How can there be a current without a voltage?
You are mixing up steady-state solutions with transient solutions. The steady state solution where sinusoidal voltage leads sinusoidal current exists only after the sine wave input has been there for a very long time. That means that there was voltage just before you measure the current. 

If you have an ideal inductor (or a non-ideal one made with superconducting wire) the DC current can indeed be non-zero with exactly zero voltage but that energy has to get in there somewhow, either by applying a transient voltage or by inducing it.