带分数Brown运动和局部线性增长的随机微分方程

Abstract

Abstract:

In this paper, we discuss the existence and uniqueness of a class of stochastic differential equations driven by fractional Brown motion with Hurst parameter H ∈ ($\frac{1}{2}$, 1) and their coefficients are local linear growth. So far, there are several ways to define stochastic integrals with respect to FBM. In this paper, we define stochastic integrals with respect to FBM as a generalized Stieltjes integral. We give the existence and uniqueness theorems respectively for SDEs driven by fractional Brown motion and their coefficients are local linear growth.

Key words: Stochastic differential equations(SDE), Fractional Brownian motion, Generalized Stieltjes integral, Local linear growth, Adapted solution

CLC Number:

O211.63

Qikang Ran. SDE Driven by Fractional Brown Motion and Their Coefficients are Locally Linear Growth[J].Acta mathematica scientia,Series A, 2020, 40(1): 200-211.

Trendmd

References 17

1	Duncan T E , Hu Y Z , Pasik-Duncan B . Stochastic calculus for fractional Brownian motion, I:Theory. SIAM Journal of Control and Optimization, 2000, 38 (2): 582- 612 doi: 10.1137/S036301299834171X
2	Elliott R J , Van Der Hoek J . A general fractional white theory and applications to finance. Mathematical Finance, 2003, 13 (2): 301- 330 doi: 10.1111/1467-9965.00018
3	Hu Y Z , Peng S G . Backward stochastic differential equation driven by fractional Brownian motion. SIAM Journal of Control and Optimization,, 2009, 48 (3): 1675- 1700 doi: 10.1137/070709451
4	肖艳清, 邹捷中. 分数布朗运动驱动下z一致连续的BSDE解的存在性与唯一性. 应用数学学报, 2012, 35 (2): 245- 251 doi: 10.3969/j.issn.0254-3079.2012.02.005
	Xiao Y Q , Zou J Z . Existence and uniqueness of the solution to BSDE driven by fractional Brownian motion with the generator is uniformly continuous in z. Acta Mathematicae Applicatae Sinica, 2012, 35 (2): 245- 251 doi: 10.3969/j.issn.0254-3079.2012.02.005
5	Zähle M . Integration with respect to fractal functions and stochastic calculus I. Prob Theory Relat Fields, 1998, 111, 333- 344 doi: 10.1007/s004400050171
6	Nualart D , Rǎscanu A . Differential equations driven by fractional Brownian motion. Collect Math, 2002, 53, 55- 81
7	Maticiuc L , Nie T Y , Rǎscanu A . Fractional backward stochastic differential equations and fractional backward variational inequalities. Journal of Theoretical Probability, 2015, 28 (1): 337- 395
8	Guerra J , Nualart D . Stochastic differential equations driven by fractional brownian motion and standard Brownian motion. Stochastic Analysis and Applications, 2008, 26, 1053- 1075 doi: 10.1080/07362990802286483
9	冉启康. 分数Brown运动驱动的非Lipschitz随机微分方程. 纯粹数学与应用数学, 2016, 32 (6): 551- 561 doi: 10.3969/j.issn.1008-5513.2016.06.001
	Ran Q K . Non-Lipschitz stochastic differential equations driven by fractional Brownian. Pure and Applied Mathematics, 2016, 32 (6): 551- 561 doi: 10.3969/j.issn.1008-5513.2016.06.001
10	王赢, 王向荣. 一类非Lipschitz条件的Backward SDE适应解的存在唯一性. 应用概率统计, 2003, 19 (3): 245- 251 doi: 10.3969/j.issn.1001-4268.2003.03.004
	Wang Y , Wang X R . Adapted solutions of backward SDE with non-Lipschitz coefficients. Chinese Journal of Applied Probability and Statistics, 2003, 19 (3): 245- 251 doi: 10.3969/j.issn.1001-4268.2003.03.004
11	冉启康. 一类非Lipschitz条件的BSDE解的存在唯一性. 工程数学学报, 2006, 23 (2): 286- 292 doi: 10.3969/j.issn.1005-3085.2006.02.011
	Ran Q K . Existence and uniqueness of solutions on a class of BSDEs with non-Lipschitz coefficient. Chinese Journal of Engineering Mathematics, 2006, 23 (2): 286- 292 doi: 10.3969/j.issn.1005-3085.2006.02.011
12	毛伟. 带有非Lipschitz系数的跳扩散微分方程解的存在性. 华中师范大学学报(自然科学版), 2016, 50 (1): 10- 14 doi: 10.3969/j.issn.1000-1190.2016.01.003
	Mao W . Existence of the solutions to jump-diffusion differential equations with non-Lipschitz coefficients. Journal of Central China Normal University (Natural Sciences), 2016, 50 (1): 10- 14 doi: 10.3969/j.issn.1000-1190.2016.01.003
13	Albeverio S , Brzeniak Z , Wu J L . Existence of global solutions and invariant measures for stochastic differential equations driven by Poisson type noise with non-Lipschitz coefficients. Journal of Mathematical Analysis and Applications, 2010, 371, 309- 322 doi: 10.1016/j.jmaa.2010.05.039
14	Majka M B. A note on existence of global solutions and invariant measures for jump SDEs with locally one-sided Lipschitz drift. 2016, arXiv: 1612.03824[math.PR]
15	Samko S G , Kilbas A A , Marichev O I . Fractional Integrals and Derivatives:Theory and Applications. Yvendon:Gordon and Breach, 1993, 77- 82
16	张骅月.分数布朗运动及其在保险金融中的应用[D].天津:南开大学, 2007
	Zhang H Y. Fractional Brownian Motion and Its Application in Insurance Finance[D]. Tianjin: Nankai University, 2007
17	张卫国, 肖炜麟. 分数布朗运动下股本权证定价研究:模型与参数估计. 北京: 科学出版社, 2013
	Zhang W G , Xiao W L . Equity Warrant Pricing Under Fractional Brownian Motion:Model and Parameter Estimation. Beijing: Science Press, 2013

