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I've just learned about martingale, but i could not find any reason that we use greater than or equal to sign when we define submartingale.

In stead of using greater than or equal to symbol, can't we use greater symbol and it seems to be more intuitive

so my question is

why $E\left( {{M_{n + 1}}|{F_n}} \right) \ge {M_n}$

instead of $E\left( {{M_{n + 1}}|{F_n}} \right) \gt {M_n}$ ?

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1 Answer 1

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If $M_{n}$ is discrete random variable, then process is submartingale, if it satisfy: $$E[M_{t+1}|M_t] > M_t$$

But if $M_t$ is continuous random variable (which is assumed here), then both the expression $$E[M_{t+1}|M_t] > M_t$$ $$E[M_{t+1}|M_t] \ge M_t$$

are equivalent. You may write in either way. For a continuous variable, $X \ge K $ is same as $X >K$.

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  • $\begingroup$ Wikipedia has non-strict inequality both in the discrete and the continuous case... $\endgroup$ Mar 5, 2016 at 19:28

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