分数 L\'{e}vy过程的随机积分及其驱动的随机微分方程

数学物理学报 ›› 2013, Vol. 33 ›› Issue (6): 1022-1034.

分数 L\'{e}vy过程的随机积分及其驱动的随机微分方程

吕学斌^1,2|戴万阳²

1.南京工业大学理学院应用数学系 |南京 210009|
2.南京大学数学系 |南京 210093

收稿日期:2012-06-24 修回日期:2013-08-21 出版日期:2013-12-25 发布日期:2013-12-25
基金资助:
国家自然科学基金(10971249, 11001051, 10971076, 41101509)和教育部人文社科规划项目(11YJA910001)资助

Stochastic Integration for Fractional L\'{e}vy Processes and Stochastic Differential Equations Driven by Fractional L\'{e}vy Noises

LV Xue-Bin^1,2, DAI Wan-Yang²

1.College of Science, Nanjing University of Technology, Nanjing 210009;
2.Department of Mathematics, Nanjing University, Nanjing |210093

Received:2012-06-24 Revised:2013-08-21 Online:2013-12-25 Published:2013-12-25
Supported by:
国家自然科学基金(10971249, 11001051, 10971076, 41101509)和教育部人文社科规划项目(11YJA910001)资助

摘要/Abstract

摘要：

基于文献[1]对平方可积纯跳的L\'{e}vy过程的白噪声分析, 把由平方可积纯跳的L\'{e}vy过程定义的分数L\'{e}vy过程看作是L\'{e}vy过程轨道的泛函, 将其S -变换意义下的形式导数定义为分数L\'{e}vy噪声, 从而, 定义了分数L\'{e}vy过程的Skorohod积分. 进一步地, 提出了一类由分数L\'{e}vy噪声驱动的Volterra方程并研究了其解的存在唯一性, 同时提出了一类由分数L\'{e}vy噪声驱动的随机微分方程并在线性增长条件及Lipschtz条件下证明其解的存在唯一性.

关键词: 白噪声分析, 分数L\{e}vy过程, Skorohod积分, 随机微分方程

Abstract:

In this paper, based on the white noise analysis of square integrable pure-jump L\'{e}vy process given by [1], we define the formal derivative of fractional L\'{e}vy process defined by the square integrable pure-jump L\'{e}vy process as the fractional L\'{e}vy noises by considering fractional L\'{e}vy process as the generalized functional of L\'{e}vy process, and then we define the Skorohod integral with respect to the fractional L\'{e}vy process. Moreover, we propose a class of stochastic Volterra equations driven by fractional L\'{e}vy noises and investigate the existence and uniqueness of their solutions; In addition, we propose a class of stochastic differential equations driven by fractional L\'{e}vy noises and prove that under the Lipschtz and linear conditions there exists unique stochastic distribution-valued solution.

Key words: White noise analysis, Fractional L\{e}vy processes, Skorohod integral, Stochastic differential equations

中图分类号:

60E07

吕学斌, 戴万阳. 分数 L\'{e}vy过程的随机积分及其驱动的随机微分方程[J]. 数学物理学报, 2013, 33(6): 1022-1034.

LV Xue-Bin, DAI Wan-Yang. Stochastic Integration for Fractional L\'{e}vy Processes and Stochastic Differential Equations Driven by Fractional L\'{e}vy Noises[J]. Acta mathematica scientia,Series A, 2013, 33(6): 1022-1034.

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