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)).