Dynamic systemic risk measures for bounded discrete time processes

Abstract

The question of measuring and managing systemic risk—especially in view of the recent financial crisis—became more and more important. We study systemic risk by taking the perspective of a financial regulator and considering the axiomatic approach originally introduced in Chen et al. (Manag Sci 59(6):1373–1388, 2013) and extended in Kromer et al. (Math Methods Oper Res 84:323–357, 2016). The aim of this paper is to generalize the static approach in Kromer et al. (2016) and analyze systemic risk measures in a dynamic setting. We work in the framework of Cheridito et al. (Electron J Probab 11:57–106, 2006) who consider risk measures for bounded discrete-time processes. Apart from the possibility to consider the “evolution of financial values”, another important advantage of the dynamic approach is the possibility to incorporate information in the risk measurement and management process. In context of this dynamic setting we also discuss the arising question of time-consistency for our dynamic systemic risk measures.

A Appendix

In this section we provide technical results that are necessary for our proofs, but are not directly related to risk measures.

In the following lemma we generalize the results from Lemma A.65 in Föllmer and Schied (2011) for multidimensional spaces.

Lemma A.1

Define the set ${\mathscr {B}}_{\infty ,m}^{r}:=\{{\bar{X}}\in (L_{{\mathscr {H}}}^{\infty })^{m}\vert \bigl \Vert {\bar{X}}\bigr \Vert _{(L_{{\mathscr {H}}}^{\infty })^{m}}\le r\}$ for $r>0$ .

• 1. For every $r>0$ the set ${\mathscr {B}}_{\infty ,m}^{r}$ is closed in $(L_{{\mathscr {H}}}^{1})^{m}$.

• 2. A convex subset ${\mathscr {C}}$ of $(L_{{\mathscr {H}}}^{\infty })^{m}$ is $\sigma ((L_{{\mathscr {H}}}^{\infty })^{m},(L_{{\mathscr {H}}}^{1})^{m})$-closed if for every $r>0$ the set ${\mathscr {C}}_{r}:={\mathscr {C}}\cap {\mathscr {B}}_{\infty ,m}^{r}$ is closed in $(L_{{\mathscr {H}}}^{1})^{m}$.

Proof

1. For every sequence $({\bar{Y}}^{\left( k\right) })\subset {\mathscr {B}}_{\infty ,m}^{r}$ with ${\bar{Y}}^{\left( k\right) }\rightarrow {\bar{Y}}$ in $(L_{{\mathscr {H}}}^{1})^{m}$ we can find a subsequence ${\bar{Y}}^{\left( k_{l}\right) }$ with $({\bar{Y}}^{\left( k_{l}\right) })^{i}\rightarrow ({\bar{Y}})^{i}$$\eta$-a.s. and in $L_{{\mathscr {H}}}^{1}$ for each $i\in \{1,\ldots ,m\}$. Since

$$\begin{aligned} \vert {\bar{Y}}^{i}\vert \le \vert ({\bar{Y}}^{\left( k_{l}\right) })^{i}\vert +\vert ({\bar{Y}}^{\left( k_{l}\right) })^{i}-{\bar{Y}}^{i}\vert \le \bigl \Vert ({\bar{Y}}^{\left( k_{l}\right) })^{i}\bigr \Vert _{L_{{\mathscr {H}}}^{\infty }}+\vert ({\bar{Y}}^{\left( k_{l}\right) })^{i}-{\bar{Y}}^{i}\vert , \end{aligned}$$

we obtain $\max _{i\in \{1,\ldots ,m\}} |{\bar{Y}}^i| \le r + \max _{i\in \{1,\ldots ,m\}} |{\bar{Y}}^{\left( k_{l}\right) })^{i} - {\bar{Y}}^i|$$\eta$-a.s. Since the convergence $({\bar{Y}}^{\left( k_{l}\right) })^{i})\rightarrow {\bar{Y}}^i$ holds $\eta$-a.s., it follows $\max _{i\in \{1,\ldots ,m\}} |{\bar{Y}}^i| \le r$$\eta$-a.s. This implies $|{\bar{Y}}^i| \le r$$\eta$-a.s. for any $i\in \{1,\ldots ,m\}$ and we get $\Vert {\bar{Y}}^i\Vert _{L_{{\mathscr {H}}}^{\infty }}\le r$$\eta$-a.s. for any $i\in \{1,\ldots ,m\}$. Finally this leads to

$$\begin{aligned} \max _{i\in \{1,\ldots ,m\}}\Vert {\bar{Y}}^i\Vert _{L_{{\mathscr {H}}}^{\infty }}\le r \end{aligned}$$

which implies ${\bar{Y}}\in {\mathscr {B}}^r_{\infty ,m}$.

