[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.
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.
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.
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 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.