Stochastic Processes and Financial Mathematics
(part one)
4.2 Urn processes
Urn processes are ‘balls in bags’ processes. In the simplest kind of urn process, which we look at in this section, we have just a single urn (i.e. bag) that contains balls of two different colours.
At time
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1. Draw a ball from the urn, look at its colour, and return this ball to the urn.
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2. Add a new ball of the same colour as the drawn ball.
So, at time
Let
Our first step is to note that
However, a closely related quantity is a martingale. Let
Then
We can think of
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Remark 4.2.1 The calculation of
is written out in full as a second example of the method. In fact, we could simply have divided the equality in (4.4) by , and obtained .
On fairness
It is clear that the symmetric random walk is fair; at all times it is equally likely to move up as down. The asymmetric random walk is not fair, due to its drift (4.2), but once we compensate for drift in (4.3) we do still obtain a martingale.
Then urn process requires more careful thought. For example, we might wonder:
Suppose that the first draw is red. Then, at time
we have two red balls and one black ball. So, the chance of drawing a red ball is now . How is this fair?!
To answer this question, let us make a number of points. Firstly, let us remind ourselves that the quantity which is a martingale is
Secondly, suppose that the first draw is indeed red. So, at
which is of course equal to the proportion of red balls that we had at
Lastly, note that it is equally likely that, on the first go, you’d pick out a black. So, starting from
To sum up: in life there are different ways to think of ‘fairness’ – and what we need to do here is get a sense for precisely what kind of fairness martingales characterize. The fact that