Exercise 6.12 (Doob’s Decomposition) For an integrable process {X(t)}

adapted to the filtration {Ft}, define

with D(0) = 0. Show that the process {M(t)} defined by M(t) = Σt k=0 D(k)
is a martingale and the process {A(t)} defined by A(t) = X(t) − M(t) is predictable.

D(t) = X(t)-Et-1[X(t)], t = 1,2,..., T,