Optimal dividend payment under ratcheting and drawdown constraints on dividends

In this talk, we consider an uncontrolled surplus process that follows a Brownian motion with drift. Our goal is to maximize the expected sum of discounted dividends until ruin time in two cases: (1) under a ratcheting constraint and (2) under a drawdown constraint on dividends. A ratcheting constraint means that the dividend rate paid to the shareholders can never decrease more than its historical maximum and a drawdown one means that the dividend rate paid to the shareholders can never decrease more than a given fraction “a” of its historical maximum. Both problems are two-dimensional. We first consider the case in which a ceiling on the maximum rate of dividends is imposed. In each case, the optimal value function is identified as the unique viscosity solution of the corresponding Hamilton-Jacobi-Equation. In the ratcheting case, the natural candidate to be the optimal strategy is given by a one-curve strategy (that is obtained by solving an ODE) and in the drawdown case, the natural candidate to be the optimal strategy is given by a two-curve strategy (where the two curves are obtained solving a system of ODE’s). Moreover, we present a numerical study in which these curves give the optimal strategy and also some sufficient conditions for the optimality of these curves. Finally, we present the case when the ceiling on dividend rates goes to infinity in which case the optimal curves converge to surprising simple limits (depending on “a”), showing that for a sufficiently large current dividend rate and surplus a take-the-money-and-run strategy is optimal in the presence of constraints on dividends. This is a joint work with Hansjörg Albrecher (UNIL) and Pablo Azcue (UTDT).