Survival in time (Kaplan Meier) when start time is unknown: is it possible and what methods exist? Is it possible to do survival analysis if one does not know the time the studied subject has already been ‘at risk’?
This is question is more theoretical than practical, as in practice you would generally know the start time of a subject's inclusion in a study. However, you could imagine a situation in which you don't know how long a subject has been at risk.
Suppose an imaginary and nonsensical study:
Let's study the survival of humans until death. We include people, but don't know everybody's age or date of birth. After regular time periods we check if they are still alive or not until some end time of the study. What can we do with the subjects that have been at risk for an unknown time, but have been at risk for a certain amount of time within the time scope of the study?
To me this sounds like some sort of left censoring situation in which you know the subject was alive before the study, but don't know for how long.
In other words: right censoring happens when the subject is still at risk for an unknown time after the study ends. Looking at the other end of the time spectrum, the beginning of the study, can a subject be at risk for an unknown time before the study starts? (Note: I'm not talking about left censoring, because that would mean a failure happens before the start of the study, which is not the study I'm describing.)
In practice, one often does not know the exact inclusion time ($t_0$), but it often isn't important either. E.g. survival after some treatment: the treatment is given over the course of a roughly known period of time, but the exact $t_0$ is often rounded to the nearest hour/day/week. E.g. surgery happening in the morning or evening could both happen on day zero. This is not a problem, because the survival time is often done on a much larger time scale (e.g. weeks/months/years), thus the error of $t_0$ is of negligible order. The problem I describe is the situation when the error of $t_0$ is of large (and unknown) order compared to the range of survival times.
Is it possible to analyse survival when you don't know when your observation started? Do methods exist for this kind of analysis?
Simply excluding this data is an option, but could be considered a waste. The time these unknown-$t_0$-subjects spend at risk in the study does provide you with information. Some of them are at risk for some time and some of them might fail: what can you do with this information?
 A: I am not sure I understand your question completely, but I guess that, as long as you have a minimal period of exposure and a label to tell whether the patient has suffered the outcome, then you can proceed with the analysis.
Note indeed that the proportional hazard assumption implies that hazard ratios are constant over time, and so the differences in exposure would not meaningfully impact on effect estimates if this assumption is true (or held true).
Otherwise, you could use a dummy variable for multivariable adjustment, to highlight which patients miss their actual start date.
Consider as well that your scenario is not hypothetical nor paradoxical, and is typically approached with standard analytical methods. For instance, in an observational study on heart failure, patients are followed over time since enrolment, and outcomes appraised accordingly (for instance comparing the prognostic impact of body mass index). However, time to event is not computed as time since birth (not meaningful) nor time since heart failure occurrence (often slippery to identify), but simply and pragmatically (albeit imperfect) time since enrolment.
