We first consider the special case, summarized in the following definition, where the weights, asset holdings, systemic importance levels, capitals and exogenous losses take only finitely many different values.
Proof
We start with the proof of the lower bounds. That is, for arbitrary
\(\epsilon >0\) we will show that
and
\(\chi _n^m\ge \hat{\chi }_n^m\ge (1-\epsilon )\hat{\chi }^m\) w.h.p. We therefore consider the contagion process given by rounds (i
\(^{\prime }\)) and (ii
\(^{\prime }\)). That is, we first consider a cascade of default contagion. Once this cascade has ended (after at most
\(n-1\) steps) we start a cascade of fire sales and so on.
In order to quantify the default contagion cascade we use [
31, Theorem 3.2.4], which extends the setting in [
17] for systemic importance. That is, if we denote by
\(\hat{{\varvec{z}}}_1\in {\mathbb {R}}_{+,0}^V\) the smallest vector such that
$$\begin{aligned} {\mathbb {E}}\left[ W^{+,r,\alpha }\mathbb {P}\left( \sum _{s\in [R]}s\mathrm {Poi}\left( \sum _{\gamma \in [T]}W^{-,s,\gamma }\hat{z}_1^{s,\beta ,\gamma }\right) > C-L\right) \mathbf {1}\{A=\beta \}\right] =\hat{z}_1^{r,\alpha ,\beta } \end{aligned}$$
for all
\((r,\alpha ,\beta )\in V\), then the systemic importance of finally defaulted banks is bounded by
$$\begin{aligned} (1-\delta )n\sum _{\beta \in [T]}{\mathbb {E}}\left[ S\mathbb {P}\left( \sum _{s\in [R]}s\mathrm {Poi}\left( \sum _{\gamma \in [T]}W^{-,s,\gamma }\hat{z}_1^{s,\beta ,\gamma }\right) > C-L\right) \mathbf {1}\{A=\beta \}\right] \end{aligned}$$
from below w.h.p. for any fixed
\(\delta >0\). In fact, by finitariness of the system we can find
\(\theta >0\) small enough such that
$$\begin{aligned}&{\mathbb {E}}\left[ W^{+,r,\alpha }\mathbb {P}\left( \sum _{s\in [R]}s\mathrm {Poi}\left( \sum _{\gamma \in [T]}W^{-,s,\gamma }z^{s,\beta ,\gamma }\right) > C-L\right) \mathbf {1}\{A=\beta \}\right] \\&\quad = {\mathbb {E}}\left[ W^{+,r,\alpha }\mathbb {P}\left( \sum _{s\in [R]}s\mathrm {Poi}\left( \sum _{\gamma \in [T]}W^{-,s,\gamma }z^{s,\beta ,\gamma }\right) \ge \left\lceil C-L+\theta \right\rceil \right) \mathbf {1}\{A=\beta \}\right] \end{aligned}$$
(note that
\(\left\lceil C-L + \theta \right\rceil \) is the weak limit of
\(\left\lceil C_n-L_n + \theta \right\rceil \) again by finitariness) and we are thus in the setting of [
17]. However, while [
31, Theorem 3.2.4] focuses on the systemic damage due to defaulted banks only, here it is also important to keep track of all losses due to defaults. In fact, the proof of [
31, Theorem 3.2.4] for finitary systems shows that the number
\(q_{j,k}^\beta \) of institutions of class
j and type
\(\beta \) with a total edge weight from finally defaulted neighbors of at least
\(k\le \tilde{c}_j\) is lower bounded by
$$\begin{aligned} (1-\delta )np_j^\beta \mathbb {P}\left( \sum _{s\in [R]}s\mathrm {Poi}\left( \sum _{\gamma \in [T]}\tilde{w}_j^{-,s,\gamma }\hat{z}_1^{s,\beta ,\gamma }\right) \ge k\right) \end{aligned}$$
w.h.p. for
\(\delta >0\). Since in this part of the proof we are interested in lower bounds, we assume in the following that
\(q_{j,\lceil \tilde{c}_j-\tilde{\ell }_j\rceil }^\beta =(1-\delta )np_j^\beta \mathbb {P}(\sum _{s\in [R]}s\mathrm {Poi}(\sum _{\gamma \in [T]}\tilde{w}_j^{-,s,\gamma }\hat{z}_1^{s,\beta ,\gamma })\ge \lceil \tilde{c}_j-\tilde{\ell }_j\rceil )\),
\(q_{j,k}^\beta =(1-\delta )np_j^\beta \mathbb {P}(\sum _{s\in [R]}s\mathrm {Poi}(\sum _{\gamma \in [T]}\tilde{w}_j^{-,s,\gamma }\hat{z}_1^{s,\beta ,\gamma }) = k)\) for
\(1\le k< \lceil \tilde{c}_j-\tilde{\ell }_j\rceil \), and
\(q_{j,0}^\beta =np_j^\beta (n)-\sum _{k=1}^{\lceil \tilde{c}_j-\tilde{\ell }_j\rceil }q_{j,k}^\beta \) w.h.p. That is, we increase the losses due to default contagion.
