Finish rewrite of SDE tutorial

Author: ArnoStrouwen

docs/src/tutorials/stochastic_diffeq.md

@@ -19,19 +19,27 @@ Let's take the Lorenz equation and add noise to each of the states.
 To show the flexibility of ModelingToolkit,
 we do not use homogeneous noise, with constant variance,
 but instead use heterogeneous noise,
-where the magnitude of the noise scales with (0.1 times) the magnitude of each of the states:
+where the magnitude of the noise scales with (0.3 times) the magnitude of each of the states:
 
 ```math
 \begin{aligned}
-dx &= (\sigma (y-x))dt  &+ 0.1xdB \\
-dy &= (x(\rho-z) - y)dt &+ 0.1ydB \\
-dz &= (xy - \beta z)dt  &+ 0.1zdB \\
+\frac{dx}{dt} &= (\sigma (y-x))  &+ 0.1x\frac{dB}{dt} \\
+\frac{dy}{dt} &= (x(\rho-z) - y) &+ 0.1y\frac{dB}{dt}  \\
+\frac{dz}{dt} &= (xy - \beta z)  &+ 0.1z\frac{dB}{dt}  \\
 \end{aligned}
 ```
 
 Where $B$, is standard Brownian motion, also called the
 [Wiener process](https://en.wikipedia.org/wiki/Wiener_process).
-In ModelingToolkit this can be done by `@brownian` variables.
+We use notation similar to the
+[Langevin equation](https://en.wikipedia.org/wiki/Stochastic_differential_equation#Use_in_physics),
+often used in physics.
+By "multiplying" the equations by $dt$, the notation used in
+[probability theory](https://en.wikipedia.org/wiki/Stochastic_differential_equation#Use_in_probability_and_mathematical_finance)
+can be recovered.
+
+We use this Langevin-like notation because it allows us to extend MTK modeling capacity from ODEs to SDEs,
+using only a single new concept, `@brownian` variables, which represent $\frac{dB}{dt}$ in the above equation.
 
 ```@example SDE
 using ModelingToolkit, StochasticDiffEq
@@ -41,9 +49,9 @@ using Plots
 @parameters σ=10.0 ρ=2.33 β=26.0
 @variables x(t)=5.0 y(t)=5.0 z(t)=1.0
 @brownian B
-eqs = [D(x) ~ σ * (y - x) + 0.1B * x,
-    D(y) ~ x * (ρ - z) - y + 0.1B * y,
-    D(z) ~ x * y - β * z + 0.1B * z]
+eqs = [D(x) ~ σ * (y - x) + 0.3x * B,
+    D(y) ~ x * (ρ - z) - y + 0.3y * B,
+    D(z) ~ x * y - β * z + 0.3z * B]
 
 @mtkbuild de = System(eqs, t)
 ```
@@ -58,22 +66,29 @@ We continue by solving and plotting the SDE.
 
 ```@example SDE
 prob = SDEProblem(de, [], (0.0, 100.0), [])
-sol = solve(prob, LambaEulerHeun())
+sol = solve(prob, SRIW1())
 plot(sol, idxs = [(1, 2, 3)])
 ```
 
 The noise present in all 3 equations is correlated, as can be seen on the below figure.
-If you want uncorrelated noise for each equation,
-multiple `@brownian` variables have to be declared.
+The figure also shows the multiplicative nature of the noise.
+Because states `x` and `y` generally take on larger values,
+the noise also takes on a more pronounced effect on these states compared to the state `z`.
 
 ```@example SDE
-@brownian Bx By Bz
+plot(sol)
 ```
 
-The figure also shows the multiplicative nature of the noise.
-Because states `x` and `y` generally take on larger values,
-the noise also takes on a more pronounced effect on these states compared to the state `z`.
+If you want uncorrelated noise for each equation,
+multiple `@brownian` variables have to be declared.
 
 ```@example SDE
+@brownian Bx By Bz
+eqs = [D(x) ~ σ * (y - x) + 0.3x * Bx,
+    D(y) ~ x * (ρ - z) - y + 0.3y * By,
+    D(z) ~ x * y - β * z + 0.3z * Bz]
+@mtkbuild de = System(eqs, t)
+prob = SDEProblem(de, [], (0.0, 100.0), [])
+sol = solve(prob, SRIW1())
 plot(sol)
 ```