# 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. – Math1000 Sep 19 '16 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)

For example, if you have the Google stock, and it's \$100. You might assume the daily stock movement is a martingale. The probability of it reaching 110 is the same as the probability of it reaching 90. On average, the most likely stock price after a week is still 100. If you need to make a bet on the most likely stock price after one week of trading, you should go for 100. Note in our example, we don't need to know the past stock price, only today's price is relevant. Since your best prediction is actually today's price, you aren't actually predicting anything.

Mathematically: Semi-martingale is similar to martingale but it's not always a martingale. For example, if you can somehow use the past stock data to predict accurately Google stock price for the first week (and only the first week), it won't be a martingale process. Starting from the second week, the process becomes a martingale again.

Mathematically: You can think semi-martingale like a merge of martingale (more precisely local martingale) and a non-martingale process.