Path solution for a SDE

Question

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?

Answer 1

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 $T=\inf\{t\mid W_t=1/X_0\}$.