Repeated games without a beginning and an end
What you're asking seems to me just a matter of interpretation. Note that $\mathbb Z$ and $\mathbb Z_+$ have the same cardinality. So it makes no substantive difference which index set you use. 

Moreover, in a continuation game with finite history, any strategy can be interpreted as a (possibly non-stationary) Markov strategy in a continuation game with infinitely long history. 

If a history of infinite length is what you really care about, then may be looking into the literature of continuous-time repeated game will provide some insight.