# A Semi-Markov Dynamic Capital Injection Problem for Distressed Banks

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## Abstract

**:**

## 1. Introduction

## 2. Government Strategies

**Liberalism**: The government does not intervene to affect the shortage of liquidity in the banking system. In this case, lack of liquid reserves for a bank implies its liquidation.**Transparency**: Banks have perfect knowledge about the strategy of the government, i.e., they know under which conditions they will be rescued.**Uncertainty**: Banks do not know exactly what is the government rescuing strategy, but they can make estimations on its savage plans.

**Liberalism:**This can be summarized by equation $\Delta {G}_{{\tau}_{1}}=0$, which implies that the government will not intervene, regardless of the particular circumstances.**Transparency:**$\Delta {G}_{{\tau}_{1}}=\left({s}_{u}-{S}_{{\tau}_{1}^{-}}^{\alpha}\right){\U0001d7d9}_{\{{S}_{{\tau}_{1}^{-}}^{\alpha}>{s}_{l}\}}$; therefore, with certainty, the government will inject capital trying to align the liquid reserve back to some fixed level ${s}_{u}>{s}_{c}\ge 0$, whenever the current reserve level is greater than ${s}_{l}<0$. In this sense, the government’s strategy is transparent to the bank and the bank will make decisions based on this perfect knowledge.**Uncertainty:**Any government strategy that deals with uncertainty about a bank’s financial status falls under this category of strategies. In this specific scenario, multiple government injection strategies could be defined. For example, a straightforward strategy could be$$\Delta {G}_{{\tau}_{1}}=\left({s}_{u}-{S}_{{\tau}_{1}^{-}}^{\alpha}\right){\U0001d7d9}_{\{{S}_{{\tau}_{1}^{-}}^{\alpha}>{s}_{l}\}}{\U0001d7d9}_{\{x=1\}}\phantom{\rule{0.277778em}{0ex}},$$A more interesting and realistic setting can be defined by$$\Delta {G}_{{\tau}_{1}}=\left(R-{S}_{{\tau}_{1}^{-}}^{\alpha}\right){\U0001d7d9}_{\{{S}_{{\tau}_{1}^{-}}^{\alpha}>{s}_{l}\}},$$

#### 2.1. No Government Intervention: The Liberalism Framework

**Case 1.**$a>{v}_{0}^{*}\left({c}_{D}\right)$. It is optimal to immediately deplete the liquid reserve, meaning liquidating the auxiliary bank immediately, see Lemma A.1 in (Hugonnier and Morellec 2017, eq. 21).

**Case 2.**$a\le {v}_{0}^{*}\left({c}_{D}\right)$. Recall that ${E}_{t}^{\alpha}$ is the cumulative earning at time t. Using the same notation in Hugonnier and Morellec (2017), we first define the auxiliary function

**Remark**

**1.**

#### 2.2. One-Time Injection with Uncertainty

**Definition**

**1.**

- If ${S}_{t}^{\alpha}\in (-\infty ,{s}_{b}]$, the government will not save the bank and the bank has to declare bankruptcy.
- If ${S}_{t}^{\alpha}\in ({s}_{b},{s}_{c}]$, the bank is considered as undergoing critical financial trouble and the government will save the bank’s reserve to level R with probability $P({S}_{{t}^{-}}^{\alpha},L,D)$ depending on the current reserve level ${S}_{{t}^{-}}^{\alpha}$, total value of deposits D, and liability L. Notice that the deposit rate ${c}_{D}\left(t\right)$ only changes once, according to (6).
- If ${S}_{t}^{\alpha}\in ({s}_{c},+\infty )$, the bank is considered to be safe and the government will not intervene.

**Remark**

**2.**

**Remark**

**3.**

#### 2.2.1. Viscosity Approach

#### Particular Case: Certain Government Intervention

#### Extension to the General Uncertainty Framework

**Remark**

**4.**

**Proposition**

**1.**

**Proof.**

#### 2.2.2. Barrier Strategy Approach

**Claim**

**1.**

**Proof.**

**Remark**

**5.**

## 3. Semi-Markov Multiple Injection

**Remark**

**6.**

#### Uniqueness of the Solution

- The jump part corresponding to the government intervention, that is$$\mathcal{G}v(s,h)={\partial}_{h}v(s,h)+\theta (s,h)\phantom{\rule{0.166667em}{0ex}}{\int}_{{s}_{u}-s}^{\infty}\left[v(s+\gamma ,0)-v(s,h)\right]\widehat{\nu}(s,h,\mathrm{d}\gamma )\phantom{\rule{0.277778em}{0ex}};$$
- The residual part which equals the one already considered in Section 2.1 and Section 2.2.

- $\theta (\xb7,\xb7)$ is measurable on its domain;
- There exists a constant dominating $\theta $ on its whole domain, i.e., $sup\theta (s,h)\le C,\phantom{\rule{0.166667em}{0ex}}\forall \phantom{\rule{3.33333pt}{0ex}}s,h$;
- $A\mapsto \widehat{\nu}(s,h,A)$ is measurable, $\forall \phantom{\rule{3.33333pt}{0ex}}A\subset [{s}_{u}-s,\infty ]$ and all $s,h$;
- ${\widehat{\nu}}^{s,h}\left(A\right):=\nu (s,h,A)$ is a probability measure on $[{s}_{u}-s,\infty ]$.

**Corollary**

**1.**

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Note

1 | Notice that, for some time t, the risky asset value ${A}_{t}$ could become negative. Negative asset values are never considered in the model since the optimal control problem is considered for times prior to ${T}^{\alpha}$. See later sections for a formal definition of the stopping time ${T}^{\alpha}$. |

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**Figure 1.**Let us consider a red zone $RZ=[{s}_{l}=80,{s}_{u}=98]$, ${s}_{0}=100$, $\overline{\mu}=0.1$, $\sigma =2$, $T=5$. We assume that the compound Poisson process described in Equation (1) has an intensity of $\lambda =1/3$, meaning that one jump is expected every three years. The jump size is assumed to be distributed as a gamma variable with shape parameter 2 and scale parameter 2.5, resulting in a jump size expectation of 5. In the upper figure, the dynamics of the bank’s risky asset is shown, with the green circle $\tau $ representing the negative jump and the red line representing the upper bound of the red region. The lower figure shows the dynamics of $\theta (\xb7,\xb7)$ assuming no dividends and no government intervention over the 5 years considered, and with the parameters ${C}_{1}$ and ${C}_{2}$ equal to 1/100 and 1/2, respectively. For this analysis, it was assumed that ${h}_{0}=0$. It can be observed that, after one year, the government is eager to help the bank as the bank’s value is far away from the lower boundary of the red region. However, as time goes by, the government increases the intensity of intervention as the time since the last salvage event has increased.

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**MDPI and ACS Style**

Di Persio, L.; Prezioso, L.; Jiang, Y. A Semi-Markov Dynamic Capital Injection Problem for Distressed Banks. *Risks* **2023**, *11*, 67.
https://doi.org/10.3390/risks11040067

**AMA Style**

Di Persio L, Prezioso L, Jiang Y. A Semi-Markov Dynamic Capital Injection Problem for Distressed Banks. *Risks*. 2023; 11(4):67.
https://doi.org/10.3390/risks11040067

**Chicago/Turabian Style**

Di Persio, Luca, Luca Prezioso, and Yilun Jiang. 2023. "A Semi-Markov Dynamic Capital Injection Problem for Distressed Banks" *Risks* 11, no. 4: 67.
https://doi.org/10.3390/risks11040067