摘要
随机过程作为金融数学中的重要工具，广泛应用于资产定价、风险管理、投资组合优化等领域。随着金融市场复杂性和不确定性的增加，随机过程的理论和方法在金融数学中的应用越来越重要。本文综述了近年来随机过程在金融数学中的应用进展，特别是在资产定价、风险管理和市场动态行为建模中的创新应用。首先，本文回顾了随机过程的基本概念和模型，包括几何布朗运动、随机微分方程等，并分析了它们在金融市场中的实际应用。其次，详细探讨了随机过程在期权定价、风险管理、资产组合优化等方面的应用，尤其是基于随机微分方程的资产定价模型。进一步地，本文讨论了随机过程在金融市场中的动态行为建模，分析了市场波动、价格跳跃等现象。最后，本文总结了随机过程在金融数学中应用的最新成果，并展望了未来研究方向，指出了当前存在的挑战及潜在的解决方案。本文旨在为相关研究人员提供一个系统的回顾，帮助推动随机过程在金融数学中的进一步发展。
关键词： 随机过程；金融数学；资产定价；风险管理；市场动态；随机微分方程；数值方法
Abstract
Stochastic processes, as a critical tool in financial mathematics, are widely used in asset pricing, risk management, investment portfolio optimization, and more. With the increasing complexity and uncertainty of financial markets, the theory and methods of stochastic processes have become more significant in financial mathematics. This paper reviews the recent advancements in the application of stochastic processes in financial mathematics, with a particular focus on their innovative use in asset pricing, risk management, and market dynamics modeling. First, the paper revisits the basic concepts and models of stochastic processes, including geometric Brownian motion and stochastic differential equations, and discusses their practical applications in financial markets. Next, it explores in detail the application of stochastic processes in option pricing, risk management, and portfolio optimization, particularly focusing on asset pricing models based on stochastic differential equations. Furthermore, the paper examines the modeling of dynamic behaviors in financial markets, analyzing phenomena such as market volatility and price jumps. Finally, the paper summarizes the latest achievements in the application of stochastic processes in financial mathematics and looks forward to future research directions, highlighting the challenges and potential solutions. This paper aims to provide a systematic review for researchers and contribute to the further development of stochastic processes in financial mathematics.
Key words： Stochastic process; Financial mathematics; Asset pricing; Risk management; Market dynamics; Stochastic differential equations; Numerical methods
参考文献 References
[1] Liao L . The Application and Stability Analysis of Stochastic Differential Equations in Financial Mathematics[J]. Academic Journal of Mathematical Sciences,2024,5(3).
[2] Gupta R ,Szczȩśniak D A E ,Kais S , et al. Entropy corrected geometric Brownian motion.[J].Scientific reports,2024, 14(1):28384.
[3] Li A ,Wang J ,Zhou L . Parameter Estimation of Uncertain Differential Equations Driven by Threshold Ornstein–Uhlenbeck Process with Application to U.S. Treasury Rate Analysis[J].Symmetry,2024,16(10):1372-1372.
[4] Maheswari L M ,Muthusamy K . Dynamical behavior of tempered [formula omitted]-Caputo type fractional order stochastic differential equations driven by Lévy noise[J].Partial Differential Equations in Applied Mathematics, 2024, 12100938-100938.
[5] Li Y ,Hu K ,Li J , et al. A formal specification language and automatic modeling method of asset securitization contract[J].Journal of King Saud University - Computer and Information Sciences,2024,36(8):102163-102163.
[6] Valle G L ,Marcos L Á M ,Rodríguez M J . Financial boundary conditions in a continuous model with discrete-delay for pricing commodity futures and its application to the gold market[J].Chaos, Solitons and Fractals: the interdisciplinary journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena,2024,187115476-115476.
[7] Ahmad S ,Becheikh N ,Kolsi L , et al. Uncovering the stochastic dynamics of solitons of the Chaffee–Infante equation[J].Scientific Reports,2024,14(1):19485-19485.
[8] Xia X . Random Processes and Their Applications in Financial Mathematics[J].International Journal of Educational Teaching and Research,2024,1(1).
[9] Maloumian N . Wandering Drunkards Walk after Fibonacci Rabbits: How the Presence of Shared Market Opinions Modifies the Outcome of Uncertainty[J].Entropy,2024,26(8):686-686.
[10] Álvaro Guinea Juliá,Alet Roux. Higher order approximation of option prices in Barndorff-Nielsen and Shephard models[J].Quantitative Finance,2024,24(8):1057-1076.
[11] Conde C I ,Ramírez S L E ,Cumbrera F . Signal processing analysis for detection of anomalies in numerical series[J]. Expert Systems With Applications,2024,255(PD):124708-124708.
[12] Lu W ,Yan L . Dynamic Pricing and Inventory Strategies for Fashion Products Using Stochastic Fashion Level Function[J]. Axioms,2024,13(7):453-453.
[13] Keyser D S ,Gijbels I . Parametric dependence between random vectors via copula-based divergence measures[J]. Journal of Multivariate Analysis,2024,203105336-.
[14] Wolf L F ,Deelstra G ,Grzelak A L . Consistent asset modelling with random coefficients and switches between regimes[J].Mathematics and Computers in Simulation,2024,22365-85.
[15] Simon G ,Ioannis K ,Marcos C T . Generalised shot-noise representations of stochastic systems driven by non-Gaussian Lévy processes[J].Advances in Applied Probability,2024,56(4):1215-1250.
[16] Cáceres G G R ,Rivera P I F ,Gómez A B , et al. An approach to the integral optimization of investment portfolios[J].Journal of Open Innovation: Technology, Market, and Complexity,2024,10(1):100235-.
[17] 高宏,梅圣烽. 金融数学的随机变量假设错误及纠正[J].时代金融,2021,(20):92-95.
[18] 王慧蕾. 金融数学中的随机过程课程教学中的几点思考[J].科技视界,2017,(24):58-59.
[19] 刘伟.基于股票市场的随机过程的统计分析[D].华东师范大学,2007.
[20] 张友兰, 周爱民. 金融数学的研究与进展[J]. 高等数学研究, 2004, 7(4): 53-55.
[21] 蔡吉花, 丛凌博, 徐晶, 等. 金融数学方向《 随机过程》 课程建设的研究与实践[J]. 经济师, 2013 (5): 217-218.
[22] 孙宗岐, 刘宣会. 金融数学概述及其展望[J]. 重庆文理学院学报: 自然科学版, 2010, 29(6): 24-27.
[23] 费为银. 随机理论在连续时间金融市场模型中的应用[J]. 安徽机电学院学报, 2001, 16(2): 1-6.
[24] 王海民, 任九泉. 20 世纪金融数学的若干进展及前瞻[J]. 数量经济技术经济研究, 2001, 18(7): 119-122.
[25] 张开菊. 浅析数学方法在金融学中的应用[J]. 科技创新导报, 2010 (3): 162-162.
[26] 高钦姣, 张胜刚, 贾晓薇. 金融学研究中的数学方法运用举例[J]. 教育现代化 (电子版), 2016 (36): 135-136.
[27] 蔡明超, 孙培源, 经济数学. 金融数学与分析技术[M]. 复旦大学出版社, 2002.
[28] 张进浩. 浅论金融数学研究进展与展望[J]. 时代金融, 2016 (27): 256-256.
[29] 孙富. 金融对数学方法运用的探讨[J]. 农金纵横, 1992 (4): 62-63.
[30] 刘晓宇. 金融数学研究进展与展望[J]. 金融理论与教学, 2015 (3): 30-32.