Math 546 Assignment 2 (due Tues. Oct. 20)
1. Assume S and T are (Ft0 )-stopping times and Z is an FS0 -measurable random variable.
Show that Z1{S≤T } is FT0 -measurable.
2. (a) Assume T is an (Ft0 )-stopping time and A ∈ FT0 . Show that
TA (ω) =
T (ω) if ω ∈ A;
∞
otherwise,
is also an (Ft0 )-stopping time.
0
(b) Assume {Xt : 0 ≤ t ≤ ∞} is a cadlag (Ft0 )-adapted process, where F∞
= F. If for
0
every (Ft )-stopping time T , XT is an integrable r.v. such that E(XT ) = E(X0 ), show
that X is a u.i. (Ft0 )-martingale.
3. If B is a standard one-dimensional Brownian motion, and√Ta = inf{t ≥ 0 : |Bt | = a}
for a ≥ 0, prove that for all λ ≥ 0, E(exp(−λTa )) = sech( 2λ a), where sech x is the
hyperbolic secant function.
Do this by first showing that M (t) = E(t ∧ Ta ) is a u.i. (FtB )-martingale, where
E(t) = exp(αBt − α2 t/2).
4. Assume {Xt : t ≥ 0} is a continuous (Ft0 )-adapted process taking values in Rd and K is
a compact subset of Rd . Prove that DK (ω) = inf{t ≥ 0 : X(t, ω) ∈ K} is an (Ft0 )-stopping
time. Do this directly without using the Projection Theorem. Hint: K = ∩n K 1/n , where
K 1/n is the open set of points which are less than 1/n away from K. Now write {DK ≤ t}
as an event involving rational times and K 1/n .
5. Law of the iterated logarithm. The goal of this exercise is to prove that for a standard
one-dimensional Brownian motion B,
lim sup
t→0+
Bt
= 1 a.s.,
φ(t)
p
where φ(t) = 2t log log(1/t) for t < e−1 . Let Mt = sups≤t Bs .
Step 1. Let r and c be real numbers such that 1 < 1/r < c2 . Bound the probabilities
P (Mrn−1 > cφ(rn )) when n → ∞ and infer that a.s.,
lim sup
t→0+
Bt
≤ 1 a.s.
φ(t)
Step 2. Show that for each 0 < r < 1 w.p. 1 there are infinitely many n so that
Brn − Brn+1 ≥
√
1 − r φ(rn ).
Conclude that the statement at the beginning of the exercise holds.