Past exam of the stochastic differential equations course of the University of Oxford / 2025 / 2 / b

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.

ii

The relation between

B

and

W

d B_{s} = d W_{s} - β B_{s} d s

with

B_{0} = 0

e^{β s}

d (e^{β s} B_{s}) = e^{β s} d W_{s} .

Hence

B_{s} = \int_{0}^{s} e^{- β (s - u)} d W_{u} = W_{s} - β \int_{0}^{s} e^{- β (s - u)} W_{u} d u .

B

as an explicit function of the continuous path of

W

, using only an ordinary time integral. This inversion holds before changing measure;

W

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