Next, we want to use [
18, Theorem EC. 1] to quantify the impact of the round of fire sales. We need to consider losses (and defaults in particular) due to the previous default contagion cascade. That is, we need to add to the exogenous losses
\(\ell _i\) the edge weight from defaulted debtors of each bank
\(i\in [n]\). This leads to a new loss vector
\(\big (\ell _i'\big )_{i\in [n]}\). Note that we can set
\(\ell _i'=\tilde{\ell }_j+\lceil \tilde{c}_j-\tilde{\ell }_j\rceil \) if
i is of type
j and the total edge-weight
k from finally defaulted debtors of
i is larger or equal to
\(\lceil \tilde{c}_j-\tilde{\ell }_j\rceil \). Denoting by
\(L_n'\) a random vector distributed according to the empirical distribution function of
\(\big (\ell _i'\big )_{i\in [n]}\), we thus derive that w.h.p.
$$\begin{aligned}&\mathbb {P}\left( {\varvec{W}}_n^-=\tilde{{\varvec{w}}}_j^-,{\varvec{W}}_n^+=\tilde{{\varvec{w}}}_j^+,{\varvec{X}}_n=\tilde{{\varvec{x}}}_j,S_n=\tilde{s}_j,C_n=\tilde{c}_j,L_n'=\tilde{\ell }_j+k,A_n=\beta \right) \\&\quad ={\left\{ \begin{array}{ll} (1-\delta )p_j^\beta \mathbb {P}\left( \sum _{s\in [R]}s\mathrm {Poi}\left( \sum _{\gamma \in [T]}\tilde{w}_j^{-,s,\gamma }\hat{z}_1^{s,\beta ,\gamma }\right) \ge \left\lceil \tilde{c}_j-\tilde{\ell }_j\right\rceil \right) ,&{}\quad \text {if }k=\left\lceil \tilde{c}_j-\tilde{\ell }_j\right\rceil , \\ (1-\delta )p_j^\beta \mathbb {P}\left( \sum _{s\in [R]}s\mathrm {Poi}\left( \sum _{\gamma \in [T]}\tilde{w}_j^{-,s,\gamma }\hat{z}_1^{s,\beta ,\gamma }\right) = k\right) ,&{}\quad \text {if }1\le k\le \left\lceil \tilde{c}_j-\tilde{\ell }_j\right\rceil -1, \\ p_j^\beta (n) - (1-\delta )p_j^\beta \mathbb {P}\left( \sum _{s\in [R]}s\mathrm {Poi}\left( \sum _{\gamma \in [T]}\tilde{w}_j^{-,s,\gamma }\hat{z}_1^{s,\beta ,\gamma }\right) \ge 1\right) ,&{}\quad \text {if }k=0. \end{array}\right. } \end{aligned}$$
For simplicity in the notation, we assume that from
\((\tilde{{\varvec{x}}}_j,\tilde{s}_j,\tilde{c}_j,\tilde{\ell }_j)=(\tilde{{\varvec{x}}}_k,\tilde{s}_k,\tilde{c}_k,\tilde{\ell }_k)\) it follows
\(j=k\) (i.e. classes
j and
k are not distinguished by their in- and out-weights only) in the following. Otherwise consider sums over classes with the same asset holdings, systemic importance, capital and exogenous loss.
