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In this paper, stochastic Gilpin-Ayala model driven by α-stable process under regime switching is presented. Firstly, we prove the existence and uniqueness of the solution for the stochastic model. Then, under certain conditions, the almost sure exponential stability and pth moment exponential stability of the trivial solution are obtained. The results indicated that the α-stable noises can make the trivial solution stable under some conditions. Computer simulations are presented to illustrate our theory. [ABSTRACT FROM AUTHOR]

This article considers the optimal reinsurance-portfolio problem that the insurer invests in two related risky assets described by different types: constant elasticity of variance model and jump-diffusion process model, besides a risk-free asset. There is a correlation between the diffusion processes of the two models. Meanwhile, the company purchases proportional reinsurance. Specially, assume the claim process follows a Lévy process and the reinsurance's premium principle has not to be certain. Then based on stochastic control theory, a novel form of the optimal value function for solving the Hamilton-Jacobi-Bellman equation is constructed. Finally, the expressions of the optimal results are obtained under maximizing the expected exponential utility of terminal wealth. In addition, we listed several examples of the common premium principles. Numerical simulations are supplied for sensitivity analysis of parameters. [ABSTRACT FROM AUTHOR]

In this paper, the valuation of European option is investigated when the discrete dividends are described by Markov-modulated Merton jump-diffusion process. According to the dividend discount theory, we regard the stock price as the net present value of all future dividends. The regime switching Esscher transform is applied to determine a risk-neutral measure. The closed form solution of European option is obtained under the condition that the dividend payments are announced in advance. Numerical simulations for the European call option prices are provided. [ABSTRACT FROM AUTHOR]

In this paper, the optimal investment and reinsurance problem is investigated for a class of the jump-diffusion risk model. Here, the insurer can purchase excess-of-loss reinsurance and invest his or her surplus into a financial market consisting of one risk-free asset and one risk asset whose price is modeled by constant elasticity of variance (CEV) model, the net profit condition and the criterion of maximizing the expected exponential utility of terminal wealth are considered in the CEV financial market. By using stochastic control theory to solve the Hamilton-Jacobi-Bellman (HJB) equation, and the explicit form of the optimal policies and value functions can be obtained. Finally, numerical examples are presented to show the impacts of model parameters on the optimal strategies. [ABSTRACT FROM AUTHOR]

The present article deals with the problem of parametric estimation for certain nonstationary processes driven by α-stable motions. The trajectory fitting method combined with the weighted least squares technique is used to obtain an estimator of the drift parameter. The asymptotic behavior of the weighted trajectory fitting estimator for general weights in non-ergodic case, including consistency and asymptotic distribution. [ABSTRACT FROM AUTHOR]

This article investigates the European style option valuation under the condition that dividend payments follow a regime switching jump diffusion model while the first l of them have been known. Especially, when l = 0, it follows that all the dividend payments become stochastic. Using the generalized Itô formula, we obtain the explicit solution for dividend payments of the model. From Dividend Discount theory, which implies that the stock price should be equal to the net present value of its all future dividend payments, the stock price process is then deduced. Under the usual pricing framework of derivatives, closed-form solution of European-style option is derived via the characteristic function of the occupation times. [ABSTRACT FROM AUTHOR]

In this article, the valuation of power option is investigated when the dynamic of the stock price is governed by a generalized jump-diffusion Markov-modulated model. The systematic risk is characterized by the diffusion part, and the non systematic risk is characterized by the pure jump process. The jumps are described by a generalized renewal process with generalized jump amplitude. By introducing NASDAQ Index Model, their risk premium is identified respectively. A risk-neutral measure is identified by employing Esscher transform with two families of parameters, which represent the two parts risk premium. In this article, the non systematic risk premium is considered, based on which the price of power option is studied under the generalized jump-diffusion Markov-modulated model. In the case of a special renewal process with log double exponential jump amplitude, the accurate expressions for the Esscher parameters and the pricing formula are provided. By numerical simulation, the influence of the non systematic risk’s price and the index of the power options on the price of the option is depicted. [ABSTRACT FROM PUBLISHER]

The Equity-Indexed annuities (EIAs) provide investors not only a minimum rate of return but also an opportunity to gain a potential profit that is linked to the performance of an equity index, typically SP 500. These properties make EIAs become very popular in the market and receive considerable attention in the actuarial literature. In this article, we investigate the Equity-Indexed annuities valuation when the discrete dividend payments follow a Markov-modulated jump diffusion model. Using the generalized Itô formula, we obtain the explicit solution for dividend payments of the model. From Dividend Discount theory, which implies that the stock price should be equal to the net present value of all its future dividend payments, the stock price process is then deduced. Within this framework, closed-form solution to the point-to-point EIA pricing problem is derived via the characteristic function of the occupation times. [ABSTRACT FROM PUBLISHER]