# Martingales

# Introduction

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.

## 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 .

## Definition

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

## Integrals

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.