Geometric Brownian motion/stock price

Awatarn

I'm tying to solve a stochastic differential equation of stock price. The equation is
[itex]dX = X(\mu dt + \sigma dW)[/itex]
where [itex]\mu, \sigma[/itex] are constants and greater than zero.
It is easy to show analytically that the expectation value to the solution is
[itex]E[X(t)] = E[X(0)] e^{\mu t}[/itex]
Then I solved this equation numerically by standard Euler method to check my analysis.
I found that if [itex]\sigma[/itex] is less than some particular value, my numerical solution is
consistent with the analytical solution which is increasing with time. The problem is
when [itex]\sigma[/itex] is greater some number, the numerical solution tends to go to zero
instead of increasing with time. Note that [itex]\mu[/itex] is the same.
What happens?

I read another book. They told me that if [itex]\mu > \sigma^2/2[/itex] then
[itex]Prob\{X(t \rightarrow \infty )=\infty \} = 1[/itex],
and if [itex]\mu < \sigma^2/2[/itex]
[itex]Prob\{X(t \rightarrow \infty ) = 0 \} = 1[/itex],
where [itex]X(t)[/itex] is the Ito solution.
For the second case, the probabilistic value is not consistent the expectation value, isn't it?
Can you help me about these two different answers?