[Stochastic Processes] — Martingale


The Martingale is the Stochastic Process representing Brownian Motion. Its name derives from a XVIII sec betting strategy which was wrongly assumed to be a safe win strategy.

Long Version

A little bit of technicality

You already know what Brownian Motion is: a small particle in suspension in water moves around as the result of water molecules hits.

At each time the particle position can be represented with a Random Variable. Let’s take a snapshot of a particular motion at different times and we will get a succession of realizations of the Position Random Variable.

If we keep it general, we will get a succession of Random Variables indexed by time, representing the generic Brownian Motion.

Succession of Random Variables representing the Brownian Motion

So is this a Martingale?

Not yet, in order for this Discrete Time Succession to be a Discrete Time Martingle one property needs to be satisfied: the Markov Property.

Markov Property: the Expected Next Random Variable Value is the Current Value, hence the story behind the present has no effect

The Markov Property means that there is no long term relationship between the past and the present, equivalently the process has no memory effect, hence as a result of this property only the present value of the process influences the expected future value.

The History

The Martingale takes its name from a betting strategy of the XVIII sec regarding games with binary outcome like coin tossing: it basically consisted of doubling the bet at each loss. This looks like a safe bet strategy as apparently it should lead to a final win.

This practically proved to be a bad strategy due to the exponential growth of the wager with the number of lost bets: this very quick growth leads very fast to hit some hard thresholds like the amount of available capital or the maximum wager accepted.