Paper: arXiv 2505.23511

Authors: Sang Hu, Zihan Zhou

Abstract

This paper studies the dividend and capital injection problem under a diffusion risk model with general discount functions. A proportional cost is imposed when injecting capitals. For exponential discounting as time-consistent benchmark, we obtain the closed-form solutions and show that the optimal strategies are of threshold type. Under general discount function which leads to time-inconsistency, we adopt the definition of weak equilibrium and obtain the extended HJB equation system. An explicit solution is derived under pseudo-exponential discounting where three cases of the dividend and capital injection thresholds are obtained. Numerical examples show that large capital injection cost may lead to no capital injection at all, while larger difference in group discount rate leads to higher equilibrium value function.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a strong theoretical contribution to mathematical finance and actuarial science by studying equilibrium dividend and capital injection strategies under time-inconsistent preferences using extended HJB equations. However, it is purely theoretical and mathematical, lacking empirical backtesting or real-world dataset validation.

Research Flowchart

  flowchart TD
    A[Goal: Solve Dividend & Capital Injection<br/>under Time-Inconsistent Preferences] --> B[Input: Diffusion Risk Model with<br/>Proportional Capital Injection Costs]
    
    B --> C{Choose Discounting Function}
    
    C -->|Exponential Discounting| D[Time-Consistent Benchmark:<br/>Standard HJB Equation]
    C -->|General Discounting| E[Time-Inconsistent Framework:<br/>Weak Equilibrium & Extended HJB System]
    
    D --> F[Closed-Form Solutions:<br/>Threshold-Type Strategies]
    E --> G[Explicit Solutions for Pseudo-Exponential<br/>Discounting: 3 Threshold Cases]
    
    F & G --> H[Findings: High injection cost stops capital injection;<br/>larger group discount gap increases value function]