Stochastic Processes and Financial Mathematics
(part one)
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4.4 Other stochastic processes
The world of stochastic processes, like the physical world that they try to model, is many and varied. We can make more general kinds of random walk (and urn/branching processes) by allowing more complex rules
for what should happen on each new time step. Those of you who have taken MAS2003 will have seen Poisson processes and Markov chains, which are two more important types of stochastic process. There are
stochastic processes to model objects that coalesce together, objects that move around in space, objects that avoid one another, objects that repeat themselves, objects that modify themselves, etc, etc.
Most (but not quite all) types of stochastic process have connections to martingales. The reason for making these connections is that by using martingales it is possible to extract information about the behaviour of
a stochastic process – we will see some examples of how this can be done in Chapters 7 and 8.
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All the processes we have studied in this section can be represented as Markov
chains with state space \(\N \). It is possible to use the general theory of Markov chains to study these stochastic processes, but it wouldn’t provide as much detail as we will obtain (in Chapters 7 and 8) using martingales.