Profit Maximization with Bankruptcy and Variable Scale

Abstract

In a diffusion model of an enterprise with variable scale, sufficient conditions are given for the maximization of expected profit (expected total discounted withdrawals) to lead to eventual bankruptcy with probability one. The optimal withdrawal policy is an "overflow policy," in which the withdrawal rate is equal to zero if the asset level is below a "barrier," and equal to the maximum rate if the asset level is greater than or equal to the barrier. The optimal policy for the control of the drift (yield) and volatility (risk) of the earnings process is derived as the solution of a differential equation, and a formula is given for the corresponding value function. The optimality of the constructed policy is demonstrated using the standard "Bellman Conditions."