The definition I know of martingale is the following: given a probability space $(\Omega,\mathcal{F},P)$, a stochastic process $\{X_t\}_{t\geq0}$ and a filtration $\{\mathcal{F}_t\}_{t\geq0}$, we say that $\{X_t\}_{t\geq0}$ is a martingale if:

  1. $X_t\in L^1(\Omega)$ for all $t\geq0$,

  2. $X_t$ is $\mathcal{F}_t$-measurable for all $t\geq0$,

  3. $E[X_t|\mathcal{F}_s]=X_s$ for every $0\leq s\leq t$.

I would like to know if that definition could be extended to the case $X_t\geq0$ for all $t\geq0$, because the expectation for nonnegative random variables is well-defined (in $[0,+\infty]$).

Motivation: I read that, if $\{X_t\}_{t\geq0}$ is a martingale, then $\{|X_t|^p\}_{t\geq0}$ is a submartingale (by Jensen's inequality). However, there is no guarantee that $|X_t|^p\in L^1(\Omega)$, so I would like to know if the fact that $|X_t|^p\geq0$ avoids the problem.

  • $\begingroup$ Regarding the question: you could drop part 1) in the definition and replace it with positivity. You have to check (but it is actually true) that you can define the conditional expectation for general positive random variables. Regarding the motivation: usually you would pass to the submartingale $|X|^p$ to see that this has bounded mean, and thus find very nice properties for the martingale $X$. So I guess you wouldn't find immediate application of your definition. $\endgroup$ – Kore-N Nov 8 '16 at 15:27

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