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

Use the following form of the Girsanov theorem. If $W$ is Brownian motion, $\vartheta$ is predictable, andis a uniformly integrable martingale with $Z_0=1$, then the probability measure with terminal density $Z_{\infty }$ makes $W_t-{\int }_0^t{\vartheta }_{s}ds$ a Brownian motion. The analogous finite-horizon statement uses the terminal density at that horizon.

Here take $W_t=X_t-x$ and ${\vartheta }_s=1_{\{s<\tau \}}/X_s$. Before $\tau$, $X_s\in [\epsilon ,M]$, so the stochastic integral is well-defined. The assumed uniform integrability of the stopped stochastic exponential givesThus the proposed weighting really defines a probability measure, and the Girsanov theorem gives the Brownian motionunder that measure. In particularAll calculations are first made on this compact stopped interval. The integral involving $1/X$ is naturally defined on $[0,T_0)$; at $T_0$ its quadratic variation may diverge. The density will have a zero limiting extension there, as established below, rather than a finite stochastic logarithm at the hitting time.

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.

Under Wiener measure, $d{X}_t=d{W}_t$ andThe definition of the stochastic exponential givesOn the other hand, the Itô formula applied to the logarithm gives preciselyTheir initial values differ by $\log x$, soalmost surely under Wiener measure. In particular,

Write ${\tau }_{\epsilon }=T_{\epsilon }\wedge T_M$. The stopped Bessel process has the same law as the weighted stopped Brownian motion, so for every bounded measurable functional $F$ of a continuous path,The Bessel process hits $M$ in finite time and never hits zero. Its minimum before that hitting time is strictly positive. Thus for sufficiently small $\epsilon$, depending on the path, its stopped path $R^{{\tau }_{\epsilon }}$ equals $R^{T_M}$ exactly. Under Wiener measure, the events $\{T_M<T_{\epsilon }\}$ increase to $\{T_M<T_0\}$, and on that event the stopped paths eventually equal $X^{T_M}$. The remaining term is bounded by $\epsilon \Vert F{\Vert }_{\infty }/x$. The dominated convergence theorem therefore yields the precise limiting identityThe bounded stopped martingale $X_{t\wedge T_0\wedge T_M}$ givesHence the limiting identity saysThis establishes Brownian conditioning by a stopped Bessel density.

The displayed density in the question requires a stopping convention. On the full, unstopped path space, the probability measureis well-defined, but its paths continue as Brownian motion after $T_M$. Its pushforward by the stopping map $\omega \mapsto \omega (\cdot \wedge T_M(\omega ))$ is the law ${\mathbb{Q}}^M$ of the stopped Bessel process. Equivalently, if ${\mathbb{W}}_x^{0,M}$ denotes the law of Brownian motion stopped at $T_0\wedge T_M$, then the actual stopped laws satisfyLiteral absolute continuity of the stopped law with respect to unstopped Wiener measure is false: paths that become permanently constant at $M$ have probability one for the stopped law and zero under Wiener measure. The same density is also valid on the stopped sigma-algebra of the original coordinate space. Thus the corrected stopped-law formulation proves the intended conclusion without equating these different full-path measures.