Past exam of the stochastic differential equations course of the University of Oxford / 2011 / 3 / a

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- Solution a

Solution

The bounded integrand makes

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

a square-integrable martingale, with quadratic variation

⟨ 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.

Novikov's condition applies to the stochastic exponential

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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