Two mathematical questions

allenishands · May 1st, 2006, 2:19 pm

Could somebody help me to give a proof or a hint to the following two questions ?(1) If X(t) is AR(2), i.e, there exist a1, a2 such that X(t) = a1X(t-1) + a2X(t-2) + e(t) (where e(t) ~ N(0,1)),then prove that Y(t) = (X(t), X(t+1)) is Markovian.(2) If B(t) is a Brownian motion, then the process F(B(t)/sqrt(T-t)) is a martingale(sqrt denotes square root), where F(x) is the CDF of standard normal.I would be appreciated if somebody could give an answer or reply.

twofish · May 1st, 2006, 5:00 pm

Some hints....1) Set X(0)X(1) to q and calculate E[X(2)X(1)] and show that it is only a function of X(0)X(1). There are two key steps. One is that E[e(t) X(t)] is zero for a fixed value of X(t). The second is that you can write X(1)X(1) as a function of X(2)X(1) and X(0)X(1), and then you can put X(2)X(1) on one side of the equals and X(0)X(1) on the other.2) Use symmetry. Calculate F(0) and show that for every path that drifts in one direction, there is an equal and opposite path that drifts in the other.