Itô’s Formula, the Stochastic Exponential, and Change of Measure on General Time Scales

Wenqing Hu

doi:10.1155/2017/9140138

Abstract

We provide an Itô formula for stochastic dynamical equation on general time scales. Based on this Itô’s formula we give a closed-form expression for stochastic exponential on general time scales. We then demonstrate 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).

Highlights

The theory of dynamical equation on time scales ([1]) has attracted many researches recently
Attempts of extension to stochastic dynamical equations and stochastic analysis on general time scales have been made in several previous works ([2,3,4,5,6])
In the work [3] the authors mainly work with a discrete time scale; in [2] the authors introduce an extension of a function and define the stochastic as well as deterministic integrals as the usual integrals for the extended function; in [4] the authors make use of their results on the quadratic variation of a Brownian motion ([7]) on time scales and, based on this, they define the stochastic integral via a generalized version of the Itoisometry; in [6] the authors introduce the so-called ∇-stochastic integral via the backward jump operator and they derive an Itoformula based on this definition of the stochastic integral

Summary

Introduction

The theory of dynamical equation on time scales ([1]) has attracted many researches recently. Attempts of extension to stochastic dynamical equations and stochastic analysis on general time scales have been made in several previous works ([2,3,4,5,6]). We present a general Ito’s formula for stochastic dynamical equations under the framework of [2]. We would like to point out that our change of measure formula is different from the continuous process case in that the density function is not given by the stochastic exponential but rather is found by the fact that the process on the time scale can be extended to a continuous process by linear extension. We will work our Itoformula for a Brownian motion on the quantum time scale (q-time scale) case at the last section of the paper.

Set-Up

Stochastic Integrals and Stochastic Differential Equations on Time Scales

Itô’s Formula for Stochastic Integrals on Time Scales

Change of Measure and Girsanov’s Theorem on Time Scales

Application to Brownian Motion on a Quantum Time Scale