Path solution for a SDE I would like to get help in solving an Ito stochastic equation:
$dY_t=-dW_t \, (Y_t^2+1)$
The process $W_t$ is the standard Brownian motion. 
Is it possible to get a  path solution  solution in terms of Brownian motion integrals?
 A: A strong solution of $\mathrm dY_t=\sigma(Y_t)\mathrm dW_t$ defined for every $t$ small enough is $Y_t=B_{\tau(t)}$ where $B$ is a Brownian motion and $\tau(t)=\inf\{s\mid A(s)\gt t\}$ with 
$$
A(t)=\int_0^t\frac{\mathrm ds}{\sigma(B_s)^2}.
$$
In particular, $Y_t$ is defined for every $t$ such that $\tau(t)$ is finite hence $Y_t$ is defined for every $t$ if and only if $A_\infty=\sup\limits_{t\gt0}A(t)$ is infinite. 
In the present case, $\sigma(x)=1+x^2$ hence $\sigma(x)^2\leqslant4$ on $|x|\leqslant1$ and 
$$
A(t)\geqslant\frac14\,\mathrm{Leb}\{s\leqslant t\mid|B_s|\leqslant1\}.
$$ 
The Brownian motion $B$ is recurrent hence $A_\infty=\infty$ almost surely, QED.
Recall that explosions of solutions of stochastic differential equations can occur due to drift terms, for example the solutions of $\mathrm dX=X^2\mathrm dW+X^3\mathrm dt$ are given, for $t$ small enough, by
$$
\frac1{X_t}=\frac1{X_0}-W_t,
$$
hence $X$ is defined only up to the almost surely finite random time $T=\inf\{t\mid W_t=1/X_0\}$.
