Studies of the elderly in the ancient world is a field that's been emerging in the last few decades, though evidence is a bit sparse due to the marginalisation of most elderly people in the ancient world. There was certainly no pension scheme or equivalent, especially for the provinces where citizens were still taxed and did not receive grain subsidies like in Rome. The Imperial government was never much in the habit of giving out money, with half of state expenditure going to upkeep of the army and the rest going to roads, ports, salaries of officials, the grain subsidies, the postal system and court expenses (Southern 76). Something like a pension scheme could not possibly have been afforded, even if the idea ever occurred to the Romans which it probably did not.

Drawing on a review (Shaw) of Parkins (cited in full below), I can tell you that Parkins estimates that 6-8% of the Empire's population was above 60 years of age and that some elderly people could and did live to the maximum human lifespan (around 115). Mortality was much higher in the city of Rome, but in the Italian country people lived longer.

Parkins works off very little evidence to draw his conclusions, but as might be expected he emphasises the importance of pietas in Roman society to argue for the obligations of children to support their parents, even though evidence for specific legal obligations does not exist for Rome. The elderly were certainly not particularly well thought of, as they were not longer relevant to society and they were severely marginalised. According to Shaw, Parkins establishes the general hard fact that for most Romans old age was "depressing, brutish and almost unendurable." (Shaw 2005 304). Shaw adds that there was a "common philosophical judgement t that old age, along with chronic illness and extreme poverty, was held to be one of the justifiable grounds for taking one’s own life." Of Perkins' conclusions, Shaw explicitly states that "there was no system of social assistance for the elderly or any 'retirement' in our sense." The review ends as follows:

" [Shaw] argues, rightly I think, that the heavy ideological exaltation of elderly males, and the honor and respect that was granted to some older persons, was a counterweight to the much less honorable and less happy circumstances in which most of the real elderly had to complete their lives."

So no, there was no pension scheme or program to take care of old people. The elderly, once deprived of their means of income by age and/or infirmity, had to rely upon their families for assistance. This burden must have been understood and accepted by the families of the elderly, under the aegis of the very important Roman value of pietas. There was however no safety net that I am aware of for the elderly who had no family to take care of them: without the kindness of some friends or strangers, they may have had little choice but to die or kill themselves. At least, with 6-8% of the Empire's population being aged 60 or over, and a number of the elderly managing to reach very great ages, it wasn't all doom and gloom.

I suppose the main place to start for more detail on this matter would have to be Parkins, T.G., "Old Age in the Roman World. A Cultural and Social History", Baltimore and London: The Johns Hopkins University Press, 2003. You can get a copy here if you're interested.

Sources:

Shaw, Brent D. Roman Old Age (review). The Classical Review, 55.1, 2005, 302-4.

Southern, Pat. "The Roman Army: A Social & Institutional History)". Oxford: Oxford University Press, 2006.