Questions on proving a stochastic process to be a martingale

Question

I need to prove that a stochastic process $M_{t} $to be a martingale, is it necessary and sufficient to prove that $E[M_{t}]=M_{0}$ and if so, can it be proved rigorously? Thank you!

Answer 1

It is necessary but not sufficient. If the only requirement to be a martingale was to have zero expectation, there would not be much point in inventing a special word for it.

As a typical counterexample, let $X$ be a random variable with $P(X=-1) = P(X=1) = 1/2$. Let $M_t = tX$. Then $E[M_t]=M_0=0$ for all $t$, but for example, you can check that $E[M_2 \mid M_1] = 2 M_1 \ne M_1$, so this process is not a martingale.

The "fairness" requirement of a martingale is that no matter what has happened up to time $s$, the (conditional) expected change in the process's value over a future increment of time is 0. In the example above, if $M_1 = 1$ then the process is guaranteed to increase from then on, and the conditional expected change is positive.