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I' am reading Introduction to Stochastic Processes by Lawler and am struggling to understand how continuous time, discrete state space processes are defined. Quote from the book,

A (time-homogeneous) continuous-time Markov chain with rates $\alpha$ is a stochastic process $X_t$ taking values in S satisfying

$$P(X_{t+\triangle t}=x|X_t=x)=1-\alpha (x)\triangle t+o(\triangle t)$$ $$P(X_{t+\triangle t}=x|X_t=y)=1-\alpha (y,x)\triangle t+o(\triangle t),y\neq x$$

I really don't understand the usefulness of including the $o(\triangle t)$. I know how $o()$ (little o) is defined, but what is its mathematical usefulness here?

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This prevents bad things from happening, while still allowing wiggle room. Wiggle room is necessary because it is not possible to have $$P(X_{t + \Delta t} = x | X_t = x) = 1 - \alpha(x)\Delta t$$ exactly: the probability would fall outside $[0, 1]$ for large $t$. "Bad things" means that if the extra term did not shrink faster that $\Delta t$, then $\alpha(x)$ might not properly describe the local "rate" of events.

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