<p>The baking system serves as the backbone of a country’s economy and plays a crucial role in its development. The banking system supports economic growth by mobilizing savings, granting loans, handling payments, supporting government policies, and maintaining financial stability. A distressed bank is one facing financial hardship due to high Non-Performing Assets, losses, mismanagement, or liquidity problems. Such banks often require regulatory assistance from a healthy, operating bank (an undistressed bank) to regain stability. Due to the interconnectedness of banks, the failure of one bank may trigger financial instability in others, a phenomenon known as systemic risk, and may lead to a banking crisis. So, it is important to have effective policies that can mitigate the spread of banking crises and thereby stabilize the country’s overall economy. This paper aims to develop an optimal strategy to eliminate systemic risk contagion in the banking sector. For this purpose, we propose a mathematical model consisting of four ordinary differential equations with two controls: measures taken by undistressed banks to provide liquidity and to guide risk management. In contrast, the other control represents financial support extended by the central bank, such as emergency credit lines, strengthening monitoring and supervision, and strict guidelines to reduce NPA. The model’s basic properties, such as non-negativity and boundedness of solutions, are established to demonstrate its financial feasibility. The basic reproduction number <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((\mathcal {R}_0)\)</EquationSource> </InlineEquation> of the model is calculated using the next generation matrix method. The local and global stability results are obtained for both risk-free and risk equilibrium points. It is found that there is no chance of systemic risk contagion if <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal {R}_0<1\)</EquationSource> </InlineEquation>. Further, a delay model has been formulated and studied to investigate the impact of time delay in declaring an exposed bank as a distressed bank using numerical simulation. A semi-relative sensitivity analysis is performed to examine the most influential parameters affecting undistressed and distressed banks. Moreover, an optimal control problem is formulated to obtain a strategy that minimizes the number of contagious banks while incurring minimal associated costs. The existence of the optimal controls is shown, and the optimality conditions are obtained analytically. Finally, the proposed model is simulated to substantiate the analytical results, and the optimal control problem is solved numerically using a forward-backwards iterative method. This study provides an optimal control approach to mitigate the banking crisis by minimizing the number of distressed banks while incurring minimal investment.</p>

Modeling Optimal Control for Systemic Risk Contagion in the Banking Sector

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发表日期：

2026年3月23日

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Bhagya Jyoti Nath,
Barlin Bhuyan,
Zenith Mondal,
Kaushik Dehingia,
Ausif Padder

摘要

The baking system serves as the backbone of a country’s economy and plays a crucial role in its development. The banking system supports economic growth by mobilizing savings, granting loans, handling payments, supporting government policies, and maintaining financial stability. A distressed bank is one facing financial hardship due to high Non-Performing Assets, losses, mismanagement, or liquidity problems. Such banks often require regulatory assistance from a healthy, operating bank (an undistressed bank) to regain stability. Due to the interconnectedness of banks, the failure of one bank may trigger financial instability in others, a phenomenon known as systemic risk, and may lead to a banking crisis. So, it is important to have effective policies that can mitigate the spread of banking crises and thereby stabilize the country’s overall economy. This paper aims to develop an optimal strategy to eliminate systemic risk contagion in the banking sector. For this purpose, we propose a mathematical model consisting of four ordinary differential equations with two controls: measures taken by undistressed banks to provide liquidity and to guide risk management. In contrast, the other control represents financial support extended by the central bank, such as emergency credit lines, strengthening monitoring and supervision, and strict guidelines to reduce NPA. The model’s basic properties, such as non-negativity and boundedness of solutions, are established to demonstrate its financial feasibility. The basic reproduction number \((\mathcal {R}_0)\) of the model is calculated using the next generation matrix method. The local and global stability results are obtained for both risk-free and risk equilibrium points. It is found that there is no chance of systemic risk contagion if \(\mathcal {R}_0<1\) . Further, a delay model has been formulated and studied to investigate the impact of time delay in declaring an exposed bank as a distressed bank using numerical simulation. A semi-relative sensitivity analysis is performed to examine the most influential parameters affecting undistressed and distressed banks. Moreover, an optimal control problem is formulated to obtain a strategy that minimizes the number of contagious banks while incurring minimal associated costs. The existence of the optimal controls is shown, and the optimality conditions are obtained analytically. Finally, the proposed model is simulated to substantiate the analytical results, and the optimal control problem is solved numerically using a forward-backwards iterative method. This study provides an optimal control approach to mitigate the banking crisis by minimizing the number of distressed banks while incurring minimal investment.