Prove XXXXXXXXXXin Proposition 10.3. Check the consistency of the first two extensions of the definition of stochastic integrals. Show that if M is a continuous square integrable martingale, and T a...

Prove (10.5) in Proposition 10.3.

Check the consistency of the first two extensions of the definition of stochastic integrals.

Show that if M is a continuous square integrable martingale, and T a finite stopping time, then

Suppose that M and L are square integrable martingales, H is predictable and satisfies (10.2), and

  Show that

Sometimes the stochastic integral of H with respect to M is defined to be the square integrable martingale N for which (10.13) holds for all square integrable martingales L.