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

Pathwise view on solutions of stochastic differential equations

The Ito-Stratonovich theory of stochastic integration and stochastic differential equations has several shortcomings, especially when it comes to existence and consistency with the theory of Lebesque-Stieltjes integration and ordinary differential equations. An attempt is made firstly, to isolate the path property, possessed by almost all Brownian paths, that makes the stochastic theory of integration work. Secondly, to construct a new concept of solutions for differential equations, which would have the required consistency and continuity properties, within a class of deterministic noise functions, large enough to include almost all Brownian paths. The algebraic structure of iterated path integrals for smooth paths leads to a formal definition of a solution for a differential equation in terms of generalized path integrals for more general noises. This suggests a way of constructing solutions to differential equations in a large class of paths as limits of operators. The concept of the driving noise is extended to include the generalized path integrals of the noise. Less stringent conditions on the Holder continuity of the path can be compensated by giving more of its iterated integrals. Sufficient conditions for the solution to exist are proved in some special cases, and it is proved that almost all paths of Brownian motion as well as some other stochastic processes can be included in the theory