Semimartingales

A semimartingale is a stochastic process that can be decomposed into a sum of a local martingale and a predictable process. This class of processes is fundamental in the theory of stochastic calculus, particularly in the study of Itô calculus. The definition is crucial for extending integration and differentiation to random processes. Formally, a stochastic process X is a semimartingale if it can be written as X = M + A, where M is a local martingale and A is a predictable process.

The concept of a predictable process is important. It refers to a process whose value at a given time depends only on information available strictly before that time. Local martingales are processes that behave like martingales locally, meaning that if we stop them at certain random times, they exhibit martingale properties.

Semimartingales provide a unified framework for a wide range of stochastic processes encountered in probability theory and its applications, including Brownian motion (which is a martingale and thus a semimartingale) and processes with jumps. The existence of the Itô-Isami decomposition is a key theorem in this area, stating that any semimartingale can be uniquely represented in this form. This decomposition is essential for defining stochastic integrals with respect to semimartingales, which are used extensively in financial mathematics, filtering theory, and other fields. The theory of semimartingales allows for the development of powerful tools for analyzing and manipulating stochastic differential equations.