Expected discounted penalty function for a phase-type risk model with stochastic return on investment and random observation periods

Abstract

Purpose The purpose of this paper is to build a phase-type risk model with stochastic return on investment and random observation periods to characterize the ruin quantities under which the insurance company may take effective investment strategies to avoid bankruptcy. Design/methodology/approach By the Markov property and Ito’s formula, this paper derives the integro-differential equations in which the interclaim times follow a phase-type distribution. Using the sinc method, this paper obtains the approximate solutions of the expected discounted penalty function. The numerical examples are given to verify the robustness of the proposed sinc method. Findings This paper discloses the relationship between the investment strategy and initial surplus level. The insurance company with a high initial surplus level prefers high risk portfolios to earn more profit. Contrarily, the insurance company would invest low risk portfolios to avoid bankruptcy. In addition, this paper shows that a short observation period would bring higher ruin probability. Originality/value The risk model is distinct in that a phase-type risk model is constructed with stochastic return on investment and random observation periods. These considerations in the risk model are in sharp contrast to the setting in which the stochastic return on investment is observed continuously. In practice, the insurance company only can periodically observe the surplus level to check the balance of the book. This setting, therefore, is difficult to adopt. This paper develops a sinc method to solve the approximate solutions of the expected discounted penalty function.

The Gerber-Shiu discounted penalty function: A review from practical perspectives

“…The Chebyshev polynomial approximation is also employed elsewhere, such as for a perturbed Markov-modulated risk model with two-sided jumps [52] and Markov-dependent risk models with multi-layer dividend strategy [203] (Section 2.4.1). Another approach, yet in a similar direction, is based on Stenger's sinc method [205] for a phase-type risk model with stochastic investment return and random observation periods along with encouraging numerical illustrations.…”

“…To apply numerical methods based on the integro-differential equation, its solution is assumed to satisfy suitable smoothness conditions on its own, such as twice differentiability [52,205], continuous second derivatives [51] and continuity at every dividend barrier [203]. As usual in this context, moreover for the solution to be smooth, suitable regularity and integrability conditions are imposed on the problem elements, such as the claim size distribution, the reward, volatility and penalty functions [51], and the premium rate [203].…”

“…Nevertheless, numerical methods on the Gerber-Shiu function do not seem to have been adequately explored in the literature. To the best of our knowledge, there exist only a handful of developments in the context of the Gerber-Shiu function: the numerical inversion of the Laplace transform [60,87] (Section 4.1), finite truncation of the Fourier expansion [30,109,175,176] (Section 4.2), the numerical solution to the integrodifferential equation with suitable boundary conditions [51,52,71,203,205] (Section 4.3), and simulation via random number generation [43,57] (Section 4.4). As is always the case, the method needs to be carefully chosen for each problem setting, because each of those methods above has both advantages and disadvantages with different technical prerequisites for implementation and convergence (Section 4.5).…”

Numerical Method for a Perturbed Risk Model with Proportional Investment

“…There are many researchers who apply the sinc method to solve integrodifferential equations, since it was developed by Frank Stenger [16]. It is widely used for solving a wide range of linear and nonlinear optimal control problems, nonlinear boundary-value problems, ordinary differential and partial differential equations [23][24][25][26]. Its theoretical research can be referred to in [15,[17][18][19][20].…”

“…Chen and Ou [1] made use of the sinc method to solve the approximate solutions of the integro-differential equations for the expected discounted dividend payments and the expected discounted penalty function. Zhuo et al [23] studied the expected discounted penalty function by the sinc method when the inter-claim time follows a phase-type distribution. Chen et al [15] considered second-order integro-differential equations satisfied by the expected discounted dividend payments through the sinc-collocation method.…”