Prove the following which we shall need later: if is a UI family of random variables such that a.s. as Then (and thus by Exercise 3.20). A proof of these standard results can be found in Rogers...

Prove the following which we shall need later: if

is a UI family of random variables such that

a.s. as

Then

(and thus

by Exercise 3.20).

A proof of these standard results can be found in Rogers and Williams (1994).

Given any uniformly integrable martingale, we can identify it with an L₁
random variable

. Conversely, given any random variable X ∈ L₁
we can define a UI martingale via

For each t, the random variable M_t
in (3.1) is not unique, but if M_t
and M∗ t both satisfy (3.1) then

a.s. That is, if M and M∗ are two martingales satisfying (3.1) for all t, they are modifications of each other. By, if

is right-continuous and complete, and we will insist on this in practice, then there is a unique (up to indistinguishability) martingale M satisfying (3.1) that is cadllag.