Martingale (probability theory)
In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expected value of the next value in the sequence is equal to the present value, regardless of all prior values. Formally, let (Xn)n≥0 be a stochastic process with respect to a filtration (Fn)n≥0. The process (Xn) is a martingale if it satisfies the following conditions:
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Integrability: E[|Xn|] < ∞ for all n. In simpler terms, the expected value of the absolute value of each random variable Xn is finite.
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Adaptation: Xn is Fn-measurable for all n. This means that the value of Xn is known at time n based on the information available up to time n (represented by the filtration Fn).
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Martingale Property: E[Xn+1 | Fn] = Xn for all n. This is the defining property. Given all the information available up to time n (Fn), the conditional expectation of the value at time n+1 (Xn+1) is equal to the value at time n (Xn).
Intuitively, a martingale represents a fair game. Knowing the past history of the game provides no advantage in predicting future outcomes. The expected gain or loss in the next round is zero, given the current state of the game.
The concept of a martingale is fundamental in probability theory and has applications in various fields, including statistics, finance, and physics. Different types of martingales exist, such as submartingales (where E[Xn+1 | Fn] ≥ Xn) and supermartingales (where E[Xn+1 | Fn] ≤ Xn). These represent games that are favorable and unfavorable, respectively.
Important theorems related to martingales include the optional stopping theorem, which allows for the analysis of martingales at random times, and convergence theorems, which establish conditions under which martingales converge to a limit.
The filtration (Fn) represents the information available up to time n. It is a sequence of sigma-algebras, where Fn ⊆ Fn+1 for all n. Often, the filtration is generated by the process itself, i.e., Fn = σ(X0, X1, ..., Xn), which means that the information available at time n is determined by the history of the process up to time n.