Hawkes跳扩散模型下的最优投资策略
PDF下载 (100)贾沐真,王 伟.Hawkes跳扩散模型下的最优投资策略[J].宁波大学学报(理工版),2025,38(6):97-104.DOI:10.20098/j.cnki.1001-5132.2025.0528
JIA Muzhen,WANG Wei.Optimal investment strategies portfolio under the Hawkes jump-diffusion model[J].Journal of Ningbo University(Natural Science & Engineering Edition),2025,38(6):97-104.DOI:10.20098/j.cnki.1001-5132.2025.0528
| Title: | Optimal investment strategies portfolio under the Hawkes jump-diffusion model |
| 作者: | 贾沐真, 王 伟 |
| Author(s): | JIA Muzhen, WANG Wei |
| 关键词: | Hawkes跳扩散; 动态规划; 最优投资; 迭代收敛 |
| Keywords: | Hawkes jump diffusion; dynamic programming; optimal investment; iterative convergence |
| 分类号: | O211.9 |
| DOI: | 10.20098/j.cnki.1001-5132.2025.0528 |
| 文献标识码: | A |
| 摘要: | 旨在研究投资者在股票和货币市场账户之间的最优资产配置问题，为此假定股票价格服从Hawkes跳扩散模型，且跳跃幅度满足正态分布，以最大化终端财富效用为目标，运用随机动态规划方法构建相应的Hamilton-Jacobi-Bellman（HJB）方程，并利用Feynman-Kac公式获得值函数的隐式解，再基于Banach不动点定理严格证明了最优投资策略的收敛性，最后采用迭代收敛数值方法求解出最优投资策略的解析解。数值结果证实，Hawkes跳扩散模型的自激发性对最优投资策略存在显著影响。 |
| Abstract: | This paper studies the optimal asset allocation of investors between stock and money market accounts. The price dynamics of risky assets are assumed to follow a Hawkes jump diffusion model, and the jump sizes satisfy a normal distribution. Aiming to maximize the utility of terminal wealth, the corresponding Hamilton-Jacobi-Bellman (HJB) equation is constructed using the stochastic dynamic programming method. The implicit solution of the value function is obtained via the Feynman-Kac formula. Furthermore, the convergence of the optimal investment strategy is rigorously proven using the Banach fixed-point theorem. Finally, the analytical solution of the optimal investment strategy is derived through an iterative convergent numerical method. The numerical results show that the self-exciting property of the Hawkes jump diffusion model has a significant impact on the optimal investment strategy. |
| 参考文献 /References: | [1] MARKOWITZ H. Portfolio selection[J]. The journal of finance, 1952, 7(1):77-91. [2] MERTON R C. Optimum consumption and portfolio rules in a continuous-time model[J]. Journal of economic theory, 1971, 3(4):373-413. [3] BRANGER N, LARSEN L S. Robust portfolio choice with uncertainty about jump and diffusion risk[J]. Journal of banking & finance, 2013, 37(12):5036-5047. [4] WANG P Q, RONG X M, ZHAO H, et al. Robust optimal investment and benefit payment adjustment strategy for target benefit pension plans under default risk[J]. Journal of computational and applied mathematics, 2021, 391:113382. [5] AFHAMI B, REZAPOUR M, MADADI M, et al. A comonotonic approximation to optimal terminal wealth under a multivariate Merton model with correlated jump risk[J]. Applied mathematics and computation, 2023, 444:127808. [6] AÏT-SAHALIA Y, MATTHYS F. Robust consumption and portfolio policies when asset prices can jump[J]. Journal of economic theory, 2019, 179:1-56. [7] LIU H, YONG J, ZHOU X Y. Optimal consumption and portfolio policies with habit formation and a jump- diffusion model[J]. Applied mathematics & optimization, 2011, 63(1):1-36. [8] WEI P Y, YANG C, ZHUANG Y. Robust consumption and portfolio choice with derivatives trading[J]. European journal of operational research, 2023, 304(2):832-850. [9] JOSA-FOMBELLIDA R. LÓPEZ-CASADO P. Optimal investment and benefit strategies for a target benefit pension plan where the risky assets are jump diffusion processes[J]. Insurance mathematics and economics, 2025, 121:100-110. [10] ZHANG H W, DANG D M. A monotone numerical integration method for mean variance portfolio optimiza- tion under jump-diffusion models[J]. Mathematics and computers in simulation, 2024, 219:112-140. [11] HAWKES A G. Spectra of some self-exciting and mutually exciting point processes[J]. Biometrika, 1971, 58(1):83-90. [12] HAWKES A G. Hawkes processes and their applications to finance: a review[J]. Quantitative finance, 2018, 18(2):193-198. [13] ERRAIS E, GIESECKE K, GOLDBERG L R. Affine point processes and portfolio credit risk[J]. SIAM journal on financial mathematics, 2010, 1(1):642-665. [14] AÏT-SAHALIA Y, CACHO-DIAZ J, LAEVEN R J A. Modeling financial contagion using mutually exciting jump processes[J]. Journal of financial economics, 2015, 117(3):585-606. [15] STABILE G, TORRISI G L. Risk processes with non-stationary Hawkes claims arrivals[J]. Methodology and computing in applied probability, 2010, 12(3):415-429. [16] MA Y, SHRESTHA K, XU W D. Pricing vulnerable options with jump clustering[J]. The journal of futures markets, 2017, 37(12):1155-1178. [17] AÏT-SAHALIA Y, HURD T R. Portfolio choice in markets with contagion[J]. Journal of financial economics, 2016, 14(1):1-28. [18] LIU G, JIN Z, LI S M. Household lifetime strategies under a self-contagious market[J]. European journal of operational research, 2021, 288(3):935-952. |
| 备注/Memo: | 收稿日期：2025−05−27 宁波大学学报（理工版）网址：http://journallg.nbu.edu.cn/ 基金项目：国家自然科学基金数学天元基金（12226510）；宁波市自然科学基金（2024J193） 第一作者：贾沐真，硕士研究生，主要研究方向为应用概率统计。E-mail: 2311400025@nbu.edu.cn *通信作者：王 伟，教授，主要研究方向为应用概率统计。E-mail: wangwei2@nbu.edu.cn 宁波大学学报（理工版）网址：http://journallg.nbu.edu.cn/ |