In particular, for the weak limit
\(({\varvec{X}},S,C,L',A)\) of
\(({\varvec{X}}_n,S_n,C_n,L_n',A_n)\) and
\(0\le k\le \lceil \tilde{c}_j-\tilde{\ell }_j\rceil \)$$\begin{aligned}&\mathbb {P}\Big ({\varvec{X}}=\tilde{{\varvec{x}}}_j,S=\tilde{s}_j,C=\tilde{c}_j,L'=\tilde{\ell }_j+k,A=\beta \Big )\\&\quad \ge (1-\delta )p_j^\beta \mathbb {P}\Bigg (\sum _{s\in [R]}s\mathrm {Poi}\Bigg (\sum _{\gamma \in [T]}\tilde{w}_j^{-,s,\gamma }\hat{z}_1^{s,\beta ,\gamma }\Bigg ) = k\Bigg ). \end{aligned}$$
Let now
the corresponding functions as in [
18] adapted to the setting with heterogeneous sales functions and
\(\hat{{\varvec{\chi }}}_\delta \) its smallest fixed point. Then
and
. In particular,
and by [
18, Lemma EC. 2] we derive that
\(\liminf _{\delta \rightarrow 0+}\hat{{\varvec{\chi }}}_\delta \ge \hat{{\varvec{\chi }}}_1\), where
\(\hat{{\varvec{\chi }}}_1\) denotes the smallest joint root of the functions
,
\(m\in [M]_0\), for fixed
\({\varvec{z}}=\hat{{\varvec{z}}}_1\).
We can hence choose
\(\delta \) small enough such that the number of finally sold shares of asset
m is lower bounded by
\(n(1-\epsilon )\hat{\chi }_1^m\) w.h.p. by [
18, Theorem EC. 1]. Further, for
and possibly further reducing
\(\delta \), we derive
w.h.p. So if
\((\hat{{\varvec{z}}}_1,\hat{{\varvec{\chi }}}_1)=(\hat{{\varvec{z}}},\hat{{\varvec{\chi }}})\), then this finishes the proof of the lower bounds.
If
\((\hat{{\varvec{z}}}_1,\hat{{\varvec{\chi }}}_1)\ne (\hat{{\varvec{z}}},\hat{{\varvec{\chi }}})\), then by construction of
\((\hat{{\varvec{z}}},\hat{{\varvec{\chi }}})\),
\(\hat{{\varvec{z}}}_1\) and
\(\hat{{\varvec{\chi }}}_1\) it must hold that
\(\hat{{\varvec{\chi }}}_1\le \hat{{\varvec{\chi }}}\) and
\(\hat{{\varvec{z}}}_1\lneq \hat{{\varvec{z}}}_2\le \hat{{\varvec{z}}}\), where
\(\hat{{\varvec{z}}}_2\in {\mathbb {R}}_{+,0}^V\) denotes the smallest vector such that for all
\((r,\alpha ,\beta )\in V\),
$$\begin{aligned} {\mathbb {E}}\left[ W^{+,r,\alpha }\mathbb {P}\left( \sum _{s\in [R]}s\mathrm {Poi}\left( \sum _{\gamma \in [T]}W^{-,s,\gamma }\hat{z}_2^{s,\beta ,\gamma }\right) > C-L-{\varvec{X}}\cdot h(\hat{{\varvec{\chi }}}_1)\right) \mathbf {1}\{A=\beta \}\right] = \hat{z}_2^{r,\alpha ,\beta }. \end{aligned}$$
The next step in the cascade process would now be the default contagion cascade from (i
\(^{\prime }\)) starting from the state of the system after the fire sales cascade. Note, however, that (w.h.p.) we can equivalently restart the whole cascade process if for the default contagion cascade we choose capitals
\(c_i-\ell _i-{\varvec{x}}_i\cdot h((1-\epsilon )\hat{{\varvec{\chi }}}_1)\). If anything this reduces contagion effects which is alright because we are interested in lower bounds.
