E[X_{n+1}|X_1,...,X_n] = X_n
Image: Charles Simms, Public domain, via Wikimedia Commons
Local martingale
E[X_{n+1}|X_1,...,X_n] = X_n
A martingale is a sequence of random variables where the expected value of the next variable, given all previous variables, is equal to the current variable. This property implies that martingales have no predictable trend over time, making them fundamental in the study of stochastic processes.
Local martingales are a broader class that includes martingales but also allows for certain types of expectation distortion. This distinction is crucial in stochastic analysis, as it helps in understanding and modeling phenomena where the martingale property does not strictly hold.
An example of a martingale is the simple symmetric random walk, where the position at each step is determined by a fair coin flip. If the position at time n is X_n, then the expected position at time n+1, given all previous positions, is X_n. This illustrates the martingale property in a tangible way.
Understanding martingales is essential in stochastic analysis because they provide a foundational concept for modeling random processes without predictable trends.
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