Trying to understanding how finite-state space, continuous time Markov Chains are defined

Question

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?

Answer 1

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.