 Solution

ID: past-exam-of-the-stochastic-differential-equations-course-of-the-university-of-oxford/2011/3/c/solution

N_{u}

Q-integrable. For

s \leq u \leq t

, the density formula and conditioning first on

F_{u}

give

E_{Q} [N_{u} ∣ F_{s}] = \frac{E _{P} [ N _{u} Z _{t} ∣ F _{s} ]}{Z _{s}} = \frac{E _{P} [ N _{u} Z _{u} ∣ F _{s} ]}{Z _{s}} = \frac{N _{s} Z _{s}}{Z _{s}} = N_{s} .

The middle equality uses that

N_{u}

F_{u}

-measurable and

Z

is a martingale; the next uses the assumed martingale property of

NZ

. Thus $N$ is a Q-martingale.

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