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I've been studying martingales before I start graduate school in statistics, and I'd like to ask, what are the most intriguing or surprising applications of martingales that you've come across, either within mathematics or more applied fields of study?

My main motivation is simply to get a holistic picture of the subject. Some questions hint that knowing that a process is a martingale is not particularly useful if certain calculations are wanted; rather, knowing that a process is a martingale results in certain properties being true which can prove quite useful. Still other questions allude to applications in the creation of algorithms. I'd be interested in learning about any or all of these types of use cases.

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I came across the following game a while ago:

You start with $\$100$. At each timestep $t = 1,...,n$ you flip a coin. If you get heads, you receive $10\%$ of what you currently have, and if you get tails, you pay $10\%$ of what you currently have. What is the expected value of the game as $n \to \infty$?

You may be tempted to say $\$0$, since $-10\%$ at $t = k$ and then $+10\%$ at $t = k+1$ (or the reverse: $+10\%$ at $t = k$ and then $-10\%$ at $t = k+1$) will always make you worse off than before ($0.9 \times 1.1 = 0.99$).

However, if you recognize that at each stage the random variable corresponding to the value of the game at the next timestep is a martingale, it is easy to conclude that the expected value of the game is $\$100$.

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