This paper investigates the optimal dividend and capital injection problem under a two-sided jump-diffusion model, incorporating the flexibility of liquidation when the surplus becomes negative. Unlike prior studies, we relax the constraint on capital injection, allowing decision-makers to choose between rescuing the firm via capital injection or liquidating it upon financial distress. We derive the dynamic programming principle and the Hamilton–Jacobi–Bellman (HJB) equation for this control problem. To overcome the analytical challenges posed by two-sided jumps, we use the Markov chain approximation (MCA) method, constructing a discrete-time controlled Markov chain that approximates the original jump-diffusion process. We show the convergence of the value function under MCA to the original value function through the weak convergence theory. We then solve the approximated problem numerically based on a value iteration algorithm. Extensive numerical examples illustrate the optimal strategies, showcasing the interplay among dividend payments, capital injections, and liquidation decisions.
Citation: Ningning Feng, Yixuan Liu, Ran Xu. Dividend and capital injection control with liquidation under two-sided jump-diffusion models[J]. Journal of Industrial and Management Optimization, 2026, 22(7): 3419-3455. doi: 10.3934/jimo.2026125
This paper investigates the optimal dividend and capital injection problem under a two-sided jump-diffusion model, incorporating the flexibility of liquidation when the surplus becomes negative. Unlike prior studies, we relax the constraint on capital injection, allowing decision-makers to choose between rescuing the firm via capital injection or liquidating it upon financial distress. We derive the dynamic programming principle and the Hamilton–Jacobi–Bellman (HJB) equation for this control problem. To overcome the analytical challenges posed by two-sided jumps, we use the Markov chain approximation (MCA) method, constructing a discrete-time controlled Markov chain that approximates the original jump-diffusion process. We show the convergence of the value function under MCA to the original value function through the weak convergence theory. We then solve the approximated problem numerically based on a value iteration algorithm. Extensive numerical examples illustrate the optimal strategies, showcasing the interplay among dividend payments, capital injections, and liquidation decisions.
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Ningning Feng, Yixuan Liu, Ran Xu. Dividend and capital injection control with liquidation under two-sided jump-diffusion models[J]. Journal of Industrial and Management Optimization, 2026, 22(7): 3419-3455. doi: 10.3934/jimo.2026125
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