Systemic Risk and Stochastic Games with Delay

Abstract

We propose a model of inter-bank lending and borrowing which takes into account clearing debt obligations. The evolution of log-monetary reserves of banks is described by coupled diffusions driven by controls with delay in their drifts. Banks are minimizing their finite-horizon objective functions which take into account a quadratic cost for lending or borrowing and a linear incentive to borrow if the reserve is low or lend if the reserve is high relative to the average capitalization of the system. As such, our problem is a finite-player linear–quadratic stochastic differential game with delay. An open-loop Nash equilibrium is obtained using a system of fully coupled forward and advanced-backward stochastic differential equations. We then describe how the delay affects liquidity and systemic risk characterized by a large number of defaults. We also derive a closed-loop Nash equilibrium using a Hamilton–Jacobi–Bellman partial differential equation approach.

Appendix: Proof of Lemma 4.1

Proof

Assuming that $(\check{X},\check{Y}, (\check{Z}^{k})_{k=1,\ldots ,N})$ is given as an input, we solve the system (29) for $\lambda =\lambda _0$ and the processes $\phi _t$, $\psi ^k_t$, $r_t$ and the random variable $\zeta$ replaced according to the prescriptions:

$$\begin{aligned} \phi _t\leftarrow & {} \phi _t +\kappa \bigl [ \check{Y}_t- \langle \widetilde{\check{Y}}_{[t]} + q\check{X}_{[t]},\theta \rangle \bigr ]\\ \psi ^{k}_t\leftarrow & {} \psi ^{k}_t + \kappa \bigl [\check{Z}^{k}_t+\sigma \left( \frac{1}{N}-\delta _{i,k}\right) \bigr ],\quad k=1,\ldots ,N\\ r_t\leftarrow & {} r_t + \kappa \bigl [\check{X}_t+\left( 1-\frac{1}{N}\right) \bigl [q\widetilde{\check{Y}}_t+ \left( q^2-\epsilon \right) \check{X}_t\bigr ]\bigr ]\\ \zeta\leftarrow & {} \zeta + \kappa \bigl [-\check{X}_T+ c\left( 1-\frac{1}{N}\right) \check{X}_T\bigr ], \end{aligned}$$

and denote the solution by $({X},{Y}, ({Z}^{k})_{k=1,\ldots ,N})$. In this way, we defined a mapping

$$\begin{aligned} \varPhi :(\check{X},\check{Y}, (\check{Z}^{k})_{k=1,\ldots ,N})\rightarrow \varPhi (\check{X},\check{Y}, (\check{Z}^{k})_{k=1,\ldots ,N})=({X},{Y}, ({Z}^{k})_{k=1,\ldots ,N}), \end{aligned}$$

and the proof consists in proving that the latter is a contraction for small enough $\kappa >0$.

Consider $(\widehat{X},\widehat{Y},(\widehat{Z}^{k})_{k=1,\ldots ,N})=({X}-{X}^\prime ,{Y}-{Y}^\prime , (Z^{k}-{Z}^{k\prime })_{k=1,\ldots ,N})$ where $(X,Y,(Z^{k})_{k=1,\ldots ,N})$ and $({X}^\prime ,Y^\prime ,({Z^{k}}^\prime )_{k=1,\ldots ,N})$ are the corresponding image using inputs $(\check{X},\check{Y},(\check{Z}^{k})_{k=1,\ldots ,N})$ and $({\check{X}^{\prime }},{\check{Y}^{\prime }},(\check{Z}^{k\prime })_{k=1,\ldots ,N})$. We obtain

