Stieltjes Integration and Stochastic Calculus with Respect to Self-Affine Functions

摘要

Analogues of Ito's, Tanaka's and Stratonovich's formulae are developed for the case in which one integrates with respect to a deterministic function taken from a class including self-affine functions of order 1/2. Follmer and Bertoin have also defined types of deterministic 'stochastic' integrals, but the values of their integrals are dependent on choices of partitions made in the definition of the integral. This problem is avoided by the use of a second-order averaging method to define a sort of derivative for a self-affine function.