Suppose you play a game. We model the outcome of the game with the random variable where the probability of winning or losing is . So it is a fair game and .
Suppose now that we play multiple times. We denote by the outcome of the th game. Suppose we bet a piece of gold each time we play. So after the first round we have goldpieces. After the second one we would have .
More generally, after the -th round we would have pieces of gold.
The expected value of in the th round, i.e., after we know , …, would be . Since if we lose, we would have and if we would win, we would have .

In this example, the family of random variables is called a simple random walk. This motivates the following more general definition.


A family of random variables , , … is called a discrete martingale if for all and for the conditional expected value it holds that .

Note that the first assumption basically says that the expected value does exist.

Towards continuity

So what is a continuous martingale? Let us try to generalize our random walk. Let be as before a discrete random walk. Now define .
We need a scaling factor, such that it works. This has to do with bounding the variance and i do not want to explain it. Note that if we let then this stuff converges to something which i will denote by . It is also called Brownian motion, therefore the symbol . Please also note that is almost surely a continuous function. Furthermore because of the scaling we have that is normally distributed with mean and variance , the differences between any two increments of Brownian motion are independent and .


Let , be a family of random variables, a filtration. Then , is called a martingale if for every and furthermore for every .


Let us integrate this Brownian motion stuff. I mean, integrate as in Riemann integration. You know, we just take a discrete integral of the random walk and then do this limit stuff as explained above and then get an integral of Brownian motion.
Since Brownian motion is a stochastic process, the integral will also be a random variable. Therefore we can characterize it by computing its distribution. Obviously should be the integral , if it exists and if the last part of that notation even makes sense. We have to hack around a bit, since only Brownian increments are independent.

Where the last part follows, since the are independent. Moreover as one can show by induction or a search in your favorite collection of mystic formulae.

Letting we get .
We just integrated Brownian motion, bitches.


25 March 2015