Optimal Dividend and Capital Injection Strategies with Exit Options in Jump-Diffusion Models

This paper studies optimal dividend and capital injection strategies with active exit options under a jump-diffusion model. We introduce a piecewise terminal payoff function to capture stop-loss exits (for deficits) and profit-taking exits (for surpluses), enabling shareholders to dynamically balance risk and return. Using the dynamic programming principle, we derive the associated quasi-variational inequalities (QVIs) and characterize the value function as the unique viscosity solution. To address analytical challenges, we employ the Markov chain approximation method, constructing a controlled Markov chain that closely approximates the jump-diffusion dynamics. Numerical solutions of the approximated problem are obtained via value iteration. The numerical results demonstrate how the value function and optimal strategies respond to different claim distributions (comparing Exponential and Pareto cases), key model parameters, and exit payoff functions. The numerical study further validates the algorithm’s convergence and examines the stability of solutions with respect to domain truncation in the QVI formulation.