If W is a Brownian motion, show that is a martingale. Suppose is a filtration satisfying the usual conditions. Show that if Mt is a sub martingale and for all t, then M is a martingale. Let X...

If W is a Brownian motion, show that

is a martingale.

Suppose

is a filtration satisfying the usual conditions. Show that if Mt is a sub martingale and

for all t, then M is a martingale.

Let X be a sub martingale. Show that

if and only if

Prove all parts of Proposition 3.8.

If T_n
is defined by (3.2), show T_n
is a stopping time for each n and T_n
↓ T.

This exercise gives an alternate definition of F_T
which is more appealing, but not as useful. Suppose that {F_t} satisfies the usual conditions. Show that F_T
is equal to the σ-field generated by the collection of random variables Y_T
such that Y is a bounded process with paths that are right continuous with left limits and Y is adapted to the filtration {F_t}.