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This paper is concerned with a long standing optimal dividend payout problemin insurance subject to the so-called ratcheting constraint, that is, thedividend payout rate shall be non-decreasing over time. The surplus process ismodeled by a drifted Brownian motion process and the aim is to find the optimaldividend ratcheting strategy to maximize the expectation of the totaldiscounted dividend payouts until the ruin time. Due to the path-dependentconstraint, the standard control theory cannot be directly applied to tacklethe problem. The related Hamilton-Jacobi-Bellman (HJB) equation is a new typeof variational inequality. In the literature, it is only shown to have aviscosity solution, which is not strong enough to guarantee the existence of anoptimal dividend ratcheting strategy. This paper proposes a novel partialdifferential equation method to study the HJB equation. We not only prove thethe existence and uniqueness of the solution in some stronger functional space,but also prove the monotonicity, boundedness, and $C^{\infty}$-smoothness ofthe dividend ratcheting free boundary. Based on these results, we eventuallyderive an optimal dividend ratcheting strategy, and thus solve the open problemcompletely. Economically, we find that if the surplus volatility is above anexplicit threshold, then one should pay dividends at the maximum rate,regardless the surplus level. Otherwise, by contrast, the optimal dividendratcheting strategy relays on the surplus level and one should only ratchet upthe dividend payout rate when the surplus level touches the dividend ratchetingfree boundary.

Optimal ratcheting of dividend payout under Brownian motion surplus