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Optimal investment and premium control in a nonlinear diffusion model

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Abstract

This paper considers the optimal investment and premium control problem in a diffusion approximation to a non-homogeneous compound Poisson process. In the nonlinear diffusion model, it is assumed that there is an unspecified monotone function describing the relationship between the safety loading of premium and the time-varying claim arrival rate. Hence, in addition to the investment control, the premium rate can be served as a control variable in the optimization problem. Specifically, the problem is investigated in two cases: (i) maximizing the expected utility of terminal wealth, and (ii) minimizing the probability of ruin respectively. In both cases, some properties of the value functions are derived, and closed-form expressions for the optimal policies and the value functions are obtained. The results show that the optimal investment policy and the optimal premium control policy are dependent on each other. Most interestingly, as an example, we show that the nonlinear diffusion model reduces to a diffusion model with a quadratic drift coefficient when the function associated with the premium rate and the claim arrival rate takes a special form. This example shows that the model of study represents a class of nonlinear stochastic control risk model.

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Acknowledgements

The authors thank two anonymous referees for their careful reading and useful comments that greatly improved the presentation of the paper.

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Authors and Affiliations

  1. China Institute for Actuarial Science, Central University of Finance and Economics, Beijing, 100081, China

    Ming Zhou

  2. Department of Statistics and Actuarial Science, the University of Hong Kong, Pokfulam, Hong Kong, China

    Kam Chuen Yuen

  3. School of Statistics, Qufu Normal University, Shandong, 273165, China

    Chuan-cun Yin

Authors
  1. Ming Zhou
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  2. Kam Chuen Yuen
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  3. Chuan-cun Yin
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Corresponding author

Correspondence to Ming Zhou.

Additional information

The research of Ming Zhou was supported by the National Natural Science Foundation of China (11571388), the MOE Project of Key Research Institute of Humanities and Social Sciences at Universities (15JJD790036), and the 111 Project (B17050). The research of Kam Chuen Yuen was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU17329216). The research of Chuancun Yin was supported by the National Natural Science Foundation of China (11571198, 11701319).

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Zhou, M., Yuen, K.C. & Yin, Cc. Optimal investment and premium control in a nonlinear diffusion model. Acta Math. Appl. Sin. Engl. Ser. 33, 945–958 (2017). https://doi.org/10.1007/s10255-017-0709-7

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  • DOI: https://doi.org/10.1007/s10255-017-0709-7

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