Inverse Bessel Process as continuous local martingale Let $B$ be a $n$-dimensional brownian motion. This question shows, that
$$\Big(\sum_{i=1}^n\int_0^t\frac{B^i_s}{||Bs||}dB^i_s,||B_t||\Big)$$
is a weak solution of $dX=\frac{n-1}{2X}dt+dW$.
Now I would like to show, that $||B||^{-1}$ is a continuous local martingale iff $n=3$.
Let $n=3$. Using the SDE, $$||B_t||^{-1}=\int_0^t||B_s||ds+W_t$$
Therefore $||B||^{-1}$ is a continuous local martingale, since this is sum of two continous local martingales? But wouldn't this apply to any other dimension as well?
How could I deal with the other direction?
I'd appreciate some hint or help on this problem. Thanks for your attention, again!
The function $$h:(0,\infty)\rightarrow(0,\infty),\quad h(x):=x^{-1}$$ has derivatives
$$\begin{align}
\frac{\partial h}{\partial x}(x)&amp;=-x^{-2}\\
\quad\frac{\partial^2 h}{\partial^2 x}(x)&amp;=2x^{-3}
\end{align}$$
Using both Ito's lemma and the SDE, one finds
$$\begin{align}
||B_t||^{-1}=h(||B_t||)&amp;=h(||B_0||)-\int_0^t||B_s||^{-2}d||B_s||+\frac{1}{2}\int_0^t2||B_s||^{-3}d[||B||]_s\\
&amp;=h(||B_0||)-\int_0^t\frac{n-3}{2}||B_s||^{-3}ds-\int_0^t ||B_s||^{-2} dW_s
\end{align}$$
The first and last term are local martingales. The middle term is a local martingale iff $n=3$.
The function $$h:(0,\infty)\rightarrow(0,\infty),\quad h(x):=x^{-1}$$ has derivatives
$$\begin{align}
\frac{\partial h}{\partial x}(x)&amp;=-x^{-2}\\
\quad\frac{\partial^2 h}{\partial^2 x}(x)&amp;=2x^{-3}
\end{align}$$
Using both Ito's lemma and the SDE, one finds
$$\begin{align}
||B_t||^{-1}=h(||B_t||)&amp;=h(||B_0||)-\int_0^t||B_s||^{-2}d||B_s||+\frac{1}{2}\int_0^t2||B_s||^{-3}d[||B||]_s\\
&amp;=h(||B_0||)-\int_0^t\frac{n-3}{2}||B_s||^{-3}ds-\int_0^t ||B_s||^{-2} dW_s
\end{align}$$
The first and last term are local martingales. The middle term is a local martingale iff $n=3$.