 Solution

ID: past-exam-of-the-stochastic-differential-equations-course-of-the-university-of-oxford/2025/2/b/i/solution

The expression obtained in part (a)(iii) gives a direct deterministic bound. For every fixed

T < \infty

and

0 \leq s \leq T

0 < Z_{s} = exp [\frac{β s}{2} - \frac{β}{2} B_{s}^{2} - \frac{β ^{2}}{2} \int_{0}^{s} B_{u}^{2} d u] \leq e^{βT /2} .

Z

a true martingale on every finite time interval. More explicitly, if

σ_{n}

E [Z_{t \land σ_{n}} ∣ F_{s}] = Z_{s \land σ_{n}}

pass to the limit by dominated convergence, using the displayed bound for

t \leq T

. Therefore

E [Z_{t} ∣ F_{s}] = Z_{s}, E Z_{t} = 1 (0 \leq s \leq t < \infty) .

This establishes exactly the finite-horizon property needed for the subsequent change of measure.

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