By the finitariness of the system, if we choose
\(\epsilon \) small enough, then
\(\hat{{\varvec{z}}}_2\) is also the smallest solution of
$$\begin{aligned}&{\mathbb {E}}\left[ W^{+,r,\alpha }\mathbb {P}\left( \sum _{s\in [R]}s\mathrm {Poi}\left( \sum _{\gamma \in [T]}W^{-,s,\gamma }\hat{z}_2^{s,\beta ,\gamma }\right) > C-L-{\varvec{X}}\cdot h((1-\epsilon )\hat{{\varvec{\chi }}}_1)\right) \mathbf {1}\{A=\beta \}\right] \\&\quad = \hat{z}_2^{r,\alpha ,\beta }. \end{aligned}$$
and we can hence use [
31, Theorem 3.2.4] to quantify the default cascade. By exactly the same means as above we can then translate the losses due to default contagion into exogenous losses and investigate the fire sales process by [
18, Theorem EC. 1]. We derive that the vector of finally sold shares is lower bounded by
\(n(1-\epsilon )\hat{{\varvec{\chi }}}_2\) w.h.p., where
\(\hat{{\varvec{\chi }}}_2\) denotes the smallest joint root of the functions
for
\({\varvec{z}}=\hat{{\varvec{z}}}_2\), and
w.h.p.
Again, if \((\hat{{\varvec{z}}}_2,\hat{{\varvec{\chi }}}_2)=(\hat{{\varvec{z}}},\hat{{\varvec{\chi }}})\), then this finishes the proof of the lower bounds. Otherwise we can continue on for \(t\ge 3\). Note, however, that \(\hat{{\varvec{z}}}\gneq \hat{{\varvec{z}}}_t\) is only possible if \(\hat{{\varvec{\chi }}}_{t-1}\) and \(\hat{{\varvec{\chi }}}_t\) are separated by a hyperplane of discontinuity of \(f^{r,\alpha ,\beta }\) for some \((r,\alpha ,\beta )\in V\) (and hence the fire sales lead to further defaults in the system). However, as remarked earlier, there can only be finitely many such hyperplanes for finitary systems. Hence by the procedure outlined above, we will reach \(\hat{{\varvec{z}}}\) in finitely many steps and hence the end results still hold w.h.p.
We can now turn to the second part of the proof. We consider the contagion process in rounds (i) and (ii) to derive upper bounds on
\(\overline{\mathcal {S}}_n\) and
\(\overline{{\varvec{\chi }}}_n\). Let
\((\tilde{{\varvec{z}}}(\delta ),\tilde{{\varvec{\chi }}}(\delta ))_{\delta >0}\) be the constructing sequence of
\(({\varvec{z}}^*,{\varvec{\chi }}^*)\) analogue to [
18, Remark EC. 1]. Then note that by upper semi-continuity and the discrete nature of
\(f^{r,\alpha ,\beta }\) we can find
\(\Delta >0\) such that
\(f^{r,\alpha ,\beta }({\varvec{z}},\tilde{{\varvec{\chi }}}(\delta ))=f^{r,\alpha ,\beta }({\varvec{z}},{\varvec{\chi }}^*)\) for all
\(0\le \delta <\Delta \),
\((r,\alpha ,\beta )\in V\) and
\({\varvec{z}}\in {\mathbb {R}}_{+,0}^V\).
Fix now some
\(\delta \in (0,\Delta )\) and consider the financial system with reduced capital values
\(c_i-\ell _i-{\varvec{x}}_i\cdot h(\tilde{{\varvec{\chi }}}(\delta ))\) for each bank
\(i\in [n]\). We only want to consider the default contagion process in this new financial system and we are hence in the setting of [
16] with limiting random variables
\(({\varvec{W}}^-,{\varvec{W}}^+,S,\lceil C-L-{\varvec{X}}\cdot h(\tilde{{\varvec{\chi }}}(\delta ))\rceil ^+,A)\). Note that by finitariness the regularity transfers. By the choice of
\(\delta \) above, we derive that for
