Math 546 Assignment 4 (due November 12)
1. [Proof of Proposition 4.24 (a,b).] Let M ∈ cM0,loc .
(a) Prove that if E([M ]t ) < ∞ for all t > 0, then M is a square integrable
(i.e., E((Mt )2 ) < ∞ for all t) (Ft )-martingale and Mt2 − [M ]t is an (Ft )-martingale.
(b) Prove that for any a, b > 0, P (Mt∗ ≥ a, [M ]t ≤ b) ≤ ab2 . Hint: One approach is to
consider M T , where T = inf{t ≥ 0 : [M ]t ≥ b}.
2. [Proof of Theorem 4.22] If M, N ∈ cM0,loc , define [M, N ] = 1/4([M + N ] − [M − N ])
Pn2n
and let An (t) = i=1 (M (tni ) − M (tni−1 ))(N (tni ) − N (tni−1 )), where tni = i2−n ∧ t.
(a) Show that [M, N ] is the unique process in F V0 s.t. M N − [M, N ] ∈ cM0,loc .
(b) Show that for any K > 0, supt≤K |An (t) − [M, N ]t | → 0 in probability.
3. Assume B is a standard one-dimensional (Ft )-Brownian motion and H is a continuous,
adapted, and bounded process. Prove that for each t ≥ 0,
R t+h
Hs dBs
converges in probability to Ht as h ↓ 0.
(B(t + h) − B(t))
t
4. Let B be a standard (F√
t )-Brownian motion and τ be an N-valued r.v. which is F0 measurable and satisfies E( τ ) = ∞. Let M (t) = B(τ t).
(a) Show that for each t ≥ 0, τ t is an (Ft )-s.t.
(b) Show that M is a continuous (Fτ t )-local martingale but not a martingale.