d X_{t} = d W_{t}

and

[M]_{t} = \int_{0}^{t} \frac{d s}{X _{s}^{2}} (t \leq τ) .

d lo g Z_{t} = \frac{d X _{t}}{X _{t}} - \frac{d t}{2 X _{t}^{2}} .

On the other hand, the Itô formula applied to the logarithm gives precisely

d lo g X_{t} = \frac{d X _{t}}{X _{t}} - \frac{d t}{2 X _{t}^{2}} .

Their initial values differ by

lo g x

, so

lo g Z_{t} = lo g X_{t} - lo g x, Z_{t} = X_{t} / x (t \leq τ),

Z_{τ} = \frac{M}{x} 1_{{T_{M} < T_{ε}}} + \frac{ε}{x} 1_{{T_{ε} < T_{M}}} .

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