$$\begin{aligned} {\hbox {d}} \widehat{X}_t= & {} \bigl [-(1-\lambda _0)\widehat{Y}_t -\lambda _0<\widetilde{\widehat{Y}}_{[t]}+q\widehat{X}_{[t]},\theta> + \kappa \bigl [\widehat{\check{Y}_t}-<\widetilde{\widehat{\check{Y}}}_{[t]}+q\widehat{\check{X}}_{[t]},\theta >\bigr ]\bigr ]{\hbox {d}}t \nonumber \\&+\sum _{k=1}^N[-(1-\lambda _0) \widehat{Z}^{k}_t +\kappa \widehat{\check{Z}}^{k}_t\bigr ]{\hbox {d}}W^k_t\nonumber \\ {\hbox {d}}\widehat{Y}_t= & {} \Bigl [-(1-\lambda _0)\widehat{X}_t+\lambda _0\bigl (1-\frac{1}{N}\bigr )\bigl [q\widetilde{\widehat{Y}}_t+ (q^2-\epsilon )\widehat{X}_t\bigr ]\nonumber \\&+\kappa \bigl [\widehat{\check{X}}_t + \bigl (1-\frac{1}{N}\bigr )\bigl [q\widetilde{\widehat{\check{Y}}}_t+ (q^2-\epsilon )\widehat{\check{X}}_t\bigr ]\bigr ]\Bigr ]{\hbox {d}}t +\sum _{k=1}^N\widehat{Z}^{k}_t{\hbox {d}}W^k_t, \end{aligned}$$

(61)

with initial condition $\widehat{X}_0=0$ and terminal conditions $\widehat{Y}_T=(1-\lambda _0)\widehat{X}_T+\lambda _0c\left( 1-\frac{1}{N}\right) \widehat{X}_T-\kappa \widehat{\check{X}}_T+\kappa c(1-\frac{1}{N})\widehat{\check{X}}_T$ and $\widehat{Y}_t=0$ for $t\in (T,T+\tau ]$ in the case of $c >0$, and $\widehat{Y}_T=0$ and $\widehat{Y}_t=0$ for $t\in (T,T+\tau ]$ in the case of $c=0$. As we stated in the text, we only give the proof in the case $c=0$ to simplify the notation. The proof of the case $c>0$ is a easy modification. Using the form of the terminal condition and It$\hat{\mathrm {o}}$’s formula, we get

$$\begin{aligned} 0= & {} \mathbb {E}[\widehat{Y} _T\widehat{X} _T]\nonumber \\= & {} \mathbb {E}\int _0^T\Bigg \{ \widehat{Y}_t \bigg [-(1-\lambda _0)\widehat{Y}_t -\lambda _0\langle \widetilde{\widehat{Y}}_{[t]}+q\widehat{X}_{[t]},\theta \rangle + \kappa \left[ \widehat{\check{Y}_t}-\langle \widetilde{\widehat{\check{Y}}}_{[t]}+q\widehat{\check{X}}_{[t]},\theta \rangle \right] \bigg ]\nonumber \\&+ \widehat{X}_t \bigg [ -(1-\lambda _0)\widehat{X}_t+\lambda _0\left( 1-\frac{1}{N}\right) \left[ q\widetilde{\widehat{Y}}_t+ (q^2-\epsilon )\widehat{X}_t\right] \nonumber \\&+\kappa \left[ \widehat{\check{X}}_t + \left( 1-\frac{1}{N}\right) \left[ q\widetilde{\widehat{\check{Y}}}_t+ (q^2-\epsilon )\widehat{\check{X}}_t\right] \right] \bigg ] -(1-\lambda _0)\sum _{k=1}^N|\widehat{Z}^{k}_t|^2+\kappa \sum _{k=1}^N\widehat{Z}^{k}_t\widehat{\check{Z}}^{k}_t \Bigg \}{\hbox {d}}t \end{aligned}$$

(62)

$$\begin{aligned}= & {} -(1-\lambda _0)\mathbb {E}\int _0^T|\widehat{Y}_t|^2{\hbox {d}}t -\lambda _0\mathbb {E}\int _0^T\widehat{Y}_t\langle \widetilde{\widehat{Y}}_{[t]}+q\widehat{X}_{[t]},\theta \rangle {\hbox {d}}t\nonumber \\&+ \kappa \mathbb {E}\int _0^T\widehat{Y}_t \left[ \widehat{\check{Y}_t}-\langle \widetilde{\widehat{\check{Y}}}_{[t]}+q\widehat{\check{X}}_{[t]},\theta \rangle \right] {\hbox {d}}t\nonumber \\&-(1-\lambda _0)\mathbb {E}\int _0^T|\widehat{X}_t|^2{\hbox {d}}t +\lambda _0\left( 1-\frac{1}{N}\right) \mathbb {E}\int _0^T\widehat{X}_t\left[ q\widetilde{\widehat{Y}}_t+ (q^2-\epsilon )\widehat{X}_t\right] {\hbox {d}}t\nonumber \\&+\kappa \mathbb {E}\int _0^T\widehat{X}_t\left[ \widehat{\check{X}}_t + \left( 1-\frac{1}{N}\right) \left[ q\widetilde{\widehat{\check{Y}}}_t+ (q^2-\epsilon )\widehat{\check{X}}_t\right] \right] {\hbox {d}}t\nonumber \\&-(1-\lambda _0)\mathbb {E}\int _0^T \sum _{k=1}^N|\widehat{Z}^{k}_t|^2{\hbox {d}}t+\kappa \sum _{k=1}^N\widehat{Z}^{k}_t\widehat{\check{Z}}^{k}_t {\hbox {d}}t \end{aligned}$$

