Why Submartingales Use a Non-Strict Conditional Expectation Inequality

Summary

This note asks why the submartingale condition uses a conditional expected future value greater than or equal to the current value, rather than requiring it to be strictly greater. It gives a brief answer distinguishing discrete and continuous random variables, and claims that the strict and non-strict forms coincide in the continuous case because equality has probability zero.

The explanation is only a short exchange and provides no derivation or supporting examples. Its claim needs qualification: continuity of a random variable alone does not generally make conditional equality impossible, and the usual definition allows equality so that constant martingales are also submartingales. The document is useful as a prompt about the definition, but its answer should not be taken as a general equivalence without additional assumptions.

Key ideas

• The standard submartingale condition permits conditional expectation to equal the current value.
• The note claims strict and non-strict inequalities are equivalent for continuous variables.
• Continuity alone does not establish that equality has probability zero.
• Allowing equality includes martingales as submartingales.

Tags

# why do we use greater than or equal to for submartingale?


# why do we use greater than or equal to for submartingale?












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

## Answer by Neeraj (score 0, accepted)

https://quant.stackexchange.com/a/24711

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.