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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2010

Paper 29

Under <Wiener measure>, $dX_t=dW_t$ and
$$
[M]_t=\int_0^t\frac{ds}{X_s^2}\qquad(t\le\tau).
$$
The definition of the <stochastic exponential> gives
$$
d\log Z_t=\frac{dX_t}{X_t}-\frac{dt}{2X_t^2}.
$$
On the other hand, the <Itô formula> applied to the logarithm gives precisely
$$
d\log X_t=\frac{dX_t}{X_t}-\frac{dt}{2X_t^2}.
$$
Their initial values differ by $\log x$, so
$$
\boxed{\log Z_t=\log X_t-\log x,\qquad Z_t=X_t/x\quad(t\le\tau),}
$$
<almost surely> under <Wiener measure>. In particular,
$$
Z_\tau=\frac Mx\mathbf1_{\{T_M<T_\varepsilon\}}
+\frac\varepsilon x\mathbf1_{\{T_\varepsilon<T_M\}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 29 / 3 / c / Solution

Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 29 / 3 / c / Solution

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Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 29 / 3 / c / Solution

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