Stochastic control models of optimal dividend and capital financing

Stochastic control models are considered for valuing a company whose capital evolves according to an arithmetic Brownian motion. The objective of the associated optimal control problem is to maximize the value of a company by controlling the flow of capital through paying out dividends either with or without a recapitalization option taken into account. Solutions to the optimal control problem are obtained by solving a Hamilton-Jacobi-Bellman (HJB) equation or a system of quasi-variational inequalities (QVI).; The optimal policy for the problem is of a barrier type. An interesting aspect of the control problem is that a company would ultimately ruin in finite time with probability one under a barrier control policy. Using asymptotic analysis, new results are obtained about the expected lifetime and the total dividend payout in a finite-time horizon under a barrier policy. The first behavior of the expected lifetime for a small discount factor rho is obtained by O(1/rho2). Hence the expected lifetime becomes increasingly large for rho small, growing as a second power of 1/rho. However, the expected present value of the total dividend payout converges to its asymptote when t >> 1/rho for rho small; therefore, most dividends are paid out in the time frame of 1/rho.; In the model of capital financing and dividend distribution, a company controls the flow of capital by paying out dividends as well as by issuing new capital. This model incorporates market friction factors including the delay (Delta) and costs (K) of raising capital. The current study shows that a unique solution to the set of QVI exists for all values of K > 0 and Delta ≥ 0. The solution, the so-called value function, is twice-differentiable (C2) at all nonnegative points except at an impulse control barrier where the solution is no longer C2. The main mathematical fact to be used in this study is that a solution of the heat equation has at most one change of sign for all t > 0 when its initial data has just one sign change. The sign change theorem is proved by using the maximum principles for differential equations.