On integration with respect to the q-Brownian motion

TL;DR

This paper develops a q-analog of stochastic integration with respect to q-Brownian motion, establishing key properties like the L_2-isometry and Ito formula, extending the theory of stochastic calculus in the q-framework.

Contribution

It introduces a new integration theory for q-Brownian motion using Jackson q-integral, including q-analogs of fundamental stochastic calculus results.

Findings

Established the q-analog of the L_2-isometry.

Derived the Ito formula for polynomial integrands.

Extended the integral to more general functions.

Abstract

For a parameter 0<q<1, we use the Jackson q-integral to define integration with respect to the so called q-Brownian motion. Our main results are the q-analogs of the L_2-isometry and of the Ito formula for polynomial integrands. We also indicate how the L_2-isometry extends the integral to more general functions.