(63)

and rearranging the terms we find: and rearranging the terms we find:

$$\begin{aligned}&(1-\lambda _0)\bigl [\mathbb {E}\int _0^T|\widehat{X}_t|^2{\hbox {d}}t + \mathbb {E}\int _0^T|\widehat{Y}_t|^2 {\hbox {d}}t + \mathbb {E}\int _0^T\sum _{k=1}^N|\widehat{Z}^{k}_t|^2\;{\hbox {d}}t\bigr ]\\&\quad =\kappa \mathbb {E}\int _0^T \widehat{X}_t\widehat{\check{X}}_t {\hbox {d}}t -\lambda _0\mathbb {E}\int _0^T\widehat{Y}_t\langle \widetilde{\widehat{Y}}_{[t]}+q\widehat{X}_{[t]},\theta \rangle {\hbox {d}}t \\&\qquad + \kappa \mathbb {E}\int _0^T\widehat{Y}_t \bigl [\widehat{\check{Y}_t}-\langle \widetilde{\widehat{\check{Y}}}_{[t]}+q\widehat{\check{X}}_{[t]},\theta \rangle \bigr ]{\hbox {d}}t +\lambda _0\bigl (1-\frac{1}{N}\bigr )\mathbb {E}\int _0^T\widehat{X}_t\bigl [q\widetilde{\widehat{Y}}_t+ (q^2-\epsilon )\widehat{X}_t\bigr ]{\hbox {d}}t \\&\qquad +\kappa \bigl (1-\frac{1}{N}\bigr )\mathbb {E}\int _0^T\widehat{X}_t\bigl [q\widetilde{\widehat{\check{Y}}}_t+ (q^2-\epsilon )\widehat{\check{X}}_t\bigr ] \bigr ]{\hbox {d}}t+\kappa \mathbb {E}\int _0^T\sum _{k=1}^N\widehat{Z}^{k}_t\widehat{\check{Z}}^{k}_t \; {\hbox {d}}t \end{aligned}$$

Letting $\mu = \epsilon (1-\frac{1}{N})-q^2(1-\frac{1}{2N})^2>0$, we obtain:

$$\begin{aligned}&(1-\lambda _0+\lambda _0\mu )\mathbb {E}\int _0^T|\widehat{X}_t|^2{\hbox {d}}t +(1-\lambda _0)\mathbb {E}\int _0^T|\widehat{Y}_t|^2{\hbox {d}}t+(1-\lambda _0)\mathbb {E}\int _0^T\sum _{k=1}^N|\widehat{Z}^{k}_t|^2{\hbox {d}}t\\&\quad \le \kappa \mathbb {E}\int _0^T\widehat{Y}_t \bigl [\widehat{\check{Y}_t}-\langle \widetilde{\widehat{\check{Y}}}_{[t]}+q\widehat{\check{X}}_{[t]},\theta \rangle \bigr ]{\hbox {d}}t\\&\qquad +\kappa \bigl (1-\frac{1}{N}\bigr )\mathbb {E}\int _0^T\bigg (\left( q^2-\epsilon \right) \widehat{\check{X}}_t+q\widetilde{\widehat{\check{Y}}}_t\bigr )\widehat{X}_t{\hbox {d}}t +\kappa \mathbb {E}\int _0^T\sum _{k=1}^N\widehat{Z}^{k}_t\widehat{\check{Z}}^{k}_t{\hbox {d}}t, \end{aligned}$$

