Math 546 Assignment 3 (due Oct. 29)
1. Assume X is an (Ft )-Poisson process with rate λ > 0 and let Mt = Xt − λt.
(a) Prove that M is and M 2 − X are (Ft )-martingales.
(b) For any constant time T > 0, verify that [M T ] = X T .
2. Let Tk be a non-decreasing sequence of stopping times such that Tk ↑ ∞ and M =
(Mt , t ∈ [0, ∞)) be a cadlag adapted process such that M0 = 0 a.s. and M Tk is a u. i.
martingale for all k. (We call M a local martingale and write M ∈ M0,loc .) If E(M ∗ ) < ∞,
show that M is a u.i. (Ft )-martingale
3. Verify that the space of continuous martingales in M20 , cM20 , is a closed subspace of
M20 .
4. [NOT TO HAND IN] Read Sec. 28 (p. 50-51) on the Kunita-Watanabe inequality. Use
(28.3) on p. 50 to prove that
Theorem 4.23. If M, N ∈ M20 , H ∈ L2 (M ) and K ∈ L2 (N ), then
Z
[H · M, K · N ]t =
Hs Ks d[M, N ]s .
0
(Yes, this is as easy as it seems.)
t