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Every sequence that was termed as a doob's martingale, I was able to deduce that it was also a martingale. So here are few of my questions:

1) Is it correct to say that every doob martingale is also a martingale? 2) What is the difference between the two then? When is it better to model a sequence of random variables as a doob martingale and when to model it as just a normal martingale?

I am new to stochastic processes so I am finding it hard to find the difference between the two and not able to find any good material on these topics!

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  • $\begingroup$ @ Tony Clayton : Would please give us of your definition of a Doob martingale. Best regards $\endgroup$
    – TheBridge
    Commented Feb 9, 2016 at 10:00

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Within the "usual" terminology every Doob martingale is a martingale but not vice versa. The Doob martingales are exactly the uniformly integrable martingales.

Here for an integrable r.v. $X$ and a filtration $(\mathcal{F}_t)_{t\in T}$ the corresponding Doob martingale $D_t$ is defined via $D_t:=E(X | \mathcal{F}_t), t\in T$.

A martingale (stochastic process) $X_t$ is uniformly integrable iff

$$ \lim_{k\to\infty}\sup_{t\in T} E(|X_t|\, 1_{|X_t|>k}) =0. $$

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