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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