\({\varvec{z}}_\delta ^*\) in this new financial system, it holds
\({\varvec{z}}_\delta ^*={\varvec{z}}^*\) and by [
31, Theorem 3.2.4] we know that the final systemic damage in the new system is upper bounded by
$$\begin{aligned}&n\sum _{\beta \in [T]}{\mathbb {E}}\left[ S\mathbb {P}\left( \sum _{s\in [R]}s\mathrm {Poi}\left( \sum _{\gamma \in [T]}W^{-,s,\gamma }(z^*)^{s,\beta ,\gamma }\right) \ge C-L-{\varvec{X}}\cdot h(\tilde{{\varvec{\chi }}}(\delta ))\right) \mathbf {1}\{A=\beta \}\right] \nonumber \\&\quad +o_p(n). \end{aligned}$$
(8.3)
The proof of [
31, Theorem 3.2.4] actually shows that the number of banks of type
\(\beta \) and class
j with at least an edge-weight of
k from defaulted neighbors at the end of the default contagion process is upper bounded by
$$\begin{aligned} (1+\epsilon )np_j^\beta \mathbb {P}\left( \sum _{s\in [R]}s\mathrm {Poi}\left( \sum _{\gamma \in [T]}\tilde{w}_j^{-,s,\gamma }(z^*)^{s,\beta ,\gamma }\right) \ge k\right) \end{aligned}$$
w.h.p. for any fixed
\(\epsilon >0\). Similarly, as in the first part of this proof (for the lower bounds) we can then construct a fire sales system with limiting random vector
\(({\varvec{X}},S,C,L')\) such that
$$\begin{aligned}&\mathbb {P}({\varvec{X}}=\tilde{{\varvec{x}}}_j,S=\tilde{s}_j,C=\tilde{c}_j,L'=\tilde{\ell }_j+k,A=\beta )\\&\quad = (1+\epsilon ) p_j^\beta \mathbb {P}\left( \sum _{s\in [R]}s\mathrm {Poi}\left( \sum _{\gamma \in [T]}\tilde{w}_j^{-,s,\gamma }(z^*)^{s,\beta ,\gamma }\right) = k\right) ,\\&\qquad 0\le k<\left\lceil \tilde{c}_j-\tilde{\ell }_j-\tilde{{\varvec{x}}}_j\cdot h(\tilde{{\varvec{\chi }}}(\delta ))\right\rceil ,\\&\mathbb {P}\left( {\varvec{X}}=\tilde{{\varvec{x}}}_j,S=\tilde{s}_j,C=\tilde{c}_j,L'=\tilde{\ell }_j+\left\lceil \tilde{c}_j-\tilde{\ell }_j-\tilde{{\varvec{x}}}_j\cdot h(\tilde{{\varvec{\chi }}}(\delta ))\right\rceil ,A=\beta \right) \\&\quad = p_j^\beta - \sum _{k=0}^{\left\lceil \tilde{c}_j-\tilde{\ell }_j-\tilde{{\varvec{x}}}_j\cdot h(\tilde{{\varvec{\chi }}}(\delta ))\right\rceil -1}(1+\epsilon ) p_j^\beta \mathbb {P}\left( \sum _{s\in [R]}s\mathrm {Poi}\left( \sum _{\gamma \in [T]}\tilde{w}_j^{-,s,\gamma }(z^*)^{s,\beta ,\gamma }\right) = k\right) \\&\quad \le (1+\epsilon ) p_j^\beta \mathbb {P}\left( \sum _{s\in [R]}s\mathrm {Poi}\left( \sum _{\gamma \in [T]}\tilde{w}_j^{-,s,\gamma }(z^*)^{s,\beta ,\gamma }\right) \ge \left\lceil \tilde{c}_j-\tilde{\ell }_j-\tilde{{\varvec{x}}}_j\cdot h(\tilde{{\varvec{\chi }}}(\delta ))\right\rceil \right) \end{aligned}$$
which dominates the stochastic final state after the default contagion cascade w.h.p.
Let now
$$\begin{aligned} f_\epsilon ^m({\varvec{\chi }}) := \sum _{\beta \in [T]}{\mathbb {E}}\left[ X^m\rho ^m_{\beta }\left( \frac{L'+{\varvec{X}}\cdot h({\varvec{\chi }})}{C}\right) \mathbf {1}\{A=\beta \} \right] - \chi ^m \end{aligned}$$
the corresponding functions for the fire sales system as in [
18] adapted to the setting with heterogeneous sales functions and
\({\varvec{\chi }}_\epsilon ^*\) the corresponding value for
\({\varvec{\chi }}^*\) in [
18]. Then for
\(f^m(z,{\varvec{\chi }})\) as in Sect.