and a straightforward computation using repeatedly Cauchy–Schwarz and Jensen’s inequalities leads to the existence of a positive constant $K_1$ such that

$$\begin{aligned}&(1-\lambda _0+\lambda _0\mu )\mathbb {E}\int _0^T|\widehat{X}_t|^2{\hbox {d}}t +(1-\lambda _0)\mathbb {E}\int _0^T|\widehat{Y}_t|^2{\hbox {d}}t+(1-\lambda _0)\mathbb {E}\int _0^T\sum _{k=1}^N|\widehat{Z}^{k}_t|^2{\hbox {d}}t\\&\quad \le \kappa K_1\bigg \{ \mathbb {E}\int _0^T|\widehat{X}_t|^2{\hbox {d}}t+\mathbb {E}\int _0^T|\widehat{Y}_t|^2{\hbox {d}}t+\mathbb {E}\int _0^T\sum _{k=1}^N|\widehat{Z}^{k}_t|^2{\hbox {d}}t\\&\qquad + \mathbb {E}\int _0^T|\widehat{\check{X}}_t|^2{\hbox {d}}t+\mathbb {E}\int _0^T|\widehat{\check{Y}}_{t}|^2{\hbox {d}}t+\mathbb {E}\int _0^T\sum _{k=1}^N|\widehat{\check{Z}}^{k}_t|^2{\hbox {d}}t\bigg \}. \end{aligned}$$

Referring to [27], applying It$\hat{\mathrm {o}}$’s formula to $|\widehat{X}_t|^2$ and $|\widehat{Y}_t|^2$, Gronwall’s inequality, and again Cauchy–Schwarz and Jensen’s inequalities, owing to $0\le \lambda _0\le 1$, we obtain a constant $K_2>0$ independent of $\lambda _0$ so that

$$\begin{aligned} \sup _{0\le t\le T}\mathbb {E}|\widehat{X}_t|^2\le & {} \kappa K_2 \left\{ \mathbb {E}\int _0^T|\widehat{\check{X}}_t|^2+|\widehat{\check{Y}}_t|^2+\sum _{k=1}^N|\widehat{\check{Z}}^{k}_t|^2{\hbox {d}}t\right\} \nonumber \\&+K_2\left\{ \mathbb {E}\int _0^T|\widehat{Y}_t|^2+\sum _{k=1}^N|\widehat{Z}^{k}_t|^2{\hbox {d}}t\right\} ,\nonumber \\ \mathbb {E}\int _0^T|\widehat{X}_t|^2{\hbox {d}}t\le & {} \kappa K_2T\left\{ \mathbb {E}\int _0^T|\widehat{\check{X}}_t|^2+|\widehat{\check{Y}}_t|^2+\sum _{k=1}^N|\widehat{\check{Z}}^{k}_t|^2{\hbox {d}}t\right\} \nonumber \\&+K_2T\left\{ \mathbb {E}\int _0^T|\widehat{Y}_t|^2+\sum _{k=1}^N|\widehat{Z}^{k}_t|^2{\hbox {d}}t\right\} ,\nonumber \\ \mathbb {E}\int _0^T|\widehat{Y}_t|^2+\sum _{k=1}^N|\widehat{Z}^{k}_t|^2{\hbox {d}}t\le & {} \kappa K_2\left\{ \mathbb {E}\int _0^T|\widehat{\check{X}}_t|^2+|\widehat{\check{Y}}_t|^2+\sum _{k=1}^N|\widehat{\check{Z}}^{k}_t|^2{\hbox {d}}t\right\} \nonumber \\&+ K_2\mathbb {E}\int _0^T|\widehat{{X}}_t|^2{\hbox {d}}t . \end{aligned}$$

(64)

