Stochastic Process 4 -- Martingale
Definition and Theorem in this chapter mainly comes from Stochastic Processes, Second Edition, Sheldon M. Ross
Definition:
A stochastic process \(\{Z_n\}_{n\ge 1}\) is said to be a martingale process if \[ E[|Z_n|] < \infty, \forall n \] and \[ E[Z_{n+1} \mid Z_1, Z_2, ..., Z_n] = Z_n. \] Martingale can interpret fair game in this way. If we consider \(Z_n\) as fortune after \(n\)-th game, then we know that the expected fortune after \(n+1\)-th game is equal to his fortune after \(n\)-th game.
From definition we immediately get that \[ E[Z_n] = E[Z_1], \forall n. \]
Definition(Random Time and Stopping Time):
The positive integer-valued, possibly infinite, random variable \(N\) is said to be a random time for the process \(\{Z_n, n \ge 1\}\) if event \(\{N = n\}\) is determined by random variables \(Z_1, ..., Z_n.\)
If further \(P(N < \infty) = 1,\) then the random variable \(N\) is said to be a stopping time.
Definition(Stopped Process):
Let \(N\) be a random time for the process \(\{Z_n, n \ge 1\}\) and let \[ \bar{Z_n} = \left\{ \begin{array}{cc} Z_n & \text{if} \quad n \le N \\ Z_N & \text{if} \quad n > N \end{array}\right. \]\(\{\bar{Z_n}, n \ge 1\}\) is called a stopped process.
Proposition: If \(\{Z_n\}\) is a martingale, then so is \(\{\bar{Z_n}\}.\)
Theorem(Optional Stopping Theorem):
Let \(\{Z_n, n \ge 1\}\) be a martingale and \(N\) be a stopping time. If either (i) \(\bar{Z_n}\) is uniformly bounded, or (ii) \(N\) is bounded, or (iii) \(E[N] < \infty,\) and there is an \(M < \infty\) such that \[ E[|Z_{n+1} - Z_n| Z_1,...,Z_n] < M. \]then \(E[\bar{Z_n}] \rightarrow E[Z_N]\) as \(n \rightarrow \infty.\) Thus, \[ E[Z_N] = E[Z_1]. \]
What does this theorem tell us?
In a fair game, if a gambler uses a stopping time to decide when to quit, then his expected final fortune is equal to his expected initial fortune. For application, please refer to [[Expectation of Stopping Time of Martingale Problems]].
Definition:
A stochastic process \(\{Z_n\}_{n\ge 1}\) having \(E[|Z_n|] < \infty, \forall n\) is said to be a submartingale process if \[ E[Z_{n+1} \mid Z_1, Z_2, ..., Z_n] \ge Z_n, \]is said to be a supermartingale if \[ E[Z_{n+1} \mid Z_1, Z_2, ..., Z_n] \le Z_n. \]
From definition we immediately have for submartingale, \[ E[Z_{n+1}] \ge E[Z_n], \]and for supermartingale, \[ E[Z_{n+1}] \le E[Z_n]. \]
Theorem(Martingale Convergence Theorem):
If \(\{Z_n\}_{n\ge 1}\) is a martingale such that for some \(M < \infty,\) \[ E[|Z_n|] \le M, \forall n \] then, with probability 1, \(\lim_{n \rightarrow \infty} Z_n\) exists and is finite.
Definition(Uniformly Integrable):
The sequence of random variables \(X_n, n \ge 1,\) is said to be uniformly integrable if for every \(\varepsilon > 0\) there is a \(y_{\varepsilon},\) such that \[ \int_{|x| > y_{\varepsilon}}|x|dF_n(x) < \varepsilon, \forall n, \]where \(F_n\) is the distribution function of \(X_n.\)
Lemma:
If \(X_n, n \ge 1\) is uniformly integrable then there exists \(M < \infty\) such that \(E(|X|) < M, \forall n.\)
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