It\^o's formula, the stochastic exponential and change of measure on general time scales

TL;DR

This paper extends Itô's formula, stochastic exponential, and Girsanov's change of measure to general time scales, including quantum time scales, broadening stochastic calculus applications.

Contribution

It introduces a unified Itô's formula and stochastic exponential framework on general time scales, including quantum time scales, with applications to Brownian motion.

Findings

Derived Itô's formula for general time scales

Established stochastic exponential expression on time scales

Applied results to Brownian motion on q-time scale

Abstract

We provide an It\^{o}'s formula for stochastic dynamical equation on general time scales. Based on this It\^{o}'s formula we give a closed form expression for stochastic exponential on general time scales. We then demonstrate a Girsanov's change of measure formula in the case of general time scales. Our result is being applied to a Brownian motion on the quantum time scale (q-time scale).