Nonuniqueness of Stochastic Differential Equation 
Let $B_t$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)\ne 0$ are real valued continuous functions where $|\mu(t,x)|+|\sigma(t,x)|$ is NOT Lipschitz continuous, and
  $$dX_t = \mu(t,X(t))dt+\sigma(t,X(t))dB_t,\ X(0)=X_0.$$
  Can one give an example where the solution $X(t)$ is not unique? Please note $\sigma(t,x)\ne 0$.

I know for $\sigma=0, \mu(t,x)=\sqrt{x}, X_0=0$, this ordinary differential equation does not have a unique solution. I would think a nonzero $\sigma(t,x)$ would only make the nonuniqueness still worse. How does one show that? 
 A: Check out Girsanov's paper from 1962 (here or here) for the proof of why his counterexample works; he gives the counterexample:

$$ dX_t = \frac{|X_t|^{\alpha}}{1+|X_t|^{\alpha}}dB_t, \quad 0\le \alpha < \frac{1}{2} $$

where $B_t$ is Brownian motion/standard Wiener process.
A: Here is another answer that is more straightforward -- as @cccccp rightly points out, my previous example was wrong to ignore the positivity of $\sqrt X_t$. $\mu(t,x)=3x^\frac13,\,\sigma(t,x)=3x^\frac23$. For any $\alpha\ge 0$, $\tau_\alpha:=\inf\{t\ge\alpha: B_t=0\}$, by Ito's Lemma,
$$X_t(\alpha)=\begin{cases}
0, & t<\tau_\alpha \\
B_t^3, &t\ge\tau_\alpha
\end{cases}
$$
is a solution of the above stochastic differential equation. Any odd natural number as the power of $B_t$ would produce the same conclusion. 
Similar examples can be constructed using Ito's Lemma on any analytic functions of $X_t$.
A: One can also consider $X$ defined by $X_t=f(B_t)$ where $f: x \mapsto (\max(x,0))^3$ is a function of class ${\cal C}^2$. An application of Ito's theorem gives
$$
dX_t=3(\sqrt[3]{X_t})^2 dB_t+3\sqrt[3]{X_t} dt;
$$
which proves that the associated stochastic equation (with $0$ initial value) possesses at least two solutions ($0$ and $X$).
