Here, I give a quick review of the concept of a Martingale. A Martingale is a sequence of random variables satisfying a specific expectation conservation law. If one can identify a Martingale relating to some other sequence of random variables, its use can sometimes make quick work of certain expectation value evaluations.
This note is adapted from Chapter 2 of Stochastic Calculus and Financial Applications, by Steele.
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Definition
| Often in random processes, one is interested in characterizing a sequence of random variables ${X_i}$. The example we will keep in mind is a set of variables $X_i \in {-1, 1}$ corresponding to the steps of an unbiased random walk in one-dimension. A Martingale process $M_i = f(X_1, X_2, \ldots X_i)$ is a derived random variable on top of the $X_i$ variables satisfying the following conservation law\begin{eqnarray} \tag{1}E(M_i | X_1, \ldots X_{i-1}) = M_{i-1}.\end{eqnarray}For example, in the unbiased random walk example, if we take $S_n = \sum_{i=1}^n X_i$, then $E(S_n) = S_{n-1}$, so $S_n$ is a Martingale. If we can develop or identify a Martingale for a given ${X_i}$ process, it can often help us to quickly evaluate certain expectation values relating to the underlying process. Three useful Martingales follow. |
Again, the sum $S_n = \sum_{i=1}^n X_i$ is a Martingale, provided $E(X_i) = 0$ for all $i$.
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The expression $S_n^2 – n \sigma^2$ is a Martingale, provided $E(X_i) = 0$ and $E(X_i^2) = \sigma^2$ for all $i$. Proof: \begin{eqnarray} \tag{2}E(S_n^2 X_1, \ldots X_{n-1}) &=& \sigma^2 + 2 E(X_n) S_{n-1} + S_{n-1}^2 – n \sigma^2\& =& S_{n-1}^2 – (n-1) \sigma^2.\end{eqnarray}