I’ve only had a brief introduction to martingales and was wondering if there are applications of the theory in other areas of mathematics and in real world applications other than in finance (finance being the obvious one)?
Do they come up at all in differential equations, statistics, physics or biology, for example?
A lot of things can be seem as a martingale, personally, I think it is even annoying that good probability bounds appears from such a simple structure as integrability and the condition $$ \mathbb{E}(X_{n+1}| \mathcal{F}_n) = X_n. $$
A beautiful application on graph theory/computer science is the diferential equation method for random graphs developed by Wormald, see this PDF. In chapter four, for example, the author uses martigale bounds and stopping times to study how many time will it take until a huge connected component appears on a random graph. So, if you believe that graphs, and mainly huge graphs, have applications in other areas and in the real world, we are done.