May 7th, 2009, 9:33 pm
Hello! I just found these wonderful forums. I'm self studying with Volume I and II of Shreve.I'm currently studying the material in Chapter 3 of Volume 2. Here is the problem I am working on:Let W(t) be a brownian motion, let F(t) be a filtration for this brownian motion, and 0 <= t. Show that W^3(t) - 3tW(t) is a martingale.I found that E[W^3(t) | F(t)] = W^3(s).I need to show that E[-3tW(t) | F(t)] = -3sW(s), for 0 <= s <= t and i'll be done. The 't' is bothering me... I know I can pull the -3 out of the Expectation, but I'm having trouble showing E[tW(t)|F(t)] = sW(s). It's clear to me that E[W(t) | F(t)] = W(s)... I'm just wondering what property I'm missing in regards to the t inside the expectation with the given filtration. Thanks for your help, this is probably a pretty trivial problem.