A risk minimization problem for finite horizon semi-Markov decision processes with loss rates

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doi:10.3934/jdg.2018009

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2018, Volume 5, Issue 2: 143-163. Doi: 10.3934/jdg.2018009

This issue Previous Article Constrained stochastic differential games with additive structure: Average and discount payoffs Next Article Stable manifold market sequences

A risk minimization problem for finite horizon semi-Markov decision processes with loss rates

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1.
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
2.
School of Economics and Statistics, Guangzhou University, Guangzhou 510006, China

* Corresponding author. Emails: liuql@m.scnu.edu.cn; zouxiaol@gzhu.edu.cn
* Corresponding author. Emails: liuql@m.scnu.edu.cn; zouxiaol@gzhu.edu.cn

Received: May 31, 2017

Revised: June 30, 2017

Published: March 2018

Research supported by Natural Science Foundation of Guangdong Province (Grant No.2014A030313438), Zhujiang New Star (Grant No. 201506010056) and Guangdong Province outstanding young teacher training plan (Grant No. YQ2015050)

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Abstract

This paper deals with the risk probability for finite horizon semi-Markov decision processes with loss rates. The criterion to be minimized is the risk probability that the total loss incurred during a finite horizon exceed a loss level. For such an optimality problem, we first establish the optimality equation, and prove that the optimal value function is a unique solution to the optimality equation. We then show the existence of an optimal policy, and develop a value iteration algorithm for computing the value function and optimal policies. We also derive the approximation of the value function and the rules of iteration. Finally, a numerical example is given to illustrate our results.

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Mathematics Subject Classification: Primary: 90C40; Secondary: 93E20.

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Figure 1. The function $F^*_{(n_0+1)k}(1, t, \lambda)$

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Figure 2. The function $F^*_{(n_0+1)k}(2, t, \lambda)$

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Figure 3. The function $H^aF^*_{(n_0+1)k-1}(i, 10, \lambda)$

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Figure 4. The function $H^aF^*_{(n_0+1)k-1}(i, 15, \lambda)$

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Figure 5. The function $\lambda^*(i, t)$

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