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-jupytext:
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-  text_representation:
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-  display_name: Python 3
-  language: python
-  name: python3
----
-
-
-
-## Markov Dynamics
-
-Let's start by thinking about how to represent continuous time Markov chains
-and how to visualize them.
-
-
-### Informal Definition
-
-Let $S = \{x_1, \ldots, x_n\}$, where each $x_i$ is a real number.
-
-Informally, a continuous time $S$-valued Markov chain $\{X_t\}$ is an $S$-valued stochastic process indexed by $t \in \mathbb R_+$ that has the Markov property.
-
-Having the Markov property means that $\{X_t\}_{t < r}$ and $\{X_t\}_{t \geq r}$ are independent given $X_r$.
-
-Because the state space is discrete, all movements between states takes the
-form of jumps.
-
-We use $J_0, J_1, J_2, \ldots$ to record the jump times, with $J_0 := 0$.
-
-The process is constant between jumps. 
-
-Here is a visualization with $S = \{1, 2, 3\}$.
-
-```{code-cell} ipython3
-XJs = 1, 3, 2, 1
-Js = 0, 0.3, 1.5, 2.1
-n = len(Js)
-
-fig, ax = plt.subplots()
-
-ax.plot(Js, XJs, 'o')
-ax.hlines(XJs, Js[:-1], Js[1:], label='$X_t$')
-ax.vlines(Js, (0, XJs[0], XJs[1], XJs[2]), XJs, alpha=0.25)
-
-ax.set(xticks=Js,
-       xticklabels=[f'$J_{k}$' for k in range(n)],
-       yticks=(0, 1, 2, 3, 4),
-       xlabel='$t$')
-
-ax.legend()
-plt.show()
-
-```
-
-
-
-### The Jump Chain Representation
-
-There is a standard way to represent the process $\{X_t\}$ that is convenient for
-computation.
-
-Let $\{Y_k\}$ with $k=0, 1, \ldots$ record the sequence of distinct values
-for the chain.
-
-(That is, $Y_k = X_{J_k}$ for $k=0, 1, \ldots$)
-
-Then 
-
-$$
-    X_t = Y_{N_t}
-    \quad \text{where} \quad
-    N_t := \sum_{k \geq 0} k \mathbb 1\{J_k \leq t < J_{k+1} \}
-$$ (xfromy)
-
-In particular, we can reconstruct $\{X_t\}$ from the **jump chain sequence** $(J_k, Y_k)$ via {eq}(xfromy).
-
-We can also plot it easily using Matplotlib.
-
-Here's a simulation that illustrates with 
-
-* steps between jumps drawn from an exponential distribution and
-* the sequence $\{Y_k\}$ drawn from a discrete time Markov chain.
-
-```{code-cell} ipython3
-n = 100
-
-# Generate J sequence randomly using exponential step sizes
-J = np.zeros(n)
-T = np.random.exponential(size=n-1)
-J[1:] = np.cumsum(T)
-
-# Generate Y sequence from discrete approximation of an AR(1)
-mc = qe.tauchen(0.5, 0.5, n=10)  
-Y = mc.simulate(n)
-
-fig, ax = plt.subplots()
-ax.step(J, Y, label="$X_t$")
-
-ax.set(xlabel="time", ylabel="state")
-ax.legend()
-plt.show()
-
-```
-
-
-
-
-
-## Implications of the Markov Property
-
-Below we will find out that this particular simulation is very representative.
-
-It's a remarkable fact that *every* continuous time Markov chain on $S$ can be
-constructed as follows:
-
-1. Set $J_0 = 0$ and then $J_{k+1} = J_k + \tau_{k+1}$ where each $\tau_i$ is an independent exponential draw.
-1. Draw $\{Y_k \}$ from a discrete time Markov chain on $S$.
-1. Take $(J_k, Y_k)$ as the jump chain sequence and build $X_t$ as above.
-
-Below, we call this the **jump chain construction**.
-
-
-