Let dMi = i dBi for i = 1,2 and Brownian motions B1 and B2. Suppose 1

Let dMi = θi dBi for i = 1,2 and Brownian motions B1 and B2. Suppose

θ1 and θ2 satisfy condition (12.5), so M1 and M2 are finite-variance martingales.

Consider discrete dates s = t0 < t1 < ··· < tN = u for some s < u. Show that covs(M1u − M1s,M2u − M2s) = Es

⎡

⎣

N j=1

(M1tj − M1tj−1 )(M2tj − M2tj−1 )

⎤

⎦ .

Hint: This istrue of discrete-time finite-variance martingales, andthe assumption that the Mi are stochastic integrals is neither necessary nor helpful in this exercise.

However, it is interesting to compare this to (12.30).