According to the Amtrak Tickets Page, you can file an application for a refund on lost Amtrak tickets. If you lost your tickets less than a year ago, you can file an application and Amtrak will hold onto it. From there, you have 5 months to track down your tickets, and if recovered, will be eligible for a refund. If you find your tickets after this 5-month period, they can’t be refunded.