Journal of Statistical Physics

, Volume 148, Issue 3, pp 579–590 | Cite as

Relation of a New Interpretation of Stochastic Differential Equations to Ito Process

  • Jianghong Shi
  • Tianqi Chen
  • Ruoshi Yuan
  • Bo YuanEmail author
  • Ping AoEmail author


Stochastic differential equations (SDE) are widely used in modeling stochastic dynamics in literature. However, SDE alone is not enough to determine a unique process. A specified interpretation for stochastic integration is needed. Different interpretations specify different dynamics. Recently, a new interpretation of SDE is put forward by one of us. This interpretation has a built-in Boltzmann-Gibbs distribution and shows the existence of potential function for general processes, which reveals both local and global dynamics. Despite its powerful property, its relation with classical ones in arbitrary dimension remains obscure. In this paper, we will clarify such connection and derive the concise relation between the new interpretation and Ito process. We point out that the derived relation is experimentally testable.


Stochastic differential equation Boltzmann-Gibbs distribution Fokker-Planck equation Potential function 



The authors would like to express their sincere gratitude for the helpful discussions with Song Xu, Xinan Wang, Yian Ma, Ying Tang. This work was supported in part by the National 973 Project No. 2010CB529200 (P.A.); by the Natural Science Foundation of China No. NFSC91029738 (P.A.) and No. NFSC61073087 (R.Y., J.S. and B.Y.).


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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringShanghai Jiao Tong UniversityShanghaiChina
  2. 2.Shanghai Center for Systems Biomedicine, Key Laboratory of Systems Biomedicine of Ministry of EducationShanghai Jiao Tong UniversityShanghaiChina
  3. 3.Institute of Theoretical PhysicsShanghai Jiao Tong UniversityShanghaiChina

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