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

For $w\ne 0$ in ${\mathbb{R}}^3$, the radial function $r(w)=\Vert w\Vert$ hasThe multidimensional Itô formula, initially stopped away from zero, givesThe martingale term has quadratic variation $dt$, since ${\sum }_i(W_t^i/R_t{)}^2=1$. The Lévy characterization of Brownian motion says that a continuous local martingale starting at zero with quadratic variation $t$ is a standard Brownian motion. Consequently the radial equation isOn $[\epsilon ,M]$ the drift $1/r$ is Lipschitz continuous. Extend it to a bounded globally Lipschitz function on the real line. The standard existence and pathwise uniqueness theorem for stochastic differential equations with globally Lipschitz coefficients applies to this extension, and gives uniqueness in law up to the first exit from $[\epsilon ,M]$. Both this radial process and the process in part (a) solve the same equation from $x$. Thuswhere each exit time is computed from its own coordinate path.

For completeness, the radial equation is valid globally: the three-dimensional Bessel process does not hit zero. Apply the Itô formula to $1/R$ on the annulus. Its drift is zero, and the stopped process is a bounded martingale. At the first exit,soThe exit time is finite because one coordinate of the underlying Brownian motion reaches $M$ in finite time. Letting $\epsilon \downarrow 0$ and then considering arbitrarily large $M$ proves that zero is never reached. The radial martingale therefore has quadratic variation $t$ on the whole time axis, completing the Lévy characterization of Brownian motion argument.