华东师范大学学报(自然科学版) ›› 2012, Vol. 2012 ›› Issue (1): 130-137.

• 应用数学与基础数学 • 上一篇    下一篇

具有单调、H\"{o}lder~连续及可积参数的一维倒向随机微分方程

肖立顺,李慧颖,范胜君   

  1. 中国矿业大学 理学院, 江苏 徐州 221116
  • 收稿日期:2011-04-01 修回日期:2011-07-01 出版日期:2012-01-25 发布日期:2012-01-26

One-dimensional BSDEs with monotonic, H\"{o}lder continuous and Integrable parameters

XIAO Li-shun,LI Hui-ying,FAN Sheng-jun   

  1. College of Science, China University of Mining and Technology, Xuzhou, Jiangsu 221116,  China
  • Received:2011-04-01 Revised:2011-07-01 Online:2012-01-25 Published:2012-01-26

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摘要： 建立了具有可积参数的一维倒向随机微分方程~(BSDE)~解的一个存在唯一性结果, 其中生成元~$g$~关于~$y$~单调且关于~$z$~是~$\alpha-$H\"{o}lder($0<\alpha<1$)~连续的. 利用~Tanaka~公式及~Girsanov~变换建立~BSDE~的~$L^1$~解的一个比较定理, 从而得到解的唯一性. 使用卷积技术给出生成元~$g$~的一个一致逼近序列并借助于它构造出~BSDE~的~$L^1$~解的一个序列, 然后证明其极限即为所需的解, 从而证明解的存在性.

关键词: 倒向随机微分方程, 可积参数, 单调生成元, H\"{o}lder~连续, 存在唯一性

Abstract: This paper established a new existence and uniqueness result for solutions to one-dimensional backward stochastic differential equations (BSDEs) with only integrable parameters, where the generator $g$ is monotonic in $y$ and $\alpha$-H\"{o}lder ($0<\alpha<1$) continuous in $z$. By Tanaka's formula and Girsanov's theorem we established a comparison theorem for solutions in $L^1$ to BSDEs, from which the uniqueness follows. By convolution technique we obtained a uniform approximation sequence of the generator $g$ and then constructed a sequence of solutions in $L^1$ for BSDEs. Finally, we proved the limitation of this sequence of solutions is the desired solution. This proved the existence.

Key words: backward stochastic differential equation, integrable parameters, monotonic generator, H\"{o}lder continuous, existence and uniqueness

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