Indicate whether or not each of the

following statements is true or false. If it is true, explain why. If it is false,

either explain why or give a counter example. Answers with no explanation

get no credit!

a) If W and B are independent Brownian Motions then the average of W

and B given byX

t

=(1/2)(W

t

+B

t

)

Xt=(1/2)(Wt+Bt)is again a Brownian Motion.

b) If X and Y are martingales then the average of X and Y give byZ

t

=(1/2)(X

t

+Y

t

)

Zt=(1/2)(Xt+Yt)is again a martingale.

c) If X has finite, non-zero quadratic variation : i.e. 0 < [X,X] <

then

X has infinite first variation : i.e. FV (X) =

.