Definition

is a stochastic process, integrable and adapted to filtration . Then is a martingale if almost surely whenever .

It is natural to think about a martingale as defined in the context of a process evolving in time. Then this definition is very reasonable:

• Integrable: the entire construction is about taking expectations. So these need to exist.
• Adapted: can’t be affected by what happens after time n. It must be defined by what has happened up to time n.
• Expectation condition: if you look at the process at a given time m, the best estimate for what will be in the future is in fact what it is now. As you might expect, it is sufficient that  almost surely for each n.

Motivation

There are many situations where the expected change in a variable over a time period is zero, whatever the value of the variable is at the start. For example, gambling. For illustration, assume we are speculating on the outcomes of tossing a coin repeatedly. You might have a complicated strategy, for example ‘double your stake when you lose’ (the so-called martingale strategy), or anything else. But ultimately, you can’t see into the future. So before every coin toss, you have to decide your stake, based on what’s happened up until now, and you will win or lose with equal probability, so your expected gain is 0, regardless of how you make your stake choice. Thus under any strategy determined without looking into the future (called previsible), the process recording your winnings is a martingale. Continue reading →