I am beginning to watch a video playlist on the subject of time series analysis, and it seems pretty clear both from notation and some of the terminology (such as "characteristic equation") that time series have a deep connection with recurrence relations.
However, in this video, the time series $y_t = \beta_0 + \beta_1 t + \epsilon_t$ is introduced, which I found strange because it does not relate $y_t$ to any of its lagged inputs. I assume this is because it is a continuous version of a time series, whereas all prior time series discussed have been discrete, but then I would expect something that looks analogous to a differential equation to be the continuous version.
I think I have a pretty good handle on differential equations and recurrence relations. Can someone explain what their connection is to time series analysis, at a high level, so that I might be able to port over some of the intuition?