Some capital injection problems for insurance type risk model

Abstract

Solvency problems for insurance companies have drawn much attention from insurers and also regulators, which reveals the importance of the capital injection strategies as a risk management tool that can be applied along the decision making processes in insurance companies. In actuarial science literature, many researchers have discussed various risk models with different capital injection strategies, where the problems are investigated within risk theory framework or connected to stochastic optimal control in dividend maximization problem. Therefore, in the first part of this thesis, the optimal dividend and capital injection problem with penalty payment at ruin is studied. By considering the surplus process killed at ruin, the problem is transferred to a combined stochastic and impulse control one up to ruin with free boundary at zero. The theoretical verifications are presented for a different type of capital injection strategies comparing to the conventional results in the literature, where the capital injections are made before the time of ruin. Under the restricted dividend density assumption, it can be proved that the value function is the unique increasing, bounded, Lipschitz continuous and upper semi-continuous at zero viscosity solution to the corresponding quasi-variational Hamilton-Jacobi-Bellman (HJB) equation. The uniqueness for such class of viscosity solutions is proved by taking into account the boundary condition at infinity. The optimality of certain type of band-type strategies are proved for the case when premium rate is greater than and equal to or less than the ceiling dividend rate respectively. Later, the results are extended to the scenario with unrestricted dividend rate. The corresponding value function and the quasi-variational HJB equation can be obtained by taking limit in the restricted dividend density case. Through a similar analysis, the value function is proved as the smallest viscosity supersolution of the corresponding quasi-variational HJB equation. Further, a comparison principle is presented in order to obtain the uniqueness of the viscosity solution with certain growth condition. In the second part, a new type of capital injection strategy is proposed, which is periodically implemented based on the number of claims. Specifically, capital injection decisions are made at predetermined accumulated number of claim instants, if the surplus is lower than a minimum required level. By assuming a combination of exponentials for the claim severities, an explicit expression for the discounted density of the surplus level after a certain number of claims is derived under the condition that ruin has not yet occurred. Utilizing this result, the expected total discounted capital injection until the first ruin time is studied. Similarly, an expression for the Laplace transform of the time to ruin is also explicitly found. Finally, the applicability of the present capital injection strategy and methodologies developed previously is illustrated through various numerical examples. In particular, for exponential claim severities, some optimal capital injection strategies for different initial surpluses which minimize the expected capital spending per unit time are numerically studied.