Is it possible to offer a trial period for subscriptions?

Yes, it is possible to offer a trial period. The minimum trial period is one month (30 days). The feature is called "First Month Free" and can be added to the subscription system for a small integration fee.

How it works for new subscribers: 

Your subscribers would fill out the subscription registration form and are asked to add a working credit card. They won't be charged until the trial period ends (after one month). 
 

Does this affect your current subscribers?

No, the change would not affect your current subscribers. The trial period would only apply for new subscribers.