semimartingal

A semimartingale is a stochastic process that can be decomposed into the sum of a local martingale and a predictable process. In simpler terms, it is a process that behaves similarly to a martingale, but with an additional drift component. The theory of semimartingales is a fundamental concept in modern probability theory, particularly in the study of stochastic calculus and its applications in finance and other fields.

Local martingales are processes where the expected value of the process at a future time, conditional on the present information, is equal to the present value, but this property only holds up to stopping times. A stopping time is a random time at which the decision to stop the process depends only on the past history of the process. Predictable processes are a class of processes that can be approximated by processes that are known in advance.

The decomposition of a semimartingale into a local martingale and a predictable process is unique. This decomposition is crucial because it allows us to extend the tools of martingale theory, such as Itô calculus, to a broader class of processes. For instance, the Itô integral, a fundamental tool for defining stochastic differential equations, can be rigorously defined for semimartingales.

Semimartingales are widely used in mathematical finance to model asset prices. The Black-Scholes model, for example, assumes that asset prices follow a geometric Brownian motion, which is a type of semimartingale. The theory of semimartingales provides the mathematical framework for analyzing and pricing financial derivatives. Beyond finance, semimartingales find applications in areas like statistical physics, filtering theory, and the study of random walks.