Definition:Supermartingale/Discrete Time/Definition 1
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Definition
Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.
Let $\sequence {X_n}_{n \ge 0}$ be an adapted stochastic process.
We say that $\sequence {X_n}_{n \ge 0}$ is a $\sequence {\FF_n}_{n \ge 0}$-supermartingale if and only if:
- $(1): \quad$ $X_n$ is integrable for each $n \in \Z_{\ge 0}$
- $(2): \quad \forall n \in \Z_{\ge 0}: \expect {X_{n + 1} \mid \FF_n} \le X_n$
Equation $(2)$ is understood as follows:
- for any version $\expect {X_{n + 1} \mid \FF_n}$ of the conditional expectation of $X_{n + 1}$ given $\FF_n$, we have:
- $\expect {X_{n + 1} \mid \FF_n} \le X_n$ almost surely.
Sources
- 1991: David Williams: Probability with Martingales ... (previous) ... (next): $10.2$: Adapted process