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

This paper establishes the existence and uniqueness of a strong solution to the Hamilton-Jacobi-Bellman equation for an optimal dividend ratcheting problem with costly capital injections under the Cramér-Lundberg model, thereby deriving an explicit optimal feedback strategy that advances the field beyond standard viscosity solution frameworks.

The Gist

Imagine you are the captain of a ship (an insurance company) sailing through a stormy sea. Your goal is to deliver as much treasure (dividends) as possible to the shareholders back at the port, but you must keep the ship afloat.

This paper solves a very tricky navigation problem involving three specific rules:

1. The "No Downgrade" Rule (Ratcheting): Once you decide to increase the treasure you send back to the shareholders, you can never lower it again. It's like a "ratchet" wrench; it only turns one way. If you promise a higher payout, you are stuck with it forever.
2. The "Emergency Fuel" Rule (Capital Injection): If a giant wave (a big insurance claim) threatens to sink the ship, you are allowed to buy emergency fuel (inject capital) to stay afloat. But this fuel is expensive—it costs more than the treasure you're trying to save.
3. The "Random Storms" Rule (Cramér-Lundberg Model): The sea isn't calm. You have a steady income (the engine), but random, unpredictable waves (claims) crash against you.

The Problem: How to Sail?

The authors ask: What is the perfect strategy?

• When should you send treasure home?
• When should you increase the payout?
• When should you buy that expensive emergency fuel?

If you send too much treasure, the ship sinks. If you send too little, the shareholders get angry. If you buy fuel too often, the cost eats up all your profits. And because of the "No Downgrade" rule, you can't just lower the payout when things get tough; you have to be very careful about when you decide to turn the ratchet up.

The Solution: A Mathematical Map

The paper uses advanced math (specifically something called a "Hamilton-Jacobi-Bellman equation") to draw a map for the captain. Here is how they did it, explained simply:

1. The "Guess and Check" Trap
Usually, mathematicians try to guess a solution and check if it works. But because of the "No Downgrade" rule and the random waves, the math is too messy for simple guessing. The equation is like a tangled knot of ropes.

2. The "Pixelated" Approach
To untangle the knot, the authors used a clever trick. Imagine the payout rate isn't a smooth slider, but a staircase with many tiny steps. They first solved the problem for a staircase with just a few steps. Then, they added more and more steps, making the staircase look more and more like a smooth ramp.

• Analogy: It's like watching a low-resolution video that slowly becomes high-definition. By solving the "low-res" version first, they could prove that a perfect, "high-definition" solution exists.

3. The "Switching Boundary" (The Magic Line)
The most important result is the discovery of a Magic Line on their map.

• Below the Line: If your ship is safe and your current payout is low, you keep things steady.
• Above the Line: If your ship is very safe and your current payout is low, it's time to turn the ratchet up. You increase the dividend to the maximum safe level for that moment.
• The Wall: If a wave hits and your ship gets too low, you hit the "Wall." You immediately buy the expensive emergency fuel just enough to keep the ship from sinking, but no more.

Why This Matters

Before this paper, we knew how to sail if the rules were simple (like Brownian motion, which is like a calm, predictable drift). But real insurance is full of sudden, massive storms (jumps).

This paper is the first to give a complete, step-by-step instruction manual for a captain dealing with:

• Sudden, massive storms.
• The rule that you can never lower your payout.
• The need to buy expensive emergency fuel.

The Takeaway:
The authors proved that there is a single, perfect strategy. It tells the insurance company exactly when to say "Yes, we can afford to pay more!" and exactly how much emergency money to buy to survive a disaster without going broke. They didn't just say "a solution exists"; they built the actual map so companies can use it in the real world.

Technical Summary

1. Problem Formulation

The paper addresses an optimal dividend payout problem for an insurance company operating under the classical Cramér-Lundberg model. The surplus process $X_t$ is governed by a deterministic income stream ($\mu$), a compound Poisson process representing random claims ($Z_i$), and two control mechanisms:

1. Dividend Payout ($C_t$): The rate of dividends paid to shareholders.
2. Capital Injection ($D_t$): Cumulative capital injected to prevent ruin, incurring a proportional cost $\ell > 1$.

Key Constraints and Features:

• Ratcheting Constraint: The dividend rate $C_t$ must be non-decreasing over time ($C_t \geq C_s$ for $t > s$). This reflects managerial reluctance or contractual obligations to cut dividends.
• Capital Injection: The company can inject capital at a cost $\ell$ per unit to keep the surplus non-negative ($X_t \geq 0$). The cost $\ell > 1$ ensures external capital is more expensive than retained earnings.
• Objective: Maximize the expected present value of discounted cumulative dividends minus the discounted cost of capital injections:
  $$V(x, c) = \sup_{\{(C_t, D_t)\}} \mathbb{E} \left[ \int_0^\infty e^{-rt} C_t \, dt - \ell \int_0^\infty e^{-rt} \, dD_t \right]$$
  where $x$ is the initial surplus and $c$ is the initial dividend rate.

