Journal of Applied Mathematics
Volume 2007 (2007), Article ID 29343, 22 pages
Research Article

Maximizing Banking Profit on a Random Time Interval

J. Mukuddem-Petersen,1 M. A. Petersen,1 I. M. Schoeman,1 and B. A. Tau2

1Department of Mathematics and Applied Mathematics, Faculty of Science, North-West University (Potchefstroom Campus), Private Bag X 6001, Potchefstroom 2520, South Africa
2School of Modeling Sciences, North-West University (Vaaldriehoek Campus), Private Bag X 6001, P.O. Box 1174, Vanderbijlpark 1900, South Africa

Received 2 March 2007; Accepted 1 April 2007

Academic Editor: Ibrahim Sadek

Copyright © 2007 J. Mukuddem-Petersen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We study the stochastic dynamics of banking items such as assets, capital, liabilities and profit. A consideration of these items leads to the formulation of a maximization problem that involves endogenous variables such as depository consumption, the value of the bank's investment in loans, and provisions for loan losses as control variates. A solution to the aforementioned problem enables us to maximize the expected utility of discounted depository consumption over a random time interval, [t,τ], and profit at terminal time τ. Here, the term depository consumption refers to the consumption of the bank's profits by the taking and holding of deposits. In particular, we determine an analytic solution for the associated Hamilton-Jacobi-Bellman (HJB) equation in the case where the utility functions are either of power, logarithmic, or exponential type. Furthermore, we analyze certain aspects of the banking model and optimization against the regulatory backdrop offered by the latest banking regulation in the form of the Basel II capital accord. In keeping with the main theme of our contribution, we simulate the financial indices return on equity and return on assets that are two measures of bank profitability.