If X is a stochastic process, let be defined by (1.1), (1.2), and (1.3), respectively. Show that {Ft} is the minimal augmented filtration generated by X . Let be a filtration satisfying the usual...

If X is a stochastic process, let

be defined by (1.1), (1.2), and (1.3), respectively. Show that {F_t} is the minimal augmented filtration generated by X .

Let

be a filtration satisfying the usual conditions and let

be the Borel σ-field on

(1) If X is adapted to

and we define

show that X⁽ⁿ⁾
is progressively measurable for each n ≥ 1.

(2) Use (1) to show that if X is adapted to {Ft} and has left continuous paths, then X is progressively measurable.

(3) If X is adapted to {Ft} and we define

show that for each

the map

to R is measurable with respect to

(4) Show that if X is adapted to

and has right continuous paths, then X is progressively measurable.