Variation of local time and new extensions to Ito's formula

In this doctoral thesis, first we prove the continuous semimartingale local time Lt is of bounded p-variation in the space variable in the classical sense for any p > 2 a.s., and based on this fact we define the integral of local time in the sense of Young integral, and in the sense of Lyons' rough path integral, so that we obtain the new extensions to Tanaka–Meyer's formula for more classes of f. We also give new conditions to two-parameter Young integral and extend Elworthy–Truman–Zhao's formula. In the final part we define a new integral, i.e. stochastic Lebesgue–Stieltjes integral and extend Tanaka–Meyer's formula to two dimensions.