Optimal dividend problems for a jump-diffusion model with capital injections and proportional transaction costs

TL;DR

This paper investigates optimal dividend and capital injection strategies for a company with a jump-diffusion surplus process, considering transaction costs and different control constraints, providing explicit solutions for various optimization problems.

Contribution

It introduces a comprehensive model for surplus control with jump-diffusion dynamics, deriving explicit optimal strategies under proportional transaction costs and different capital injection constraints.

Findings

Explicit optimal strategies identified for each problem type.

Value functions characterized for jump-diffusion surplus processes.

Strategies adapt to different constraints and profit distributions.

Abstract

In this paper, we study the optimal control problem for a company whose surplus process evolves as an upward jump diffusion with random return on investment. Three types of practical optimization problems faced by a company that can control its liquid reserves by paying dividends and injecting capital. In the first problem, we consider the classical dividend problem without capital injections. The second problem aims at maximizing the expected discounted dividend payments minus the expected discounted costs of capital injections over strategies with positive surplus at all times. The third problem has the same objective as the second one, but without the constraints on capital injections. Under the assumption of proportional transaction costs, we identify the value function and the optimal strategies for any distribution of gains.