2. See Lemma A.65 in Föllmer and Schied (2011). $\square$

Because of the following lemma, we can define the topology $\sigma ({\mathscr {R}}_{\tau ,\theta }^{\infty ,m},{\mathscr {A}}_{\tau ,\theta }^{1,m})$ on ${\mathscr {R}}_{\tau ,\theta }^{\infty ,m}$ and the topology $\sigma ({\mathscr {A}}_{\tau ,\theta }^{1,m},{\mathscr {R}}_{\tau ,\theta }^{\infty ,m})$ on ${\mathscr {A}}_{\tau ,\theta }^{1,m}$ analogously to $\sigma ({\mathscr {R}}^{\infty ,m},{\mathscr {A}}^{1,m})$ on ${\mathscr {R}}^{\infty ,m}$ and $\sigma ({\mathscr {A}}^{1,m},{\mathscr {R}}^{\infty ,m})$ on ${\mathscr {A}}^{1,m}$ .

Lemma A.2

${\mathscr {A}}_{\tau ,\theta }^{1,m}$ and ${\mathscr {R}}_{\tau ,\theta }^{\infty ,m}$ satisfy the following properties:

• 1. ${\mathscr {R}}_{\tau ,\theta }^{\infty ,m}$ separates points of ${\mathscr {A}}_{\tau ,\theta }^{1,m}$ under $\left\langle \cdot ,\cdot \right\rangle _{m}\vert _{{\mathscr {R}}_{\tau ,\theta }^{\infty ,m}\times {\mathscr {A}}_{\tau ,\theta }^{1,m}}$, i.e. if ${\bar{\xi }}\in {\mathscr {A}}_{\tau ,\theta }^{1,m}$ and $\bigl \langle {\bar{X}},{\bar{\xi }}\bigr \rangle _{m}=0$ for all ${\bar{X}}\in {\mathscr {R}}_{\tau ,\theta }^{\infty ,m}$, then ${\bar{\xi }}=0$.

• 2. ${\mathscr {A}}_{\tau ,\theta }^{1,m}$ separates points of ${\mathscr {R}}_{\tau ,\theta }^{\infty ,m}$ under $\left\langle \cdot ,\cdot \right\rangle _{m}\vert _{{\mathscr {R}}_{\tau ,\theta }^{\infty ,m}\times {\mathscr {A}}_{\tau ,\theta }^{1,m}}$, i.e. if ${\bar{X}}\in {\mathscr {R}}_{\tau ,\theta }^{\infty ,m}$ and $\bigl \langle {\bar{X}},{\bar{\xi }}\bigr \rangle _{m}=0$ for all ${\bar{\xi }}\in {\mathscr {A}}_{\tau ,\theta }^{1,m}$, then ${\bar{X}}=0$.

Proof

Note that ${\mathscr {R}}^{\infty ,m}$ separates points of ${\mathscr {A}}^{1,m}$ and ${\mathscr {A}}^{1,m}$ separates points of ${\mathscr {R}}^{\infty ,m}$ under $\left\langle \cdot ,\cdot \right\rangle _{m}$. Fix ${\bar{\xi }}\in {\mathscr {A}}_{\tau ,\theta }^{1,m}$ and suppose that $\bigl \langle {\bar{X}},{\bar{\xi }}\bigr \rangle _{m}=0$ for all ${\bar{X}}\in {\mathscr {R}}_{\tau ,\theta }^{\infty ,m}$. For each ${\bar{Y}}\in {\mathscr {R}}^{\infty }$, we obtain $\bigl \langle {\bar{Y}},{\bar{\xi }}\bigr \rangle _{m}=\bigl \langle {\bar{Z}},{\bar{\xi }}\bigr \rangle _{m}$ for ${\bar{Z}}:={\bar{Y}}I_{\left[ \tau ,\theta \right] }+{\bar{Y}}_{\theta }I_{\left( \theta ,\infty \right) }\in {\mathscr {R}}_{\tau ,\theta }^{\infty ,m}$. This implies that $\bigl \langle {\bar{Y}},{\bar{\xi }}\bigr \rangle _{m}=0$ for all ${\bar{Y}}\in {\mathscr {R}}^{\infty }$, and thus ${\bar{\xi }}=0$.

To show the second assertion, consider ${\bar{X}}\in {\mathscr {R}}_{\tau ,\theta }^{\infty ,m}$ and suppose that $\bigl \langle {\bar{X}},{\bar{\phi }}\bigr \rangle _{m}=0$ for all ${\bar{\phi }}\in {\mathscr {A}}_{\tau ,\theta }^{1,m}$. For every ${\bar{\xi }}\in {\mathscr {A}}^{1,m}$, we have $\bigl \langle {\bar{X}},{\bar{\xi }}\bigr \rangle _{m}=\sum _{i=1}^{m}{\mathbb {E}}[\sum _{t\in {\mathbb {N}}_{0}}{\bar{X}}_{t}^{i}\varDelta \xi _{t}]$ and there exists a ${\bar{\phi }}\in {\mathscr {A}}_{\tau ,\theta }^{1,m}$ with $\bigl \langle {\bar{X}},{\bar{\xi }}\bigr \rangle _{m}=\bigl \langle {\bar{X}},{\bar{\phi }}\bigr \rangle _{m}$. Therefore, $\bigl \langle {\bar{X}},{\bar{\psi }}\bigr \rangle _{m}=0$ for all ${\bar{\psi }}\in {\mathscr {A}}^{1,m}$, which implies that ${\bar{X}}=0$. $\square$