一种基于多水平蒙特卡罗方法的最优控制在金融上的应用
Application of Optimal Control Based on Multilevel Monte Carlo Method in Finance
摘要: 本文使用多水平蒙特卡洛梯度投影优化方法求解一类随机最优控制问题。蒙特卡洛方法，是一种广泛使用的方法，是求解最优控制问题的一种常用方法，但它有自身的局限性，收敛速度慢。我们选取收敛速度较快的多水平蒙特卡洛方法。对这种方法的理论进行分析，之后将该方法用于具体实例，求解金融问题，并通过数值实验验证方法的有效性。
Abstract: This paper uses a multi-level Monte Carlo gradient projection optimization method to solve a class of stochastic optimal control problems. Monte Carlo method is a widely used method for solving optimal control problems, but it has its own limitations and slow convergence speed. We selected a multi-level Monte Carlo method with fast convergence speed to analyze the theory of this method, and then applied it to specific examples to solve financial problems. The effectiveness of the method was verified through numerical experiments.
文章引用：雷鹏. 一种基于多水平蒙特卡罗方法的最优控制在金融上的应用[J]. 运筹与模糊学, 2023, 13(3): 1610-1617. https://doi.org/10.12677/ORF.2023.133162

参考文献

[1] Heitsch, H. and Romisch, W. (2003) Scenario Reduction Algorithms in Stochastic Programming. Computational Optimization and Applications, 24, 187-206.
[2] Raffard, R., Hu, J. and Tomlin, C. (2005) Adjoint-Based Optimal Control of the Expected Exit Time for Stochastic Hybrid Systems. In: Morari, M. and Thiele, L., Eds., Hybrid Systems: Computation and Control, Lecture Notes in Computer Science, Vol. 3414, Springer-Verlag, Berlin. [Google Scholar] [CrossRef] 
[3] Archibald, R., Bao, F., Yong, J. and Zhou, T. (2020) An Efficient Numerical Algorithm for Solving Data Driven Feedback Control Problems. Journal of Scientific Computing, 85, Article Number: 51. [Google Scholar] [CrossRef] 
[4] Kushner, H.J. and Dupuis, P. (2001) Numerical Methods for Stochastic Control Problems in Continuous Time. 2nd Edition, Springer-Verlag, New York. [Google Scholar] [CrossRef] 
[5] Korn, R. and Kraft, H. (2001) A Stochastic Control Approach to Portfolio Problems with Stochastic Interest Rates. SIAM Journal on Control and Optimization, 40, 1250-1269. [Google Scholar] [CrossRef] 
[6] Giles, M.B. (2015) Multilevel Monte Carlo Methods. Acta Numerica, 24, 259-328. [Google Scholar] [CrossRef] 
[7] 谢非. 资产定价与数据科学中的拟蒙特卡洛方法[D]: [硕士学位论文]. 北京: 清华大学, 2019.
[8] Gong, B., Liu, W., Tang, T., et al. (2017) An Efficient Gradient Pro-jection Method for Stochastic Optimal Control Problems. SIAM Journal on Numerical Analysis, 55, 2982-3005. [Google Scholar] [CrossRef] 
[9] 姜礼尚. 期权定价的数学模型与方法[M]. 北京: 高等教育出版社, 2008.
[10] 蒲兴成, 张毅. 随机微分方程及其在数理金融中的应用[M]. 北京: 科学出版社, 2010.

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