Let B be a Brownian motion. Define Yt = B2 t −t.

(a) Use the fact that a Brownian motion has independent zero-mean increments with variance equal to the length of the time interval to show that Y is a martingale.

(b) Apply Itô’s formula to calculate dY and verify condition (12.5) to show that Y is a martingale. Hint: To verify (12.5) use the fact that E



T 0

B2 t dt



=

T 0

E[B2 t ]dt .

(c) Let dM = θ dB for a Brownian motion B. Use Itô’s formula to show that M2 t −

t 0

(dMs)

2 is a local martingale.

(d) Let dMi = θi dBi for i = 1,2, and Brownian motions B1 and B2. Use Itô’s formula to show that M1tM2t −

t 0

(dM1s) (dM2s)

is a local martingale.