Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 35 / 3

Because $W$ is continuous and adapted, the Borel function $\mathrm{sgn}(W)$ is predictable. Its value at zero is $-1$, so its square is exactly one everywhere. Consequently $A$ is a zero-starting continuous local martingale withThe Lévy characterization of Brownian motion therefore makes $A$ a Brownian motion in the same filtration. Here that characterization is the deterministic-clock case of the conditional-exponential argument in question 2.

Apply the Itô formula to $V=W^2$:Since $\sqrt{V_t}=|W_t|$ and $|W_t|\mathrm{sgn}(W_t)=W_t$, including at zero, substitution yieldsThus $W^2$ is a squared Bessel process of dimension one.

Use the common filtration in which the two driving processes are independent Brownian motions and the solutions are adapted. Let $Z=X+Y$, and define predictable bounded weightswith the ratios defined only on $\{Z>0\}$. The nonnegative solutions have $X=Y=0$ when $Z=0$. SetIndependent Brownian drivers have zero cross-variation. Since $h_1^2+h_2^2=1$ even on the zero set, $[B{]}_t=t$, and the Lévy characterization of Brownian motion makes $B$ a Brownian motion. Moreover $\sqrt{Z}{h}_1=\sqrt{X}$ and $\sqrt{Z}{h}_2=\sqrt{Y}$ everywhere. Adding the two original equations therefore proves additivity of independently driven squared Bessel processes:The unit-vector fill on $\{Z=0\}$ is essential: simply dividing by $\sqrt{Z}$ there would leave the Brownian construction undefined.

Localize to compact subintervals of $(0,\infty )$, so that the Itô formula can be applied to $f(z)=\sqrt{z}$. Its derivatives are $f^{\prime }(z)=1/(2\sqrt{z})$ and $f^{\prime \prime }(z)=-1/(4z^{3/2})$, while $d[Z{]}_t=4Z_{t}dt$. Thus, before $\zeta$,Hence the Bessel process equation isThe inverse-power drift is interpreted only while $R>0$; its boundary behaviour is justified next, not assumed in this calculation.

Uniqueness in law means that, for a fixed initial value, every two weak stochastic solutions of the stochastic differential equation have the same distribution as entire solution paths, even if their filtered probability spaces and driving Brownian motions differ. For an equation on $(0,\infty )$, initially consider the stopped path and lifetime at the boundary.

We first justify that the boundary cannot intervene when $\gamma \ge 2$. For any solution of the displayed Bessel equation, let ${\tau }_{\epsilon ,L}$ be its first exit from $(\epsilon ,L)$, where $0<\epsilon <r<L$. Applying Itô to $R^2$ givesThus the annulus exit has finite expectation and is almost surely finite. The one-dimensional generator is $\frac{1}{2}{\partial }_{rr}+\frac{\gamma -1}{2r}{\partial }_r$. Its harmonic scale functions are $\log r$ for $\gamma =2$ and $r^{2-\gamma }$ for $\gamma >2$, as direct differentiation verifies. They are bounded on the stopped annulus, so optional sampling givesBoth tend to zero as $\epsilon \downarrow 0$. A continuous path hitting zero before ${\tau }_L$ would hit every inner level first, so that event has probability zero. Exhausting the upper levels proves that zero is not hit at any finite time. Nor can an upper explosion occur: localization of $R^2$ gives $\mathbb{P}({\tau }_L\le t)\le (r^2+\gamma t)/L^2\to 0$. These are the Hitting-zero classification for a Bessel process arguments needed for the dimension-at-least-two case.

Now suppose $\gamma$ is an integer and take a $\gamma$-dimensional Brownian motion $U$ from any deterministic vector $u$ with $|u|=r$. For $\rho =|U|>0$, differentiation of the Euclidean norm givesDefine the radial noise $\beta ={\sum }_i\int U_i/\rho d{W}_i$, filling the integrand by a fixed unit vector on the zero set, as in part (b). It has bracket $t$ and is a Brownian motion. the Itô formula, initially before a possible zero, givesThe boundary argument just given makes this a global positive weak stochastic solution. The assumed uniqueness in law therefore identifies every solution's path law with that of $|U|$. Orthogonal invariance of Brownian motion makes this radial law independent of the selected point $u$ on the sphere. ConsequentlyThis proves equality of process laws, not merely equality of one-time marginal distributions.