Adapted Stochastic Process is Martingale iff Supermartingale and Submartingale/Discrete Time
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Theorem
Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \mathop \ge 0}, \Pr}$ be a discrete-time filtered probability space.
Let $\sequence {X_n}_{n \mathop \ge 0}$ be a discrete-time $\sequence {\FF_n}_{n \mathop \ge 0}$-adapted stochastic process.
Then $\sequence {X_n}_{n \mathop \ge 0}$ is a $\sequence {\FF_n}_{n \mathop \ge 0}$-martingale if and only if it is a $\sequence {\FF_n}_{n \mathop \ge 0}$-supermartingale and a $\sequence {\FF_n}_{n \mathop \ge 0}$-submartingale.
Proof
For each $n \in \Z_{\ge 0}$, we have:
- $\expect {X_{n + 1} \mid \FF_n} = X_n$ almost surely
- $\expect {X_{n + 1} \mid \FF_n} \le X_n$ almost surely
and:
- $\expect {X_{n + 1} \mid \FF_n} \ge X_n$ almost surely.
That is:
- $\sequence {X_n}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \mathop \ge 0}$-martingale if and only if it is a $\sequence {\FF_n}_{n \mathop \ge 0}$-supermartingale and a $\sequence {\FF_n}_{n \mathop \ge 0}$-submartingale.
$\blacksquare$
Sources
- 1991: David Williams: Probability with Martingales ... (previous) ... (next): $10.3$: Martingale, supermartingale, submartingale