Can I claim Makrov Chain or Hidden Markov Model are simple Recurrent Neural Networks? If we focus on sequence modeling (discrete time and discrete observations), Can I claim Makrov Chain or Hidden Markov Model are simple Recurrent Neural Networks (Because both of them have "time dependency components")? 
 A: No. A hidden markov model is a graphical model - or a bivariate stochastic process - and a markov chain is a stochastic process. They lack many of the properties that a typical RNN would have -- ability to train via backpropagation and ability to model non-markovian processes. A "node" in a hidden markov model is a random variable, while a node in an RNN has no such meaning. Just about the only way they are related is their ability to model sequential data.
A: No.
An important feature for Markov or Hidden Markov model is the Markovian property. Where given the current, past and future are independent. In the case of the first order Markov model, it can be described as 
$$
P(X_i|X_1,\cdots, X_{i-1})= P(X_i|X_{i-1})
$$
On the other hand, for recurrent neural network, we can view the neural network is a very complicated function that is trying to estimate 
$$
P(X_i|X_1,\cdots, X_{i-1})
$$
using large amount of the data.

In The Unreasonable Effectiveness of Recurrent Neural Networks, Fun with RNN part, we can see the model can generate a valid (but random) xml. 
<page>
  <title>Antichrist</title>
  <id>865</id>
  <revision>
    <id>15900676</id>
    <timestamp>2002-08-03T18:14:12Z</timestamp>
    <contributor>
      <username>Paris</username>
      <id>23</id>
    </contributor>
    <minor />
    <comment>Automated conversion</comment>
    <text xml:space="preserve">#REDIRECT [[Christianity]]</text>
  </revision>
</page>

This would be extremely hard to use Markov or Hidden Markov model on character level. Since we need to remember things happens in far past, e.g.,  <page> and '' tag are 300 + characters apart. If we are building a 300th order Markov model, the transition matrix will be $26^{300}$, even if we only assume there are 26 letters.
