Stochastic calculus for fractional Brownian motion - I. Theory

被引：387

作者：

Duncan, TE ^{[1
]}

Hu, YZ ^{[1
]}

Pasik-Duncan, B ^{[1
]}

机构：

[1] Univ Kansas, Dept Math, Lawrence, KS 66045 USA

来源：

SIAM JOURNAL ON CONTROL AND OPTIMIZATION | 2000年 / 38卷 / 02期

关键词：

fractional Brownian motion; stochastic calculus; Ito integral; Stratonovich integral; Ito formula; Wick product; Ito calculus; multiple Ito integrals; multiple Stratonovich integrals;

D O I：

10.1137/S036301299834171X

中图分类号：

TP [自动化技术、计算机技术];

学科分类号：

0812 ;

摘要：

In this paper a stochastic calculus is given for the fractional Brownian motions that have the Hurst parameter in (1/2, 1). A stochastic integral of Ito type is defined for a family of integrands so that the integral has zero mean and an explicit expression for the second moment. This integral uses the Wick product and a derivative in the path space. Some Ito formulae (or change of variables formulae) are given for smooth functions of a fractional Brownian motion or some processes related to a fractional Brownian motion. A stochastic integral of Stratonovich type is defined and the two types of stochastic integrals are explicitly related. A square integrable functional of a fractional Brownian motion is expressed as an infinite series of orthogonal multiple integrals.

引用

页码：582 / 612

页数：31

共 50 条

[31] Impulsive stochastic fractional differential equations driven by fractional Brownian motion [J].

Abouagwa, Mahmoud ;

Cheng, Feifei ;

Li, Ji .

ADVANCES IN DIFFERENCE EQUATIONS, 2020, 2020 (01)

[32] The application of fractional derivatives in stochastic models driven by fractional Brownian motion [J].

Longjin, Lv ;

Ren, Fu-Yao ;

Qiu, Wei-Yuan .

PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIONS, 2010, 389 (21) :4809-4818

[33] Stochastic fractional evolution equations with fractional brownian motion and infinite delay [J].

Xu, Liping ;

Li, Zhi .

APPLIED MATHEMATICS AND COMPUTATION, 2018, 336 :36-46

[34] On null Controllability of fractional stochastic differential system with fractional Brownian motion [J].

Ahmed, Hamdy M. .

JOURNAL OF CONTROL AND DECISION, 2021, 8 (01) :50-63

[35] Stochastic delay fractional evolution equations driven by fractional Brownian motion [J].

Li, Kexue .

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, 2015, 38 (08) :1582-1591

[36] Nonlocal fractional stochastic differential equations driven by fractional Brownian motion [J].

Jingyun Lv ;

Xiaoyuan Yang .

Advances in Difference Equations, 2017

[37] Impulsive stochastic fractional differential equations driven by fractional Brownian motion [J].

Mahmoud Abouagwa ;

Feifei Cheng ;

Ji Li .

Advances in Difference Equations, 2020

[38] Nonlocal fractional stochastic differential equations driven by fractional Brownian motion [J].

Lv, Jingyun ;

Yang, Xiaoyuan .

ADVANCES IN DIFFERENCE EQUATIONS, 2017,

[39] Approximating a geometric fractional Brownian motion and related processes via discrete Wick calculus [J].

Bender, Christian ;

Parczewski, Peter .

BERNOULLI, 2010, 16 (02) :389-417

[40] Neutral stochastic differential equations driven by Brownian motion and fractional Brownian motion in a Hilbert space [J].

Liu, Weiguo ;

Luo, Jiaowan .

PUBLICATIONES MATHEMATICAE-DEBRECEN, 2015, 87 (1-2) :235-253

← 1 2 3 4 5 →