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

The strictly increasing continuous clock $[M]$ has a finite continuous inverse ${\tau }_s$. The time change of a continuous process therefore preserves continuity and adaptedness. Moreover,The optional time-change theorem makes $M_{{\tau }_s}$ a continuous local martingale for ${\tilde{\mathcal{F}}}_s={\mathcal{F}}_{{\tau }_s}$, while composition with the increasing map $s\mapsto {\tau }_s$ preserves the finite variation of $A$. Hence $\tilde{X}$ is a continuous semimartingale in the time-changed filtration.

For $Z_t=e^{B_t+at}$, the Itô formula givesThus its local-martingale part is $M_t={\int }_0^t{Z}_s,d{B}_s$, its finite-variation part is $(a+1/2){\int }_0^t{Z}_{s}ds$, and its quadratic variation clock is

Let $C_t={\int }_0^t{Z}_s^{2}ds$ 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 $du=Z_t^{2}dt$ transforms the decomposition in part b intoComparison with the Bessel process equation $d{Y}_u=d{W}_u+(d-1)(2Y_u{)}^{-1}du$ givesThus 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.

The lifetime of the time-changed process isBy the strong law for Brownian motion, $B_t/t\to 0$ almost surely. If $a<0$, the exponent is eventually at most $at<0$, so the integral is finite. If $a>0$, it is eventually positive and grows linearly; if $a=0$, the recurrence of one-dimensional Brownian motion makes $B$ spend infinite total time in, for example, $[-1,1]$, so the integral is infinite. Consequently

For $Y=X^{\alpha }$, the Itô formula and the Bessel equation giveUse the clockand its inverse. The Dambis-Dubins-Schwarz theorem turns the first term into Brownian motion, while division of the drift by the clock rate givesHence $\tilde{Y}$ is a Bessel process of dimensionThis is the Power time change of a Bessel process.