Math 546 Assignment 1 (due Tues. Oct. 6)
1. If X = (Xt , t ≥ 0) and Y = (Yt , t ≥ 0) are a.s. right-continuous stochastic processes
such that Y is a version of X, prove that X and Y are indistinguishable.
2. Let (Ft0 )t≥0 be a filtration on a complete probability space (Ω, F, P ) and let N be the
0
set of P -null sets in F. Show that Ft ≡ Ft+
∨ N is right-continuous (and hence satisfies
the usual hypotheses).
0
∨ N iff
Hint: You may assume the standard measure theoretic result that A ∈ Ft+
1
2
i
0
A = (B ∪ N ) − N for some N ∈ N and B ∈ Ft+ .
3. Let (Bt , t ≥ 0) be an (Ft0 )-Brownian motion. Prove it is also an (Ft )-Brownian motion
where Ft = Ft0 .
4 (Time reversal). If B is a d-dimensional Brownian motion with initial law µ and B 0 (t) =
B(1) − B(1 − t), show that B 0 is a standard d-dimensional Brownian motion.
5. [Not to hand in] Let {Xn : n ∈ Z+ } be a reverse submartingale satisfying
inf n E(Xn ) > −∞. Prove that {Xn } is uniformly integrable.
Hint: We proved in class that the result holds for the non-negative reverse submartingale
Xn+ and therefore that
lim sup E(Xn 1(Xn ≥ λ))dP = 0.
λ→∞ n
Next, prove that for λ > 0, n > k
E(Xn 1(Xn ≤ −λ)) ≥ E(Xn ) − E(Xk ) + E(Xk 1(Xn ≤ −λ)).