  We study online auction settings in which agents arrive and depart
dynamically in a random (secretary) order, and each agent's private type
consists of the agent's arrival and departure times, value and budget. We
consider multi-unit auctions with additive agents for the allocation of both
divisible and indivisible items. For both settings, we devise truthful
mechanisms that give a constant approximation with respect to the auctioneer's
revenue, under a large market assumption. For divisible items, we devise in
addition a truthful mechanism that gives a constant approximation with respect
to the liquid welfare --- a natural efficiency measure for budgeted settings
introduced by Dobzinski and Paes Leme [ICALP'14]. Our techniques provide
high-level principles for transforming offline truthful mechanisms into online
ones, with or without budget constraints. To the best of our knowledge, this is
the first work that addresses the non-trivial challenge of combining online
settings with budgeted agents.
