A Generalized Ito's Formula in Two-Dimensions and Stochastic Lebesgue-Stieltjes Integrals

Chunrong Feng (Loughborough University)
Huaizhong Zhao (Loughborough University)


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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Pages: 1568-1599

Publication Date: December 23, 2007

DOI: 10.1214/EJP.v12-468


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