Dividend ratcheting and capital injection under the Cramér-Lundberg model: Strong solution and optimal strategy

Abstract: We consider an optimal dividend payout problem for an insurance company whose surplus follows the classical Cramér-Lundberg model. The dividend rate is subject to a ratcheting constraint (i.e., it must be nondecreasing over time), and the company may inject capital at a proportional cost to avoid ruin. This problem gives rise to a stochastic control problem with a self-path-dependent control constraint, costly capital injections, and jump-diffusion dynamics. The associated Hamilton-Jacobi-Bellman (HJB) equation is a partial integro-differential variational inequality featuring both a nonlocal integral term and a gradient constraint. We develop a systematic probabilistic and PDE-based approach to solve this HJB equation. By discretizing the space of admissible dividend rates, we construct a sequence of approximating regime-switching systems of ordinary integro-differential equations. Through careful a priori estimates and a limiting argument, we prove the existence and uniqueness of a \emph{strong solution} in a suitable space. This regularity result is fundamental: it allows us to characterize the optimal dividend policy via a switching free boundary and to construct an explicit optimal feedback control strategy. To the best of our knowledge, this is the first complete solution -- comprising both the value function and an implementable optimal strategy -- for a dividend ratcheting problem with capital injection under the Cramér-Lundberg model. Our work advances the mathematical theory of optimal stochastic control beyond the standard viscosity solution framework, providing a rigorous foundation for dividend policy design in economics.

• The paper introduces a novel strong solution for the HJB variational inequality addressing dividend ratcheting and capital injection constraints in insurance risk models.
• It utilizes regime-switching discretization and free boundary analysis to derive an optimal feedback strategy that balances dividend payouts and minimal capital injections.
• Results reveal that the value function exhibits concavity and regularity, ensuring a robust framework for practical insurance dividend policy optimization.

Optimal Dividend Ratcheting with Capital Injections in the Cramér-Lundberg Model

Problem Setting and Novelty

The paper "Dividend ratcheting and capital injection under the Cramér-Lundberg model: Strong solution and optimal strategy" (2604.04641) investigates a stochastic control problem motivated by insurance risk management, specifically optimal dividend payout strategies constrained by ratcheting and subject to costly capital injections. The surplus process follows the classical Cramér-Lundberg compound Poisson model. The main control problem is to maximize the expected present value of dividends minus the cost from capital injections, with the following core constraints:

• Dividend ratcheting: The dividend payout rate is non-decreasing, capturing managerial reluctance or contractual constraints against dividend cuts.
• Capital injection: The insurer can inject capital at any time to avoid ruin, but this comes at a proportional premium, reflecting transaction and agency costs.
• Path-dependence & Jump risk: The surplus is impacted by discrete jumps due to claims, and the ratcheting introduces a self-exciting, history-dependent admissible set for controls.

This formulation integrates three sources of complexity rarely treated in unison: jump-process surplus dynamics, path-dependent ratcheting constraints on dividends, and capital injections with strict costs and a solvency requirement. Prior literature either omits the ratcheting aspect, handles Brownian rather than Poisson risk, or is limited to weaker (viscosity) solution frameworks.

HJB Characterization and Main Results

The control problem leads to a novel partial integro-differential Hamilton-Jacobi-Bellman variational inequality (HJB-VI), which embodies three critical elements: a nonlocal integral operator due to jump claims, a ratcheting-driven gradient (w.r.t. dividend rate) constraint, and a reflecting (w.r.t. capital injection) gradient constraint:

$$\min\big\{ \mathcal{L}_c v - \lambda v + h(x) - c,\; \ell - v_x(x,c),\; -v_c(x,c) \big\} = 0.$$

Here, $v(x,c)$ is the value function for initial surplus $x$ and current dividend rate $c$, $\mathcal{L}_c$ is the generator for the surplus with control $c$, $\ell>1$ the per-unit cost of capital, and $h(x)$ a term induced by the jump distribution.

Existence and Uniqueness of Strong Solutions

A central technical achievement of the paper is establishing both existence and uniqueness of a strong solution to the HJB-VI in suitable function spaces. In contrast to the viscosity solution approach, the strong solution constructed possesses higher regularity, enabling constructive feedback policy design. The analysis involves:

• Decomposition of the control space via regime-switching discretization (finite ratcheting increments), reducing the VI to a sequence of coupled ordinary integro-differential boundary problems.
• Uniform a priori estimates and compactness arguments to pass to the limit and recover a strong solution in the full (continuous control) setting.
• Verification of regularity and monotonicity properties: the value function is shown to be concave in surplus, nonincreasing in the dividend rate, Lipschitz, and bounded between tight analytic barriers.

Free Boundary and Optimal Control

The ratcheting constraint naturally partitions the state space into two regions: one where the dividend rate should be increased ("switching"), and one where it should be held constant ("non-switching"). These are separated by a free boundary $X(c)$ for each $c<\overline{c}$, where $v(x,c)$0 is the maximal rate.

• In the switching region, surplus exceeding $v(x,c)$1 triggers a dividend rate increase, precisely to the maximally admissible value.
• The optimal feedback strategy is to continuously pay dividends at the current maximal rate consistent with the ratcheting trajectory (as dictated by a function $v(x,c)$2), raising it only when the running surplus maximum increases.
• Capital is injected minimally (just-in-time barrier strategy) to reflect surplus at zero and prevent bankruptcy, but only as a last resort due to the assumption $v(x,c)$3.

The free boundary $v(x,c)$4 is shown to be continuous and finite, and its construction is rigorously justified through regularity and monotonicity properties deduced from the strong solution.

Implications and Analytical Structure

Practical Relevance

The mathematical rigor and explicit feedback characterization provide operational tools for insurance firms designing robust dividend distribution policies under realistic regulatory and behavioral constraints. Ratcheting reflects the practical reality that dividend cuts are typically penalized by capital markets, and the ability to inject costly external capital links the analysis with solvency regulatory frameworks (such as Solvency II/IFRS 17).

Theoretical Significance

Establishing the strong solution for the HJB-VI with both nonlocal terms and coupled gradient constraints pushes the boundary of stochastic control theory. The PDE regularity enables not only existence proofs but also verification of optimality, free boundary identification, and explicit policy construction, going well beyond what is achievable with weak solution frameworks.

Structural and Analytical Insights

The work demonstrates that the value function's sensitivity to the dividend rate is both bounded and regular, ensuring stability and well-posedness of the optimal control. The regime-switching approximation and passage to the limit is an effective analytic blueprint for other path-dependent or hybrid control problems.

Directions for Future Research

• Extension to spectrally negative Lévy risk models or models with stochastic interest rates.
• Incorporation of regulatory capital requirements or risk-based solvency constraints.
• Study of numerical schemes for the HJB-VI exploiting strong solution regularity.
• Sensitivity analysis with respect to jump distribution tails and the capital injection cost $v(x,c)$5.
• Investigation into alternative path-dependent constraints, such as moving-average or drawdown-limited dividends.

Conclusion

This paper presents a comprehensive mathematical treatment of the optimal dividend ratcheting problem with costly capital injection, under the classical Cramér-Lundberg model. The establishment of a unique strong solution to the associated HJB variational inequality, together with an explicit and implementable feedback control, provides a rigorous analytic foundation for practical corporate finance strategies under realistic operational constraints. The methodology and technical contributions extend the frontier of strong solution techniques in singular, path-dependent, and nonlocal stochastic control.

Reference:

"Dividend ratcheting and capital injection under the Cramér-Lundberg model: Strong solution and optimal strategy" (2604.04641)