2. Mathematical Framework

The problem is formulated as a stochastic control problem with self-path-dependent constraints (due to the ratcheting condition) and jump-diffusion dynamics (due to the compound Poisson process).

• Hamilton-Jacobi-Bellman (HJB) Equation: The associated HJB equation is a partial integro-differential variational inequality (PIDVI). It features:
  • A non-local integral term ($T v$) arising from the jump component.
  • Two gradient constraints:
    1. $v_x \leq \ell$ (from the cost of capital injection).
    2. $-v_c \geq 0$ (from the ratcheting constraint, implying the value function is non-increasing in the initial dividend rate).
• Domain: The problem is analyzed on the domain $Q^+_\infty = \mathbb{R}_+ \times (-\infty, \bar{c}]$, where $\bar{c}$ is the upper bound on the dividend rate.

3. Methodology

The authors develop a systematic probabilistic and PDE-based approach to solve the HJB equation, moving beyond the standard viscosity solution framework.

Step 1: Boundary Case Analysis

• The authors first solve the boundary case where the dividend rate is fixed at its maximum $\bar{c}$. This reduces the problem to a single Ordinary Integro-Differential Equation (OIDE).
• They prove the existence of a unique, bounded, $C^1$ solution $g(x)$ with specific regularity properties (concavity, bounded derivatives).

Step 2: Discretization and Regime-Switching Approximation

• To handle the continuous ratcheting constraint, the authors discretize the space of admissible dividend rates into a finite set $\{c_i\}$.
• This transforms the original PIDVI into a regime-switching system of OIDEs. The system couples equations for different dividend rates via a "min" operator representing the switching logic.
• They establish uniform a priori estimates (bounds on the solution, its first derivative, and second derivative) for this discrete system using comparison principles and penalty methods.

Step 3: Limiting Argument and Strong Solution Construction

• By passing to the limit as the discretization step size goes to zero, they construct a strong solution to the original HJB variational inequality.
• Regularity: Unlike viscosity solutions, this strong solution is continuously differentiable in the surplus variable $x$ and possesses sufficient regularity to define explicit feedback controls.

Step 4: Verification and Optimal Strategy

• Using the regularity of the strong solution, the authors verify that the constructed value function coincides with the true value function of the control problem.
• They characterize the optimal strategy via a free boundary $X(c)$ and an equivalent maximum rate function $M(x, c)$.

4. Key Results

A. Existence and Uniqueness of Strong Solution
The paper proves that the HJB variational inequality admits a unique bounded strong solution $v(x, c)$. This solution satisfies:

• $v \in C(Q^+_\infty)$ and $v(\cdot, c) \in C^1(\mathbb{R}_+)$.
• Gradient constraints: $0 \leq v_x \leq \ell$ and $v_c \leq 0$.
• Concavity in $x$ and specific bounds on second derivatives.

B. Structure of the Optimal Strategy
The optimal policy is characterized by a switching free boundary $X(c)$:

1. Non-Switching Region ($NS$): When the current surplus $x < X(c)$, the company maintains the current dividend rate $C_t = c$. The HJB equation holds as an equality: $L_c v - T v + h - c = 0$.
2. Switching Region ($S$): When $x > X(c)$, the company increases the dividend rate immediately to a new level determined by the function $M(x, c)$. In this region, the gradient constraint $-v_c = 0$ is active (meaning the value is constant with respect to increasing $c$).
3. Capital Injection: Capital is injected only when a claim would otherwise drive the surplus negative. The injection amount is the minimal amount required to keep $X_t \geq 0$.

C. Properties of the Free Boundary

• The free boundary $X(c)$ is continuous and non-decreasing in $c$.
• It separates the state space into regions where the dividend rate should be increased versus kept constant.

5. Significance and Contributions

1. First Complete Solution under Cramér-Lundberg: This is the first work to provide a complete solution (both value function and implementable optimal strategy) for a dividend ratcheting problem with capital injection under the Cramér-Lundberg model. Previous works were limited to Brownian motion settings.
2. Beyond Viscosity Solutions: The paper advances the mathematical theory by establishing a strong solution rather than just a viscosity solution. This is crucial because viscosity solutions often lack the differentiability required to construct explicit feedback controls in complex jump-diffusion settings.
3. Handling Self-Path-Dependence: The methodology successfully tackles the "self-path-dependent" nature of the ratcheting constraint combined with jump dynamics, which creates a significantly more complex variational inequality than standard problems.
4. Practical Applicability: The derived strategy offers a rigorous foundation for dividend policy design in economics, explicitly accounting for the trade-off between the desire to increase dividends (ratcheting), the cost of external capital, and the risk of ruin in a realistic jump-risk environment.

In summary, the paper bridges a gap between theoretical stochastic control and practical insurance management by providing a rigorous, analytically tractable solution to a highly constrained, realistic optimization problem.