Generalized Riemann Approach to Stochastic Integral - An Innovative Approach

Project Number
RP 4/14 TTL

Project Duration
November 2014 - October 2017

Status
In-Progress

Abstract
This project investigates the use of the generalized Riemann approach to the study of stochastic integrals and stochastic differential equations. It aims to extend the knowledge we have used so far in Ito-Henstock integral to the study of other family of stochastic integrals. Further, the theory and application of stochastic differential equations will be interpreted using the generalized Riemann approach, as it was done in the classical non-stochastic cases. As the generalized Riemann integral encompasses a much larger class of integrable functions since it is able to integrate highly oscillatory functions in the classical non-stochastic integration theory, the second part of this research project is to apply the generalized Riemann approach for the exploration of a much wider class of stochastic differential equations. Note that there is high relevance of stochastic differential equations in the real-world and in many other branches and disciplines such as Engineering, Life Science and Finance. Many functions and stochastic processes could be highly oscillating; hence the generalized Riemann approach holds promise for the study. The classical way of teaching and learning stochastic calculus is technically involved, as it involves sophisticated knowledge of measure theory and functional analysis. Being more ambitious, the generalized Riemann approach could be the way to teach the entire stochastic calculus, as it is technically less involved and more encompassing, and requires less pre-requisites. As a product of this research project, we hope to produce a textbook based on sound mathematical research in this project can be developed in this area, thereby developing an undergraduate and early postgraduate mathematics module for the University.

Funding Source
NTU

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