We consider the spectrally negative L@vy processes and determine the joint laws for the quantities such as the first and last passage times over a fixed level, the overshoots and undershoots at first passage, the mini...We consider the spectrally negative L@vy processes and determine the joint laws for the quantities such as the first and last passage times over a fixed level, the overshoots and undershoots at first passage, the minimum, the maximum, and the duration of negative values. We apply our results to insurance risk theory to find an explicit expression for the generalized expected discounted penalty function in terms of scale functions. Furthermore, a new expression for the generalized Dickson＇s formula is provided.展开更多
Differential evolution （DE） has become a very popular and effective global optimization algorithm in the area of evolutionary computation. In spite of many advantages such as conceptual simplicity, high efficiency a...Differential evolution （DE） has become a very popular and effective global optimization algorithm in the area of evolutionary computation. In spite of many advantages such as conceptual simplicity, high efficiency and ease of use, DE has two main components, i.e., mutation scheme and parameter control, which significantly influence its performance. In this paper we intend to improve the performance of DE by using carefully considered strategies for both of the two components. We first design an adaptive mutation scheme, which adaptively makes use of the bias of superior individuals when generating new solutions. Although introducing such a bias is not a new idea, existing methods often use heuristic rules to control the bias. They can hardly maintain the appropriate balance between exploration and exploitation during the search process, because the preferred bias is often problem and evolution-stage dependent. Instead of using any fixed rule, a novel strategy is adopted in the new adaptive mutation scheme to adjust the bias dynamically based on the identified local fitness landscape captured by the current population. As for the other component, i.e., parameter control, we propose a mechanism by using the Levy probability distribution to adaptively control the scale factor F of DE. For every mutation in each generation, an Fi is produced from one of four different Levy distributions according to their historical performance. With the adaptive mutation scheme and parameter control using Levy distribution as the main components, we present a new DE variant called Levy DE （LDE）. Experimental studies were carried out on a broad range of benchmark functions in global numerical optimization. The results show that LDE is very competitive, and both of the two main components have contributed to its overall performance. The scalability of LDE is also discussed by conducting experiments on some selected benchmark functions with dimensions from 30 to 200.展开更多
Let （Xt）t≥0 be a Lévy process taking values in R^d with absolutely continuous marginal distributions. Given a real measurable function f on R^d in Kato＇s class, we show that the empirical mean 1/t ∫ f（Xs）d...Let （Xt）t≥0 be a Lévy process taking values in R^d with absolutely continuous marginal distributions. Given a real measurable function f on R^d in Kato＇s class, we show that the empirical mean 1/t ∫ f（Xs）ds converges to a constant z in probability with an exponential rate if and only if f has a uniform mean z. This result improves a classical result of Kahane et al. and generalizes a similar result of L. Wu from the Brownian Motion to general Lévy processes.展开更多
The extinction phenomenon induced by multiplicative non-Gaussian L′evy noise in a tumor growth model with immune response is discussed. Under the influence of the stochastic immune rate, the model is analyzed in term...The extinction phenomenon induced by multiplicative non-Gaussian L′evy noise in a tumor growth model with immune response is discussed. Under the influence of the stochastic immune rate, the model is analyzed in terms of a stochastic differential equation with multiplicative noise. By means of the theory of the infinitesimal generator of Hunt processes, the escape probability, which is used to measure the noise-induced extinction probability of tumor cells, is explicitly expressed as a function of initial tumor cell density, stability index and noise intensity. Based on the numerical calculations, it is found that for different initial densities of tumor cells, noise parameters play opposite roles on the escape probability. The optimally selected values of the multiplicative noise intensity and the stability index are found to maximize the escape probability.展开更多
基金Acknowledgements The authors thank two anonymous referees for their constructive suggestions which have led to much improvement on the paper. The first author is grateful to Professor Xiaowen Zhou for useful discussion. The research of Yuen was supported by a university research grant of the University of Hong Kong. The research of Yin was supported by the National Natural Science Foundation of China （No. 11171179）, the Research Fund for the Doctoral Program of Higher Education of China （No. 20133705110002）, and the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province.
文摘We consider the spectrally negative L@vy processes and determine the joint laws for the quantities such as the first and last passage times over a fixed level, the overshoots and undershoots at first passage, the minimum, the maximum, and the duration of negative values. We apply our results to insurance risk theory to find an explicit expression for the generalized expected discounted penalty function in terms of scale functions. Furthermore, a new expression for the generalized Dickson＇s formula is provided.
基金the National Natural Science Foundation of China under Grant Nos. 71071106 and 70801062
文摘Differential evolution （DE） has become a very popular and effective global optimization algorithm in the area of evolutionary computation. In spite of many advantages such as conceptual simplicity, high efficiency and ease of use, DE has two main components, i.e., mutation scheme and parameter control, which significantly influence its performance. In this paper we intend to improve the performance of DE by using carefully considered strategies for both of the two components. We first design an adaptive mutation scheme, which adaptively makes use of the bias of superior individuals when generating new solutions. Although introducing such a bias is not a new idea, existing methods often use heuristic rules to control the bias. They can hardly maintain the appropriate balance between exploration and exploitation during the search process, because the preferred bias is often problem and evolution-stage dependent. Instead of using any fixed rule, a novel strategy is adopted in the new adaptive mutation scheme to adjust the bias dynamically based on the identified local fitness landscape captured by the current population. As for the other component, i.e., parameter control, we propose a mechanism by using the Levy probability distribution to adaptively control the scale factor F of DE. For every mutation in each generation, an Fi is produced from one of four different Levy distributions according to their historical performance. With the adaptive mutation scheme and parameter control using Levy distribution as the main components, we present a new DE variant called Levy DE （LDE）. Experimental studies were carried out on a broad range of benchmark functions in global numerical optimization. The results show that LDE is very competitive, and both of the two main components have contributed to its overall performance. The scalability of LDE is also discussed by conducting experiments on some selected benchmark functions with dimensions from 30 to 200.
文摘Let （Xt）t≥0 be a Lévy process taking values in R^d with absolutely continuous marginal distributions. Given a real measurable function f on R^d in Kato＇s class, we show that the empirical mean 1/t ∫ f（Xs）ds converges to a constant z in probability with an exponential rate if and only if f has a uniform mean z. This result improves a classical result of Kahane et al. and generalizes a similar result of L. Wu from the Brownian Motion to general Lévy processes.
基金Supported by the National Natural Science Foundation of China under Grant Nos.10932009,11172233,and 11302169
文摘The extinction phenomenon induced by multiplicative non-Gaussian L′evy noise in a tumor growth model with immune response is discussed. Under the influence of the stochastic immune rate, the model is analyzed in terms of a stochastic differential equation with multiplicative noise. By means of the theory of the infinitesimal generator of Hunt processes, the escape probability, which is used to measure the noise-induced extinction probability of tumor cells, is explicitly expressed as a function of initial tumor cell density, stability index and noise intensity. Based on the numerical calculations, it is found that for different initial densities of tumor cells, noise parameters play opposite roles on the escape probability. The optimally selected values of the multiplicative noise intensity and the stability index are found to maximize the escape probability.
