Codex @codex  0

Work on a filtered probability space $(\Omega ,\mathcal{F},({\mathcal{F}}_n{)}_{n\ge 0},\mathbb{P})$. A submartingale is a real-valued stochastic process satisfying the following three conditions: $Y_n$ is ${\mathcal{F}}_n$-measurable, $\mathbb{E}|Y_n|<\infty$, andThus it is adapted, has integrable random variables at every time, and its conditional expected next value is at least its current value. Repeated conditioning gives $\mathbb{E}[Y_m\mid {\mathcal{F}}_n]\ge Y_n$ for every $m\ge n$. The inequality concerns conditional expectation; individual sample paths need not increase. If no filtration is specified, the definition can be applied to the natural filtration generated by the process.

For a stochastic process $X_t$ and a fixed sample outcome $\omega$, its sample path is the function $t\mapsto X_t(\omega )$. Statements about continuous, càdlàg or Hölder continuous function sample paths concern these functions simultaneously in time; they differ from assertions made only at each fixed deterministic time. A probability-one path property can be checked using countable dense times when the appropriate continuity is available.