I am trying to explain the differences between the following concepts to someone and I realized I myself am super confused:


Continuous/discrete Markov Process


Continuous/Discrete Markov chains


Markov property : $P\{X\_n=i|X\_{n-1}=j,X\_{n-2}=k,...\}=P\{X\_n=i|X\_{n-1}=j\}$ ?


I used to think: Every process that has Markov property is a Markov Process. 
Every Markov process is a Markov chain and every Markov chain is a Markov process. 


But it seems crazy now when I think about it, because if they are all the same, why there are different names for them? 


And they are continuous (discrete) if their parameter set $T$ is continuous (discrete) regardless of their state space? 


I want to start with homogeneous Markov chain and process too. But since I am already too confused and Wikipedia is making me more confused, I prefer to wait till I get these basic definitions straight first. 


Also if anyone has any suggestions on how to explain these terms w/o causing any confusion I would appreciate it a lot. (I would also contact my SP teacher to teach him that, coz he has clearly taught us bad)


Thanks a lot



Markov Process, Markov Chain
Before adding the Markov property, I think it helps to distinguish between [discrete time](https://en.wikipedia.org/wiki/Discrete-time_stochastic_process) and [continuous time](https://en.wikipedia.org/wiki/Continuous-time_stochastic_process) stochastic processes with discrete or continuous state spaces. (This classification of stochastic processes is nicely summarized [here](https://en.wikipedia.org/wiki/Stochastic_process#Classification).) Suppose that $(S,\Sigma)$ is a measurable space and $(\Omega, \mathcal{F}, P)$ is a given [probability space](https://en.wikipedia.org/wiki/Probability_space). Then an $S$-valued [stochastic process](https://en.wikipedia.org/wiki/Stochastic_process) is a family of $S$-valued random variables $X\_t$ parametrized by a time-like variable $t \in T$ where $T \subset \mathbb{R}$. The space $S$ is the **state space** of the process, and it could be discrete or continuous. If $T=\mathbb{N}$ then $X\_t$ is called a **discrete time** stochastic process, and if $T$ is some interval in $\mathbb{R}$, then $X\_t$ is referred to as a **continuous time** stochastic process. 


A stochastic process with the [Markov property](https://en.wikipedia.org/wiki/Markov_property) is called either a **Markov process** or a **Markov chain**. (It seems more natural to call them the former, but Wikipedia uses the latter.) To be sure, the Markov property states that for any $A \in \Sigma$ and for each $s,t \in T$ with $s<t$, 
$$
P(X\_t \in A \mid \mathcal{F}\_s) = P(X\_t \in A \mid X\_s)
$$ where $\mathcal{F}\_t$ is the filtration of the process up to time $s$. This definition makes perfect sense for $T$ being a discrete or continuous set of times and for $S$ being a discrete or continuous state space. For Markov processes the above classification of stochastic processes is nicely summarized in the following [table](https://en.wikipedia.org/wiki/Markov_chain#Introduction). This basic framework can be used to classify all sorts of Markov processes including: [random walks](https://en.wikipedia.org/wiki/Random_walk), [Brownian motions](https://en.wikipedia.org/wiki/Brownian_motion), [Poisson processes](https://en.wikipedia.org/wiki/Jump_process), [birth-death processes](https://en.wikipedia.org/wiki/Birth%E2%80%93death_process), [diffusion processes](https://en.wikipedia.org/wiki/Diffusion_process), [jump diffusion processes](https://en.wikipedia.org/wiki/Jump_diffusion) and [branching processes](https://en.wikipedia.org/wiki/Branching_process). For an intro to Markov processes, check out: 


* Kirkwood, J. R. **Markov Processes.** Vol. 20. CRC Press, 2015.


