Introduction to Stochastic Processes
A stochastic process is a collection of random variables indexed by time, . It describes a system evolving under uncertainty, and it is the natural language for prices, rates and volatility.
The classification that matters most is time and state: discrete or continuous in each. A random walk is discrete time; Brownian motion is continuous in both.
The processes to know
Random walk. with independent steps. The discrete prototype of a price series, and it embodies the weak-form efficient market claim: the current level contains everything the history had to say.
Brownian motion . The continuous limit of a random walk, with three defining properties: independent increments, increments normally distributed with variance proportional to elapsed time, and continuous paths. The variance property is where scaling comes from.
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