Maximizing Banking Profit on a Random Time Interval

We study the stochastic dynamics of banking items such as assets, capital,liabilities and profit. A consideration of these items leads to the formulation ofa maximization problem that involves endogenous variables such as depositoryconsumption, the value of the bank's investment in loans, and provisions for loanlosses as control variates. A solution to the aforementioned problem enables usto maximize the expected utility of discounted depository consumption over arandom time interval, [t,τ], and profit at terminal timeτ. Here, the term depository consumption refers to theconsumption 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 eitherof 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 ourcontribution, we simulate the financial indices return on equity and return on assets that are two measures of bank profitability