \( (\bar{x}, \bar{\zeta }, \bar{p}, \bar{q})\). Note that \(\Vert \bar{\zeta }\Vert =M\).

Let us now show that \((\bar{x},\bar{\zeta }) \in B^0(\lambda )\).

  • \({}^tW(\bar{q})\lambda =0\) since \(\Vert {}^tW(q^{\phi (\nu )})\lambda \Vert \le \frac{1}{\phi (\nu )}\) for all \(\nu \). For all \(i\), \(\bar{x}_i \in X_i\) because \(X_i\) is closed. For all \(i\), \(\bar{\zeta }_i \in Z_i\). Indeed, \(Z_i\) is closed and \(\zeta ^{\phi (\nu )}_i=M\frac{ z^{\phi (\nu )}_i}{\left\| z^{\phi (\nu )}\right\| } \in Z_i\) since \(z_i^{\phi (\nu )} \in Z_i\), \(0 \in Z_i\), \(0<\frac{M}{\left\| z^{\phi (\nu )}\right\| }<1 \) and \(Z_i\) is convex.

  • For all \(i\), \((\bar{x}_i,\bar{\zeta }_i) \in B^i_{\fancyscript{F}}(0,\bar{q})\). Indeed, for all \(\nu \in \mathbb {N}^*\), \((x^\nu ,z^\nu ) \in B^1(\lambda )\), hence

    $$\begin{aligned} p^{\phi (\nu )} \scriptscriptstyle {\Box }\textstyle (x_i^{\phi (\nu )}-e_i) \le W(q^{\phi (\nu )})z^{\phi (\nu )}_i. \end{aligned}$$

    So \( \left( \frac{M}{\left\| z^{\phi (\nu )}\right\| }p^{\phi (\nu )}\right) \scriptscriptstyle {\Box }\textstyle (x_i^{\phi (\nu )}-e_i) \le W(q^{\phi (\nu )})\left( \frac{M}{\left\| z^{\phi (\nu )}\right\| } z^{\phi (\nu )}_i\right) \). At the limit, since \(\zeta ^{\phi (\nu )}_i = \frac{Mz^{\phi (\nu )}}{\left\| z^{\phi (\nu )}\right\| }\) and \(\frac{M}{\left\| z^{\phi (\nu )}\right\| }\) converges to \(0\), one gets, \(0\le W(\bar{q})\bar{\zeta }_i\), that is \((\bar{x}_i,\bar{\zeta }_i) \in B^i_{\fancyscript{F}}(0,\bar{q})\).

  • \(\sum _{i \in {\fancyscript{I}}}\bar{x}_i= \sum _{i \in {\fancyscript{I}}}e_i\) and \(\sum _{i \in {\fancyscript{I}}}\bar{\zeta }_i=0\) since for all \(\nu \), \(\sum _{i \in {\fancyscript{I}}}x^{\phi (\nu )}_i= \sum _{i \in {\fancyscript{I}}}e_i\) and \( \sum _{i \in {\fancyscript{I}}}\zeta ^{\phi (\nu )}_i =\sum _{i \in {\fancyscript{I}}}M\frac{ z^{\phi (\nu )}_i}{\left\| z^{\phi (\nu )}\right\| }=0\).

Hence, one gets a contradiction since \((\bar{x},\bar{\zeta }) \in B^0(\lambda )\) and \(\left\| \bar{\zeta }\right\| =M\) whereas we have chosen \(M\) large enough so that for all \((x,z) \in B^0(\lambda )\), \(\left\| z\right\| < M.\) \(\square \)

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Bonnisseau, JM., Chery, A. Stability of marketable payoffs with long-term assets. Ann Finance 10, 523–552 (2014). https://doi.org/10.1007/s10436-014-0251-z

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