What does it mean by "the process starts out in state $i$"?

Question

Let $\{X(t): t \geq 0\}$ be a continuous-time Markov chain. Let $T_{i}$ denote the holding time in state $i$. Then we have the following proposition:

Could you please explain what does it mean mathematically by "the process starts out in state $i$"? Thank you so much!

Answer 1

I convert @астон вілла олоф мэллбэрг's comment as answer to close this question.

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I think it means $X(0) = i$. You are using time homogeniety because it does not matter at what time you hit $i$, so WLOG let time itself start running once you hit $i$ : which is the same as saying $X(0) = i$.