Tanaka's Formula for Wiener Process and Convex Functions

Author: anano basilaia
Keywords: Wiener Process, local time, convex function, Tanaka formula
Annotation:

In the Bachelor thesis the local time of the Wiener process and the corresponding Tanaka's formula is studied. The derivation of this formula is based on the passing to the limit in the classical Ito's formula, when we approximate the absolute value function of x, by smooth (twice continuously differentiable) functions. In the thesis, the Tanaka's formula is generalized for arbitrary univariate convex function as well. Considering the fact that, in general, convex function does not have classical second order derivative, we have to apply the notion of second derivative measure and use the integration by parts formula for Lebesgue-Stieltjes integral. Afterwards, we construct smooth approximations for arbitrary univariate convex function and use them to pass to limit in the classical Ito's formula. Corresponding technique is non-trivial and requires the use of fundamental results of abstract analysis and modern probability theory. The novelty of the thesis is the usage of the notion of the local time of the Wiener process to prove that almost all trajectories of Wiener process have infinite variation on any bounded time interval.