[1]	Chen Yong,Li Ying,Sheng Ying,Gu Xiangmeng. Parameter Estimation for an Ornstein-Uhlenbeck Process Driven by a Type of Gaussian Noise with Hurst Parameter $H\in (0,\frac{1}{2})$ [J]. Acta mathematica scientia,Series A, 2023, 43(5): 1483-1518.
[2]	Chen Yong,Gu Xiangmeng. An Improved Berry-Esséen Bound of Least Squares Estimation for Fractional Ornstein-Uhlenbeck Processes [J]. Acta mathematica scientia,Series A, 2023, 43(3): 855-882.
[3]	Yi Ding,Jingjun Guo. Pricing Asian Options Under Time-Changed Mixed Fractional Brownian Motion with Transactions Costs [J]. Acta mathematica scientia,Series A, 2021, 41(4): 1135-1146.
[4]	Liping Xu,Zhi Li. Transportation Inequalities for Mixed Stochastic Differential Equations [J]. Acta mathematica scientia,Series A, 2021, 41(1): 227-236.
[5]	Liang Wu. Hurst Parameter Under Finite Second Moment and Under Heavy Tails [J]. Acta mathematica scientia,Series A, 2020, 40(4): 1072-1082.
[6]	Liheng Sang,Zhenlong Chen,Xiaozhen Hao. Smoothness for the Renormalized Self-Intersection Local Time of Bifractional Brownian Motion [J]. Acta mathematica scientia,Series A, 2020, 40(3): 796-810.
[7]	Zhaoqiang Yang. Pricing European Lookback Option in a Special Kind of Mixed Jump-Diffusion Black-Scholes Model [J]. Acta mathematica scientia,Series A, 2019, 39(6): 1514-1531.
[8]	Jing Cui,Qiuju Liang,Nana Bi. Asymptotic Stability of Impulsive Neutral Stochastic Functional Differential Equation Driven by Fractional Brownian Motion [J]. Acta mathematica scientia,Series A, 2019, 39(3): 570-581.
[9]	ZHANG Hua-Yue, CHEN Wan-Hua, QU Li-An. Dynamic Below-Target Semi-Variance Risk Measure in a Fractional \|Black-Scholes Market [J]. Acta mathematica scientia,Series A, 2011, 31(6): 1674-1682.
[10]	Liu Shaoyue; Yang Xiangqun. Optimal Portfolio in a Fractional Black-Scholes Model with Arbitrary Hurst Parameter [J]. Acta mathematica scientia,Series A, 2008, 28(4): 742-746.

SDE Driven by Fractional Brown Motion and Their Coefficients are Locally Linear Growth

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References 17

Related Articles 10

Metrics

Comments

Recommended 10