Stochastic Calculus for a Time-changed Semimartingale and the Associated Stochastic Differential Equations

TL;DR

This paper develops a framework for stochastic calculus involving time-changed semimartingales, extending classical Ito calculus and stochastic differential equations to include time-change effects, with applications to Brownian motion.

Contribution

It introduces a condition under which stochastic integrals with time-changed semimartingales are equivalent to time-changed integrals of the original process, extending Ito calculus and SDEs.

Findings

Derived a specialized Ito formula for time-changed semimartingales.

Extended classical SDEs driven by Brownian motion to include time-change effects.

Provided solutions and examples for the new class of stochastic differential equations.

Abstract

It is shown that under a certain condition on a semimartingale and a time-change, any stochastic integral driven by the time-changed semimartingale is a time-changed stochastic integral driven by the original semimartingale. As a direct consequence, a specialized form of the Ito formula is derived. When a standard Brownian motion is the original semimartingale, classical Ito stochastic differential equations driven by the Brownian motion with drift extend to a larger class of stochastic differential equations involving a time-change with continuous paths. A form of the general solution of linear equations in this new class is established, followed by consideration of some examples analogous to the classical equations. Through these examples, each coefficient of the stochastic differential equations in the new class is given meaning. The new feature is the coexistence of a usual drift term along with a term related to the time-change.