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2007 A Generalized Ito's Formula in Two-Dimensions and Stochastic Lebesgue-Stieltjes Integrals
Chunrong Feng, Huaizhong Zhao
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Electron. J. Probab. 12: 1568-1599 (2007). DOI: 10.1214/EJP.v12-468


In this paper, a generalized It${\hat {\rm o}}$'s formula for continuous functions of two-dimensional continuous semimartingales is proved. The formula uses the local time of each coordinate process of the semimartingale, the left space first derivatives and the second derivative $\nabla _1^- \nabla _2^-f$, and the stochastic Lebesgue-Stieltjes integrals of two parameters. The second derivative $\nabla _1^- \nabla _2^-f$ is only assumed to be of locally bounded variation in certain variables. Integration by parts formulae are asserted for the integrals of local times. The two-parameter integral is defined as a natural generalization of both the Ito integral and the Lebesgue-Stieltjes integral through a type of It${\hat {\rm o }}$ isometry formula.


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Chunrong Feng. Huaizhong Zhao. "A Generalized Ito's Formula in Two-Dimensions and Stochastic Lebesgue-Stieltjes Integrals." Electron. J. Probab. 12 1568 - 1599, 2007.


Accepted: 23 December 2007; Published: 2007
First available in Project Euclid: 1 June 2016

zbMATH: 1191.60067
MathSciNet: MR2365878
Digital Object Identifier: 10.1214/EJP.v12-468

Primary: 60H05
Secondary: 60J55

Keywords: continuous semimartingale , generalized It$hat {rm o}$'s formula , Local time , stochastic Lebesgue-Stieltjes integral

Vol.12 • 2007
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