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

Chonghu Guan, Zuo Quan Xu

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.

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.

Paper Structure

This paper contains 27 sections, 23 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Problem formulation and preliminaries
  3. Solution to the problem \ref{['value']}
  4. A comparison principle
  5. Boundary problem: $V(x,\overline{c})$
  6. HJB equation on ${\cal Q}^{+}_{\infty}$
  7. Optimal strategy to \ref{['value']}.
  8. Step 1 (Upper bound: $v\geqslant V$).
  9. Step 2 (Admissibility).
  10. Step 3 (Lower bound and optimality: $v\leqslant V$).
  11. Step 4 (Conclusion).
  12. Solvability of the HJB equation on ${\cal Q}^{+}_{\infty}$
  13. Local HJB equation
  14. Approximation by a regime switching system
  15. Construction of the solution to \ref{['v_pb']}
  16. ...and 12 more sections

Key Result

Lemma 2.1

The value function $V$ defined in value is monotonically decreasing in $c$, concave in $x$ and satisfies Moreover, for any $x< 0$, a strategy $\{(C_t,D_t)\}_{t\geqslant0}$ is an optimal for $V(0,c)$ if and only if $\{(C_t,D_t-x)\}_{t\geqslant0}$ is optimal for $V(x,c)$; in particular, it suffices to find an optimal solution for $V(0,c)$ so as to solve $V(x,c)$. $\blacktriangleleft$$\blacktriangle

Theorems & Definitions (25)

  • Lemma 2.1
  • Lemma 3.1: Comparison principle
  • Lemma 3.2
  • Theorem 3.3: Optimal strategy in the boundary case
  • Definition 3.4: Strong solution to \ref{['v_pb00']}
  • Lemma 3.5: Uniqueness of strong solution to \ref{['v_pb00']}
  • Theorem 3.6: Existence of strong solution to \ref{['v_pb00']}
  • Proposition 3.7: Property of the free boundary $\mathcal{X}(\cdot)$
  • Lemma 3.8: Property of the equivalent maximum rate $\mathfrak{M}(\cdot)$
  • Theorem 3.9: Optimal strategy in ${\cal Q}^{+}_{\infty}$
  • ...and 15 more