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I am aware of the existence of the following questions: How to discretely stochastically simulate a continuous-time Markov chain? Recipe to discretize a continuous- time markov process

I tried to understand them, but frankly, I couldn't.

Like in discreet time Markov chain, I want to find the probability of being in state $j$ at interval $k$ (discretized). I know of the rate matrix and initial probability distribution. How would I do that?

References will be appreciated.

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The following reference perfectly answers my question:

DOYTCHINOV, BOGDAN, and RACHEL IRBY. "TIME DISCRETIZATION OF MARKOV CHAINS." Pi Mu Epsilon Journal 13, no. 2 (2010): 69-82. http://www.jstor.org/stable/24340803.

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