By using (64), there exists $0<\mu '<\mu /{K_2}$ such that

$$\begin{aligned}&\lambda _0\mu 'K_2\mathbb {E}\int _0^T|\widehat{{X}}_t|^2{\hbox {d}}t \nonumber \\&\quad \ge \lambda _0\mu '\left( \mathbb {E}\int _0^T|\widehat{Y}_t|^2+\sum _{k=1}^N|\widehat{Z}^{k}_t|^2{\hbox {d}}t\right) \nonumber \\&\qquad -\lambda _0\mu ' \kappa K_2\left\{ \mathbb {E}\int _0^T|\widehat{\check{X}}_t|^2+|\widehat{\check{Y}}_t|^2+\sum _{k=1}^N|\widehat{\check{Z}}^{k}_t|^2{\hbox {d}}t\right\} \nonumber \\&\quad \ge \lambda _0\mu '\left( \mathbb {E}\int _0^T|\widehat{Y}_t|^2+\sum _{k=1}^N|\widehat{Z}^{k}_t|^2{\hbox {d}}t\right) \nonumber \\&\qquad -\mu ' \kappa K_2\left\{ \mathbb {E}\int _0^T|\widehat{\check{X}}_t|^2+|\widehat{\check{Y}}_t|^2+\sum _{k=1}^N|\widehat{\check{Z}}^{k}_t|^2{\hbox {d}}t\right\} \end{aligned}$$

(65)

Therefore, we have

$$\begin{aligned}&\bigg (1-\lambda _0+\lambda _0(\mu -K_2\mu ')\bigg )\mathbb {E}\int _0^T|\widehat{X}_t|^2{\hbox {d}}t\nonumber \\&\qquad +(1-\lambda _0+\lambda _0\mu ')\mathbb {E}\int _0^T|\widehat{Y}_t|^2{\hbox {d}}t+(1-\lambda _0+\lambda _0\mu ')\mathbb {E}\int _0^T\sum _{k=1}^N|\widehat{Z}^{k}_t|^2{\hbox {d}}t \nonumber \\&\quad \le \kappa K_1\bigg \{ \mathbb {E}\int _0^T|\widehat{X}_t|^2{\hbox {d}}t+\mathbb {E}\int _0^T|\widehat{Y}_t|^2{\hbox {d}}t+\mathbb {E}\int _0^T\sum _{k=1}^N|\widehat{Z}^{k}_t|^2{\hbox {d}}t\nonumber \\&\qquad +\mathbb {E}\int _0^T|\widehat{\check{X}}_t|^2{\hbox {d}}t+\mathbb {E}\int _0^T|\widehat{\check{Y}}_{t}|^2{\hbox {d}}t+\mathbb {E}\int _0^T\sum _{k=1}^N|\widehat{\check{Z}}^{k}_t|^2\bigg \} \nonumber \\&\qquad +\kappa K_2\mu '\left\{ \mathbb {E}\int _0^T|\widehat{\check{X}}_t|^2{\hbox {d}}t+\mathbb {E}\int _0^T|\widehat{\check{Y}}_t|^2{\hbox {d}}t+\mathbb {E}\int _0^T\sum _{k=1}^N|\widehat{\check{Z}}^{k}_t|^2{\hbox {d}}t\right\} . \end{aligned}$$

(66)

Note that since $\mu -K_2\mu '$ and $\mu '$ stay in positive, we have $(1-\lambda _0+\lambda _0(\mu -K_2\mu '))\ge \mu ''$ and $(1-\lambda _0+\lambda _0\mu ')\ge \mu ''$ where for some $\mu ''>0$. Combining the inequalities (64–66), we obtain

$$\begin{aligned}&\mathbb {E}\int _0^T|\widehat{X}_t|^2{\hbox {d}}t+\mathbb {E}\int _0^T|\widehat{Y}_t|^2{\hbox {d}}t+\mathbb {E}\int _0^T\sum _{k=1}^N|\widehat{Z}^{k}_t|^2{\hbox {d}}t\nonumber \\&\quad \le \kappa K \left( \mathbb {E}\int _0^T|\widehat{\check{X}}_t|^2{\hbox {d}}t+\mathbb {E}\int _0^T|\widehat{\check{Y}}_{t}|^2{\hbox {d}}t+\mathbb {E}\int _0^T\sum _{k=1}^N|\widehat{\check{Z}}^{k}_t|^2{\hbox {d}}t\right) , \end{aligned}$$

(67)

where the constant K depends upon $\mu '$, $\mu ''$, $K_1$, $K_2$, and T. Hence, $\varPhi$ is a strict contraction for sufficiently small $\kappa$. $\square$