We are given the Bessel process SDE $$dX_t=\frac{\delta -1}{2X_t}dt+ dB_t, X_0>0.$$ Where $B_t$ is a standard Brownian motion, at least until $X_0=0$. We need to solve four problems:
- Show that $M_t=X_t^{2-\delta}$ is a continuous local martingale.
- Let $\tau_a=\inf\{t:X_t=a\}$. Compute $\mathbb{P}(\tau_a<\tau_b)$ for $0<a<X_0<b$.
- Show for $\delta <2, b>1$ how to condition on the event $\tau_b<\tau_0$.
- Describe the law of $X|_A$ where $A=\{\tau_0>\tau_b\}$.
I manage to do the first three questions, but not the fourth one. Here are my answers up to the third one.
- $X_t$ is a semimartingale and by Ito's formula, $$M_t-M_0=(2-\delta)(X_t^{1-\delta },B_t)_t$$ which is a continuous local martingale.
- Let $\delta \neq 2$. Then stopping times of $X$ correspond to stopping times of $M_t$. We may bound $M$ by stopping it, this stopped process is of course a martingale. Using the relations between $M$ and $X$ we get $$\mathbb{P}(\tau_a<\tau_b)=\frac{x_o^{2\delta}-b^{2-\delta}}{b^{2-\delta}-a^{2-\delta}}.$$
- For $\delta<2$ we know that, by taking the limit $a\to 0$ in the formula above, $\mathbb{P}(\tau_0<\tau_b)> 0$. However, $M_{\tau_b \wedge \tau_0}$ takes values in $\{0,b^{2-\delta}\}$, so we can condition by letting $$\mathbb{E}(f | \tau_0>\tau_b)=\mathbb{E}(fM_{\tau_b \wedge \tau_0}).$$
- It is clear that we need to change to measure $\mathbb{Q}(A)=\mathbb{E}(M_{\tau_b \wedge \tau_0} 1_A)$, but how do we do this? Let's first assume that $M$ is positive, for example by considering first $M_{\tau_b \wedge \tau_a}$ where $a$ is positive and small. Let $K_t=(\frac{1}{M},M)_t$, a continuous local martingale. Then $$M_t-[M,K]_t=M_t-\left(\frac{1}{M}\cdot [M]\right)_t=M_t-[(M^{-\frac{1}{2}}\cdot M)].$$ is a local martingale with respect to measure $\mathbb{Q}$. As a covariation process has zero quadratic variation this process must have the same quadratic variation as $M$. This is where I am stuck. Any help or tips are much appreciated.