Deterministic and Stochastic Modeling of Deposit–Loan Dynamics with Optimal Regulatory Control

Abstract

Banks must balance deposit stability, loan expansion, and regulatory compliance while operating under liquidity constraints and financial risks. This study presents a mathematical model to examine the dynamics of bank deposits and loans under the influence of liquidity mechanisms and regulatory policies. The model proceeds in three stages: a deterministic nonlinear model, a dynamic optimal control model, and a stochastic model. Under the deterministic model, deposit withdrawals are liquidity-dependent, leading to a feedback mechanism in which liquidity improves deposit stability while financing loan growth. The theoretical results demonstrate the model’s positive and bounded solutions and show the existence and local stability of equilibria. Several parameters are based on regulatory policies or calibrated from Indonesian banking data, while the unknown parameters are estimated using the particle swarm optimization (PSO) algorithm. The results show that the proposed model is capable of fitting and predicting the data and has slightly lower mean absolute percentage errors for in-sample and out-of-sample compared with the benchmark model, and achieves comparable directional forecasting performance based on the index of directionality. Sensitivity analysis shows that the capital adequacy ratio supports lending, whereas an increased reserve requirement limits lending. An optimal control approach is developed by considering the reserve and capital requirements as time-varying policy variables. By applying Pontryagin’s maximum principle, we establish the necessary conditions for optimality. Numerical experiments demonstrate that the optimal control regulation enhances financial ratios, particularly the loan-to-deposit and liquidity ratios, at a reasonable cost. Finally, the stochastic model accounts for random variations in withdrawals and credit risks. Simulation-based observations reveal that although the system becomes more volatile, the mean dynamics are close to the deterministic case. Our framework offers a data-based and analytically tractable approach for studying the dynamics of banking variables and the effects of regulatory policies. The proposed model provides a mathematical tool for assessing the long-term effects of regulatory policies on banking performance and can assist bank managers and regulators in designing strategies that balance lending activity and liquidity resilience.