Optimal ratcheting of dividend payout under Brownian motion surplus

Abstract

This paper is concerned with a long-standing optimal dividend payout problem subject to the so-called ratcheting constraint, that is, the dividend payout rate shall be nondecreasing over time and is thus self-path-dependent. The surplus process is modeled by a drifted Brownian motion process and the aim is to find the optimal dividend ratcheting strategy to maximize the expectation of the total discounted dividend payouts until the ruin time. Due to the self-path-dependent control constraint, the standard control theory cannot be directly applied to tackle the problem. The related Hamilton-Jacobi-Bellman (HJB) equation is a new type of variational inequality. In the literature, it is only shown to have a viscosity solution, which is not strong enough to guarantee the existence of an optimal dividend ratcheting strategy. This paper proposes a novel partial differential equation method to study the HJB equation. We not only prove the existence and uniqueness of the solution in some stronger functional space, but also prove the strict monotonicity, boundedness, and C∞-smoothness of the dividend ratcheting free boundary. Based on these results, we eventually derive an optimal dividend ratcheting strategy, and thus solve the open problem completely. Economically speaking, we find that if the surplus volatility is above an explicit threshold, then one should pay dividends at the maximum rate, regardless of the surplus level. Otherwise, by contrast, the optimal dividend ratcheting strategy relies on the surplus level and one should only ratchet up the dividend payout rate when the surplus level touches the dividend ratcheting free boundary. Moreover, our numerical results suggest that one should invest in those companies with stable dividend payout strategies since their income rates should be higher and volatility rates smaller.