Dynamic modeling and optimal control of bank balance sheets under capital adequacy constraints

Abstract

The aim of this study is to examine the dynamics of deposits, loans, and equity on the balance sheet of the bank under capital adequacy constraints. The intention is to provide a tractable framework for assessing solvency and regulatory policies. Therefore, a nonlinear continuous-time model was developed with logistic growth, credit risk, and capital adequacy conditions. The model was calibrated using monthly data for Indonesian commercial banks from 2022 to 2024, and the parameters were estimated using particle swarm optimization. The results showed that the model replicates observed trajectories with mean absolute percentage errors below 2.1% to confirm its empirical validity. The simulations showed that stricter capital requirements slowed equity growth while moderate requirements supported long-run capitalization. A time-varying capital adequacy policy was formulated as an optimal control problem, and the Pontryagin maximum principle was applied to derive an optimal regulatory path. The results showed that adaptive regulation stabilized capitalization while limiting policy costs. The trend reflected the value of the continuous-time control theory in financial regulation.

Keywords

Bank balance sheet / Capital adequacy ratio / Nonlinear differential equations / Optimal control / Financial regulation / Particle swarm optimization

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