Explicit solution for a simple SDE?

Question

I'm not well-versed in stochastic calculus so I assume the question might be trivial.

Consider the one dimensional SDE :

$$dX_t = (1-X_t^2)dB_t $$ $$X_0 = x_0 \in [-1,1] $$ Where $B_t$ is a standard Brownian. I know from the context that $X_t \in [-1,1] $ for all $t\geq0$ actually. Then it is easy to see that $X_t \rightarrow -1$ or $1$, as the quadratic variation must be bounded essentially. However, what more can be said about the solution ? Can we get an explicit (or semi-explicit) descritpion of $X_t$ ?

Answer 1

Here we can do a time-change to get a time-changed Brownian motion solution. see here Characterization of martingale diffusions ending in $\{-1,1\}$ This is also done carefully in Shreves-Karatzas section 5.5, where they go into a lot of detail on the existence/uniqueness of these Martingale-diffusions and their time-changes.In 5.4 Theorem they show that the time change method is valid even for continuous $\sigma$ with zeroes.

We have for Brownian motion $\beta_{t}$

$$X_{t}=\beta_{A_{t}}$$

with $A_{t}:=\inf\{s\geq 0: \int_{0}^{s}\sigma(x_{0}+\beta_{s})^{-2}ds=t\}$ and $\sigma(x)=1-x^{2}$.