Topic · Mathematics
← All topicsStochastic Processes
36 articles · 8 checkpoints · 18 deeper reads · 10 reference notes
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The standard model for something on a leash. It wanders randomly like a price, but the further it strays from a home level the harder it is pulled back, which gives it a typical range and a measurable half-life instead of drifting away forever.
The family of random processes that includes both Brownian motion's gentle wandering and sudden jumps, the mathematical umbrella for modeling markets that mostly drift and occasionally gap.
The sum of squared price moves, and the one measurement of a jagged path that settles down instead of blowing up. It is why stochastic calculus needs an extra term, why volatility is observable from a single path, and what a variance swap actually pays.
The mathematics of waiting lines, how many orders are stacked up in a limit order book, how long a queued order waits for a fill, built from a simple rule: things arrive one at a time and leave one at a time.
How to make the best decision at every instant when the future is random: the Hamilton-Jacobi-Bellman equation turns a lifetime of choices into one equation about right now.
A rule of motion for something that is pushed by a predictable force and kicked by a random one at the same time. Writing that rule down gives you a whole distribution of futures instead of a single forecast, and almost every model in derivatives and rates is one of these.
The precise line between an exit rule you can actually execute and one that secretly needs tomorrow's prices. Get it right and the optional stopping theorem does your algebra for you; get it wrong and you have invented look-ahead bias.
How to add up the profit of a strategy that trades against a price so jagged it has no slope anywhere. The trick is that you must choose your position before the move, not after, and that one rule is what makes the whole of stochastic calculus work.
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