3,
$$\begin{aligned}&f_\epsilon ^m({\varvec{\chi }})+\chi ^m\\&\quad = \sum _{\beta \in [T]}\sum _{j\in J}\sum _{k\ge 0}\tilde{x}_j^m\rho ^m_{\beta }\left( \frac{\tilde{\ell }_j+k+\tilde{{\varvec{x}}}_j\cdot h({\varvec{\chi }})}{\tilde{c}_j}\right) \mathbb {P}({\varvec{X}}=\tilde{{\varvec{x}}}_j,C=\tilde{c}_j,L'=\tilde{\ell }_j+k,A=\beta )\\&\quad \le \sum _{\beta \in [T]}\sum _{j\in J}\sum _{k\ge 0}\tilde{x}_j^m\rho ^m_{\beta }\left( \frac{\tilde{\ell }_j+k+\tilde{{\varvec{x}}}_j\cdot h({\varvec{\chi }})}{\tilde{c}_j}\right) (1+\epsilon ) p_j^\beta \\&\qquad \mathbb {P}\left( \sum _{s\in [R]}s\mathrm {Poi}\left( \sum _{\gamma \in [T]}\tilde{w}_j^{-,s,\gamma }(z^*)^{s,\beta ,\gamma }\right) = k\right) \\&\quad = (1+\epsilon )(f^m({\varvec{z}}^*,{\varvec{\chi }})+\chi ^m). \end{aligned}$$
In particular, we can choose
\(\epsilon >0\) small enough such that
$$\begin{aligned} f_\epsilon ^m(\tilde{{\varvec{\chi }}}(\delta /2)) \le f^m({\varvec{z}}^*,\tilde{{\varvec{\chi }}}(\delta /2))+\epsilon \big (f^m({\varvec{z}}^*,\tilde{{\varvec{\chi }}}(\delta /2))+\tilde{\chi }^m(\delta /2)\big ) \le -\delta /2 + \epsilon {\mathbb {E}}[X^m] < 0, \end{aligned}$$
where in the last inequality it was used that
\(f^m({\varvec{z}}^*, \tilde{{\varvec{\chi }}}(\delta /2))\le f^m(\tilde{{\varvec{z}}}(\delta /2), \tilde{{\varvec{\chi }}}(\delta /2))=-\delta /2\). We can hence conclude that
\({\varvec{\chi }}_\epsilon ^* \le \tilde{{\varvec{\chi }}}(\delta /2)\) componentwise. By (a slightly tedious but simple generalization to type dependent sales functions of) [
18, Theorem EC. 1] we thus derive that the number of finally sold shares of asset
m in the fire sales system
\(({\varvec{X}},S,C,L')\) is upper bounded by
\(n((\chi _\epsilon ^*)^m+o(1))\le n(\tilde{\chi }^m(\delta /2)+o(1)) \le n\tilde{\chi }^m(\delta )\), where the last inequality holds for
n large enough since
\(\tilde{\chi }^m(\delta )>0\).
The idea for the rest of this proof is now to apply this upper bound on the number of finally sold shares inductively in each step of the following adjusted contagion process: Let
\(\overline{{\varvec{\chi }}}_0={\varvec{0}}\in {\mathbb {R}}^M\) and for
\(1\le k\) choose
\(\overline{\mathcal {D}}_k\subseteq [n]\) the smallest set such that
$$\begin{aligned} \overline{\mathcal {D}}_k = \left\{ i\in [n]:\,\sum _{j\in \overline{\mathcal {D}}_k}e_{j,i}\ge c_i-\ell _i-{\varvec{x}}_i\cdot h(\overline{{\varvec{\chi }}}_{k-1}) \right\} \end{aligned}$$
respectively
\(\overline{{\varvec{\chi }}}_k\in {\mathbb {R}}_{+,0}^M\) the smallest vector such that
$$\begin{aligned} \overline{{\varvec{\chi }}}_k = n^{-1}\sum _{i\in [n]}{\varvec{x}}_i \odot \rho _{\alpha _i}\left( \frac{\sum _{j\in \overline{\mathcal {D}}_k}e_{j,i}+\ell _i+{\varvec{x}}_i\cdot h(\overline{{\varvec{\chi }}}_k)}{c_i}\right) . \end{aligned}$$
Clearly, this process stabilizes after at most
n steps at
\((\overline{\mathcal {D}}_n,\overline{{\varvec{\chi }}}_n)\), the smallest solution to (
2.3) and (
2.4) and by Lemma
2.1 gives thus an upper bound to the actual contagion process described by (
2.1) and (
2.2), which is sufficient for our considerations here.
