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