 Solution

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

The bounded integrand makes

X_{s} = \int_{0}^{s} Y_{u} d W_{u}

⟨ X ⟩_{s} = \int_{0}^{s} Y_{u}^{2} d u \leq C^{2} t (s \leq t) .

Thus

E exp (\frac{1}{2} ⟨ X ⟩_{t}) \leq e^{C^{2} t /2} < \infty.

Z_{s} = exp (X_{s} - ⟨ X ⟩_{s} /2)

. Hence $Z$ is a true martingale with $E Z_{s} = Z_{0} = 1$ . It is strictly positive because its exponent is finite almost surely.

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