In particular, since
\({\overline{{\varvec{\chi }}}_0}={\varvec{0}}\le \tilde{{\varvec{\chi }}}(\delta )\), we derive that
\({\overline{\mathcal {D}}_1}\subseteq \mathcal {D}^\delta \), where
\(\mathcal {D}^\delta \subseteq [n]\) is the smallest set such that
$$\begin{aligned} \mathcal {D}^\delta = \left\{ i\in [n]:\,\sum _{j\in \mathcal {D}^\delta }e_{j,i}\ge c_i-\ell _i-{\varvec{x}}_i\cdot h(\tilde{{\varvec{\chi }}}(\delta )) \right\} \end{aligned}$$
(8.4)
and hence
\({\overline{{\varvec{\chi }}}_1}\le {\varvec{\chi }}^\delta \), where
\({\varvec{\chi }}^\delta \) denotes the smallest vector such that
$$\begin{aligned} {\varvec{\chi }}^\delta = n^{-1}\sum _{i\in [n]}{\varvec{x}}_i \odot \rho _{\alpha _i}\left( \frac{\sum _{j\in \mathcal {D}^\delta }e_{j,i}+\ell _i+{\varvec{x}}_i\cdot h({\varvec{\chi }}^\delta )}{c_i}\right) . \end{aligned}$$
(8.5)
However, (
8.4) is exactly the cascade of default contagion with initial capitals given by
\(c_i-\ell _i-{\varvec{x}}_i\cdot h(\tilde{{\varvec{\chi }}}(\delta ))\),
\(i\in [n]\), which we considered before and (
8.5) the subsequent cascade of fire sales for which we showed that the vector of finally sold shares is upper bounded by
\(n\tilde{{\varvec{\chi }}}(\delta )\) w.h.p. (i.e.
\({\varvec{\chi }}^\delta \le \tilde{{\varvec{\chi }}}(\delta )\)). We can then consider the second iteration and derive that w.h.p.
\({\overline{\mathcal {D}}_2}\subseteq \mathcal {D}^\delta \) and
\({\overline{{\varvec{\chi }}}_2}\le {\varvec{\chi }}^\delta \). Inductively this shows that w.h.p.
\({\overline{\mathcal {D}}_k}\subseteq \mathcal {D}^\delta \) and
\({\overline{{\varvec{\chi }}}_k}\le {\varvec{\chi }}^\delta {\le \tilde{{\varvec{\chi }}}(\delta )}\) w.h.p. for each fixed
\(k\in \mathbb {N}\) (independent of
n).
Now note that because of the finitariness of the system, the contagion process actually stabilizes already after a bounded number of iterations \(K\in \mathbb {N}\) (independent of n). We have thus shown that also for the final vector of sold shares \({\varvec{\chi }}_n\) it holds \({\varvec{\chi }}_n\le {\overline{{\varvec{\chi }}}_n=\overline{{\varvec{\chi }}}_K\le }\tilde{{\varvec{\chi }}}(\delta )\) w.h.p. Letting \(\delta \rightarrow 0\) this proves the upper bound on \({\varvec{\chi }}_n\).
For the final systemic damage note that w.h.p.
\(\mathcal {D}_n\subseteq \mathcal {D}^\delta \) and hence
$$\begin{aligned} n^{-1}\mathcal {S}_n \le n^{-1}\mathcal {S}^\delta {:=n^{-1}\sum _{i\in \mathcal {D}^\delta }s_i}. \end{aligned}$$
But
$$\begin{aligned} n^{-1}\mathcal {S}^\delta&\le {\mathbb {E}}\left[ S\mathbb {P}\left( \sum _{s\in [R]}s\mathrm {Poi}\left( \sum _{\gamma \in [T]}W^{-,s,\gamma }(z^*)^{s,\beta ,\gamma }\right) \ge C-L-{\varvec{X}}\cdot h(\tilde{{\varvec{\chi }}}(\delta ))\right) \right] + o_p(1)\\&= g({\varvec{z}}^*,\tilde{{\varvec{\chi }}}(\delta )) + o_p(1) \end{aligned}$$
by (
8.3). Using upper semi-continuity of
g and letting
\(\delta \rightarrow 0\) this finishes the proof.
\(\square \)