多维G-方程极值定理
Extreme Value Theorem of Multidimensional G-Equation
摘要: 本文基于G-期望空间理论，通过反证法，利用Hessian矩阵将G-方程的极值定理推广到多维空间和变系数的情况，得到了多维变系数G-方程极值在定义域边界取得的结论，在物理学、金融学以及计算数学领域有很高的实用性价值。
Abstract: This article is based on the theory of G-expected space and uses the method of proof to generalize the extreme value theorem of G-equations to multi-dimensional space and variable coefficient cases using Hessian matrix. The conclusion that the extreme values of multi-dimensional variable coefficient G-equations are obtained at the boundary of the domain is obtained, which has high practical value in the fields of physics, finance, and computational mathematics.
文章引用：李洋, 江继祥, 张宇, 许乾海. 多维G-方程极值定理[J]. 理论数学, 2024, 14(5): 140-144. https://doi.org/10.12677/pm.2024.145170

参考文献

[1] Peng, S. (2007) G-Expectation, G-Brownian Motion and Related Stochastic Calculus of Ito Type. Stochastic Analysis and Applications. Springer, Berlin, Heidelberg, 541-567. [Google Scholar] [CrossRef] 
[2] Peng, S. (1992) A Generalized Dynamic Programming Principle and Hamilton-Jacobi-Bellman Equation. Stochastics: An International Journal of Probability and Stochastic Processes, 38, 119-134. [Google Scholar] [CrossRef] 
[3] Pasik-Duncan, B. and Duncan, T.E. (2001) Stochastic Controls: Hamiltonian Systems and HJB Equations. IEEE Transactions on Automatic Control, 46, 1846. [Google Scholar] [CrossRef] 
[4] 彭实戈. 非线性期望的理论, 方法及意义[J]. 中国科学: 数学, 2017, 47(10): 1223-1254.
[5] Peng, S. (2008) A New Central Limit Theorem under Sublinear Expectations. arXiv:0803.2656.
[6] Peng, S. (2008) Multi-Dimensional G-Brownian Motion and Related Stochastic Calculus under G-Expectation. Stochastic Processes and Their Applications, 118, 2223-2253.
[7] Hu, M. (2012) Explicit Solutions of the G-Heat Equation for a Class of Initial Conditions. Nonlinear Analysis: Theory, Methods and Applications, 75, 6588-6595. [Google Scholar] [CrossRef] 
[8] Peng, S. and Zhou, Q. (2020) A Hypothesis-Testing Perspective on the Gnormal Distribution Theory. Statistics and Probability Letters, 156, 108623.
[9] Gong, X. and Yang, S. (2013) The Application of G-Heat Equation and Numerical Properties. arXiv:1304.1599. [Google Scholar] [CrossRef] 
[10] Lin, Q. (2010) The Tychonoff Uniqueness Theorem for the G-Heat Equation. arXiv:1006.5300.
[11] Hu, M. and Sun, Y. (2021) Explicit Positive Solutions to G-Heat Equations and the Application to G-Capacities. Journal of Differential Equations, 297, 246-276. [Google Scholar] [CrossRef] 
[12] Li, Y. and Kulperger, R. (2018) An Iterative Approximation of the Sublinear Expectation of an Arbitrary Function of G-Normal Distribution and the Solution to the Corresponding G-Heat Equation. arXiv:1804.10737.
[13] 姜礼尚, 陈亚浙, 刘西垣. 数学物理方程讲义[M]. 北京: 高等教育出版社, 2007.
[14] Songchitruksa, P. and Tarko, A.P. (2006) The Extreme Value Theory Approach to Safety Estimation. Accident Analysis & Prevention, 38, 811-822. [Google Scholar] [CrossRef] [PubMed]
[15] Gomes, M.I. and Guillou, A. (2015) Extreme Value Theory and Statistics of Univariate Extremes: A Review. International Statistical Review, 83, 263-292. [Google Scholar] [CrossRef] 
[16] 宋彬彬. G-方程的数值方法[D]: [硕士学位论文]. 苏州: 苏州大学, 2015.
[17] 王萼芳, 石生明. 高等代数[M]. 第4版. 北京: 高等教育出版社, 2013: 227-233.

为你推荐