# Optimal dividend policies with random profitability

## TL;DR

This paper investigates optimal dividend policies under bankruptcy risk with general cash flow models, providing theoretical proofs, numerical schemes, and characterizing strategies as barrier and band types with extensions.

## Contribution

It introduces a comprehensive model with general cash flow processes, proves the continuity of the value function, and characterizes optimal strategies including liquidation, with extensions like equity issuance.

## Key findings

- Optimal strategies are barrier and band strategies.
- The value function satisfies a nonlinear PDE with a gradient constraint.
- Extensions include models with equity issuance and credit lines.

## Abstract

We study an optimal dividend problem under a bankruptcy constraint. Firms face a trade-off between potential bankruptcy and extraction of profits. In contrast to previous works, general cash flow drifts, including Ornstein--Uhlenbeck and CIR processes, are considered. We provide rigorous proofs of continuity of the value function, whence dynamic programming, as well as comparison between the sub- and supersolutions of the Hamilton--Jacobi--Bellman equation, and we provide an efficient and convergent numerical scheme for finding the solution. The value function is given by a nonlinear PDE with a gradient constraint from below in one dimension. We find that the optimal strategy is both a barrier and a band strategy and that it includes voluntary liquidation in parts of the state space. Finally, we present and numerically study extensions of the model, including equity issuance and credit lines.