On the Properties of the Itô-Henstock Integral

Final Defense
On the Properties of the Itô-Henstock Integral

by Charles D. Broñosa
MS Mathematics Candidate

Date: Friday, 05 July 2024
Time: 4 pm
Venue: Online

Advisers:
Emmanuel A. Cabral, PhD
Ateneo de Manila University
Timothy Robin Y. Teng, PhD
Ateneo de Manila University

Panelists:
Jayrold P. Arcede, PhD
Caraga State University
Elvira P. De Lara-Tuprio, PhD
Ateneo de Manila University
Job A. Nable, PhD
Ateneo de Manila University

In this thesis, a definition of the Itô-Henstock Integral with respect to the Brownian motion W is introduced. Based on this definition, some useful properties such as uniqueness, additivity of the integral, integrability over subitervals, Cauchy criterion, and isometry are shown to hold true also for the Itô-Henstock integral. Furthermore, some characterizations of this integral, such as orthogonal increment property, L² martingale, and AC² property, are also established. These properties are necessary in the study of Itô stochastic integration based on the theory of Henstock integration. As a final result, the thesis ends with an integration by parts formula, where an extended definition of the Itô-Henstock integral with respect to a general process X is introduced. This formula is established using the definition and properties that were devel- oped earlier in the thesis.

Key Words: Itô-Henstock integral, Stochastic integration by parts, Henstock integration