Complete documentation fixes.

Author: chenwilliam77

docs/src/numerical_algorithms.md

@@ -35,7 +35,7 @@ such that
 ```math
 \begin{aligned}
 0 & = \mu(z_n, y_n) - z_n,\\
-0 & = \xi(z_n, y_n) + \Gamma_5 z_n + \Gamma_6 y_n + \mathscr{V}(z_{n - 1}),\\
+0 & = \xi(z_n, y_n) + \Gamma_5 z_n + \Gamma_6 y_n + \mathcal{V}(z_{n - 1}),\\
 \end{aligned}
 ```
 
@@ -46,7 +46,7 @@ Then we compute ``\Psi_n`` by solving
 
 ```math
 \begin{aligned}
-0 & = \Gamma_3 + \Gamma_4 \Psi_n + (\Gamma_5 + \Gamma_6 \Psi_n)(\Gamma_1 + \Gamma_2 \Psi_n) + J\mathscr{V}(z_{n - 1}).
+0 & = \Gamma_3 + \Gamma_4 \Psi_n + (\Gamma_5 + \Gamma_6 \Psi_n)(\Gamma_1 + \Gamma_2 \Psi_n) + J\mathcal{V}(z_{n - 1}).
 \end{aligned}
 ```
 
@@ -66,8 +66,8 @@ Let ``q`` be the embedding parameter. Then the homotopy continuation method iter
 ```math
 \begin{aligned}
 0 & = \mu(z, y) - z,\\
-0 & = \xi(z, y) + \Gamma_5 z + \Gamma_6 y + q \mathscr{V}(z),\\
-0 & = \Gamma_3 + \Gamma_4 \Psi + (\Gamma_5 + \Gamma_6 \Psi)(\Gamma_1 + \Gamma_2 \Psi) + q J\mathscr{V}(z)
+0 & = \xi(z, y) + \Gamma_5 z + \Gamma_6 y + q \mathcal{V}(z),\\
+0 & = \Gamma_3 + \Gamma_4 \Psi + (\Gamma_5 + \Gamma_6 \Psi)(\Gamma_1 + \Gamma_2 \Psi) + q J\mathcal{V}(z)
 \end{aligned}
 ```
 

docs/src/risk_adjusted_linearization.md

@@ -21,15 +21,15 @@ is described by the differentiable, conditional cumulant generating function (cc
 
 ```math
 \begin{aligned}
-\kappa[\alpha(z_t) \mid z_t] = \log\mathbb{E}_t[\exp(\alpha(z_t)' \varepsilon_{t + 1})],\quad \text{ for any differentiable map }\alpha::\mathbb{R}^{n_z}\rightarrow\mathbb{R}^{n_\varepsilon}.
+\kappa[\alpha(z_t) \mid z_t] = \log\mathbb{E}_t[\exp(\alpha(z_t)' \varepsilon_{t + 1})],\quad \text{ for any differentiable map }\alpha:\mathbb{R}^{n_z}\rightarrow\mathbb{R}^{n_\varepsilon}.
 \end{aligned}
 ```
 
 The functions
 ```math
 \begin{aligned}
 \xi:\mathbb{R}^{2n_y + 2n_z}\rightarrow \mathbb{R}^{n_y},& \quad \mu:\mathbb{R}^{n_y + n_z}\rightarrow \mathbb{R}^{n_z},\\
-\Lambda::\mathbb{R}^{n_z} \rightarrow \mathbb{R}^{n_z \times n_y}, & \quad \Sigma::\mathbb{R}^{n_z}\ \rightarrow \mathbb{R}^{n_z\times n_\varepsilon}
+\Lambda:\mathbb{R}^{n_z} \rightarrow \mathbb{R}^{n_z \times n_y}, & \quad \Sigma:\mathbb{R}^{n_z}\ \rightarrow \mathbb{R}^{n_z\times n_\varepsilon},
 \end{aligned}
 ```
 are differentiable. The first two functions characterize the effects of time ``t`` variables on the expectational and
@@ -54,31 +54,31 @@ global solutions of canonical economic models and outperforms perturbations arou
 The affine approximation of an dynamic economic model is
 ```math
 \begin{aligned}
-    \mathbb{E}[z_{t + 1}] & = \mu(z, y) + \Gamma_1(z_t - z) + \Gamma_2(y_t - y)\\
-    0                      & = \xi(z, y) + \Gamma_3(z_t - z) + \Gamma_4(y_t - y) + \Gamma_5 \mathbb{E}_t z_{t + 1} + \Gamma_6 \mathbb{E}_t y_{t + 1} + \mathscr{V}(z) + J\mathscr{V}(z)(z_t  - z),
+    \mathbb{E}[z_{t + 1}] & = \mu(z, y) + \Gamma_1(z_t - z) + \Gamma_2(y_t - y),\\
+    0                      & = \xi(z, y) + \Gamma_3(z_t - z) + \Gamma_4(y_t - y) + \Gamma_5 \mathbb{E}_t z_{t + 1} + \Gamma_6 \mathbb{E}_t y_{t + 1} + \mathcal{V}(z) + J\mathcal{V}(z)(z_t  - z),
 \end{aligned}
 ```
 
 where ``\Gamma_1, \Gamma_2`` are the Jacobians of ``\mu`` with respect to ``z_t`` and ``y_t``, respectively;
 ``\Gamma_3, \Gamma_4`` are the Jacobians of ``\xi`` with respect to ``z_t`` and ``y_t``, respectively;
-``\Gamma_5, \Gamma_6`` are constant matrices; ``\mathscr{V}(z)`` is the model's entropy;
-``J\mathscr{V}(z)`` is the Jacobian of the entropy;
-``J\mathscr{V}(z)`` is the Jacobian of the entropy;
+``\Gamma_5, \Gamma_6`` are constant matrices; ``\mathcal{V}(z)`` is the model's entropy;
+``J\mathcal{V}(z)`` is the Jacobian of the entropy;
+``J\mathcal{V}(z)`` is the Jacobian of the entropy;
 
 and the state variables ``z_t`` and jump variables ``y_t`` follow
 ```math
 \begin{aligned}
     z_{t + 1} & = z + \Gamma_1(z_t - z) + \Gamma_2(y_t - y) + (I_{n_z} - \Lambda(z_t) \Psi)^{-1}\Sigma(z_t)\varepsilon_{t + 1},\\
-    y_t       & = y + \Psi(z_t - z)
+    y_t       & = y + \Psi(z_t - z).
 \end{aligned}
 ```
 
 The unknowns ``(z, y, \Psi)`` solve the system of equations
 ```math
 \begin{aligned}
 0 & = \mu(z, y) - z,\\
-0 & = \xi(z, y) + \Gamma_5 z + \Gamma_6 y + \mathscr{V}(z),\\
-0 & = \Gamma_3 + \Gamma_4 \Psi + (\Gamma_5 + \Gamma_6 \Psi)(\Gamma_1 + \Gamma_2 \Psi) + J\mathscr{V}(z).
+0 & = \xi(z, y) + \Gamma_5 z + \Gamma_6 y + \mathcal{V}(z),\\
+0 & = \Gamma_3 + \Gamma_4 \Psi + (\Gamma_5 + \Gamma_6 \Psi)(\Gamma_1 + \Gamma_2 \Psi) + J\mathcal{V}(z).
 \end{aligned}
 ```