Let $C_t=\int_0^tZ_s^2ds$ and $\tau_u=\inf\{t:C_t>u\}$. By the <Dambis-Dubins-Schwarz theorem>, $W_u=M_{\tau_u}$ is Brownian motion. Setting $Y_u=Z_{\tau_u}$ and using $du=Z_t^2dt$ transforms the decomposition in part b into
$$
dY_u=dW_u+\frac{a+1/2}{Y_u}\,du.
$$
Comparison with the <Bessel process> equation $dY_u=dW_u+(d-1)(2Y_u)^{-1}du$ gives
$$
d=2a+2.
$$
Thus an exponential Brownian motion with drift becomes a Bessel process under its quadratic-variation time change; this is an <Exponential Brownian-to-Bessel time change>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 202 / 4 / c / Solution

{c}

For $Z_t=e^{B_t+at}$, use the clock $C_t=\int_0^tZ_s^2ds$. After the inverse time change, the <Dambis-Dubins-Schwarz theorem> gives
$$
dZ_{\tau_u}=dW_u+\frac{a+1/2}{Z_{\tau_u}}du,
$$
so the time-changed process is a <Bessel process> of dimension $d=2a+2$. Its lifetime is $C_\infty$, which is finite almost surely exactly when $d<2$.

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