Stochastic

A Generalized Itˆs Formula in o Two-Dimensions and Stochastic Lebesgue-Stieltjes Integrals
Chunrong Feng1,2 , Huaizhong Zhao1
1

2

Department of Mathematical Sciences, Loughborough University, LE11 3TU, UK. C.Feng@lboro.ac.uk, H.Zhao@lboro.ac.uk School of Mathematics and dust Sciences, Shandong University, Jinan, Shandong Province, 250100, China

Summary. A generalized Itˆ mandate for time dependent functions of categoric o continuous semi-martingales is proved. The formula uses the local time of each(prenominal) coordinate process of the semi-martingale, left space and time ?rst derivatives and twinkling derivative ? ? f only which are assumed to be of locally spring varia1 2 tion in certain variables, and stochastic Lebesgue-Stieltjes integrals of two parameters. The two-parameter integral is de?ned as a natural inductance of the Itˆ o integral and Lebesgue-Stieltjes integral through a casing of Itˆ isometry formula. o Keywords: local time, continuous semimartingale, generalized Itˆs formula, o stochastic Lebesgue-Stieltjes integral. AMS 2000 subject classi?cations: 60H05, 60J55

1 Introduction
The classical Itˆs formula for twice di?erentiable functions has been exo tended to less smooth functions by many mathematicians.



Progresses pose been do mainly in one- symmetry beginning with Tanakas pioneering workplace [24] for |Xt | to which the local time was beautifully linked. Further extensions were made to time independent convex functions in [18] and [25]; to the case of perfectly continuous function with the ?rst derivative 1,2 being locally bounded in [3]; to Wloc functions of a Brownian motion in [11] for one dimension and [12] for multidimensions. It was proved in [11] t that f (Bt ) = f (B0 ) + 0 f (Bs )dBs + 1 [f (B), B]t , where [f (B), B]t is the 2 t covariation of the processes f (B) and B and is equal to 0 f (Bs )d? Bs ? t f (Bs )dBs as a di?erence of backward and forward integrals. An inte0 ? gral ?? f (x)dx Lt (x) was introduced in [3]...If you want to get a full essay, order it on our website: Ordercustompaper.com



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