SDE and martingale

[Mathias]

Hey,
Find the values for p where (dx(t)=(-r+pr+\frac{1}{2}p(p-1)\sigma^2)x(t)dt+p\sigma x(t)dW(t)) is a martingale?
Why can't I say that if the process has zero drift then it's a martingale and solve ((-r+pr+\frac{1}{2}p(p-1)\sigma^2)=0)? or how should I go about this - is the only way, to use the properties of a martingale and show for which p the conditions are fulfilled or is there an easier way?

(The question is a subpart of another question where I found an SDE:
(dY(t)=(pr+\frac{1}{2}p(p-1)\sigma^2)Y(t)dt+p\sigma Y(t)dW(t))
and I have to find the values for p where (exp(-rt)Y(t)) is a martingale so I define (x(t)=exp(-rt)Y(t)) and use Ito, to get
(dx(t)=(-r+pr+\frac{1}{2}p(p-1)\sigma^2)x(t)dt+p\sigma x(t)dW(t)))

 

[quotes]

I guess the reason why the problem is defined by martingale term instead of simply asking for a solution to an equation is that you can simplify the problem using Ito's lemma - like the way you posted in the end.