Simplified stochastic calculus via semimartingale representations

TL;DR

This paper introduces a simplified stochastic calculus framework that unifies the treatment of predictable transformations of semimartingales, facilitating easier derivation of relationships among stochastic processes.

Contribution

It presents a new unified stochastic calculus framework for semimartingales, simplifying transformations and enabling derivation of new process relationships.

Findings

Unified treatment of real-valued and complex-valued semimartingales

Simplified rules for predictable transformations

Blueprint for deriving new stochastic process relationships

Abstract

We develop a stochastic calculus that makes it easy to capture a variety of predictable transformations of semimartingales such as changes of variables, stochastic integrals, and their compositions. The framework offers a unified treatment of real-valued and complex-valued semimartingales. The proposed calculus is a blueprint for the derivation of new relationships among stochastic processes with specific examples provided below.