A generalized itô’s formula in two-dimensions and stochastic lebesgue-stieltjes integrals

Abstract

In this paper, a generalized Itô’s formula for continuous functions of two-dimensional contin- uous 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 ∇-1 ∇-2 f, and the stochastic Lebesgue-Stieltjes integrals of two parameters. The second derivative ∇-1 ∇-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 Itô integral and the Lebesgue-Stieltjes integral through a type of Itô isometry formula. © 2007 Applied Probability Trust.