Optimization of Capital Injections and Absolutely Continuous Dividend Payments in a Diffusion Model

Battle Count: 85

This paper is primarily relevant to insurance mathematics and corporate finance rather than quantitative trading. The diffusion model and stochastic control framework share mathematical tools with quantitative finance (e.g., HJB equations, optimal stopping), but the specific problem of dividend/capital injection optimization is more applicable to insurance companies and corporate treasury management than to trading strategies. The Løkka-Zervos dichotomy and refraction-reflection strategies are not directly applicable to trading signal generation or portfolio construction.

The paper is highly theoretical with no computational implementation. Applying the results requires: (1) solving ODEs for fundamental solutions psi, phi, phi_F of the generator; (2) computing the optimal threshold b_c via smooth-fit conditions; (3) evaluating the comparison criterion V_c(0) vs 0 or V'_d(0) vs beta; (4) constructing Skorohod-reflected diffusions numerically. The viscosity solution framework and concavity proofs are mathematically sophisticated. For specific diffusion models (e.g., geometric Brownian motion, CIR), the fundamental solutions may be expressible in closed form, but for general diffusions, numerical ODE solvers would be needed.

This is a purely theoretical mathematics paper with complete proofs provided in the main text and appendices. All mathematical derivations are self-contained. However, there is no computational component, code, or numerical experiments to reproduce. The results are analytical theorems and propositions. Reproducibility depends on verifying the mathematical proofs.