# Semi-martingale vs. martingale. What is the difference?

Can someone please explain (preferably in layman's terms) what is the difference between semi-martingales and martingales?

I have found the following sentence on Wikipedia: In probability theory, a real valued process X is called a semimartingale if it can be decomposed as the sum of a local martingale and an adapted finite-variation process. I assume semi-martingale is a more general term than martingale. Is that correct?

• First you need to understand the definitions of local martingales and adapted finite-variation processes. This requires a bit of measure theory which is not a very layman-friendly topic. It is true that semimartingales are a generalization of martingales, and in fact are the class of (stochastic) integrators for which the Itô integral is defined. Sep 19, 2016 at 9:30

Martingale and semi-martingale have very precise mathematical definitions, so it's definitely not something easy to understand. I'll try to give some intuition without too much mathematical details.

Martingale is a stochastic process that it's expectation is equal to the current value. This means:

1. No clear trend (like a random process)