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Minor improvements in the documentation (sde_example.md) #708

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Minor improvements in the documentation (sde_example.md) #708

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13 changes: 6 additions & 7 deletions docs/src/tutorials/sde_example.md
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Original file line number Diff line number Diff line change
Expand Up @@ -9,18 +9,17 @@
In this example, we will solve the equation

```math
du = f(u,p,t)dt + g(u,p,t)dW
du = f(u,p,t)dt + g(u,p,t)dW,
```

where ``f(u,p,t)=αu`` and ``g(u,p,t)=βu``. We know via Stochastic Calculus that the
where ``f(u,p,t)=αu`` and ``g(u,p,t)=βu``. We know via Stochastic calculus that the
solution to this equation is

```math
u(t,Wₜ)=u₀\exp((α-\frac{β^2}{2})t+βWₜ)
u(t,Wₜ)=u₀\exp\left[\left(α-\frac{β^2}{2}\right)t+βWₜ\right].
```

To solve this numerically, we define a problem type by giving it the equation
and the initial condition:
To solve this numerically, we define a stochastic problem type using `SDEProblem` by specifying `f(u, p, t)`, `g(u, p, t)` and the initial condition:

```@example sde
using DifferentialEquations
Expand All @@ -31,7 +30,7 @@ f(u, p, t) = α * u
g(u, p, t) = β * u
dt = 1 // 2^(4)
tspan = (0.0, 1.0)
prob = SDEProblem(f, g, u₀, (0.0, 1.0))
prob = SDEProblem(f, g, u₀, tspan)
```

The `solve` interface is then the same as with ODEs. Here we will use the classic
Expand Down Expand Up @@ -82,7 +81,7 @@ plot(sol, plot_analytic = true)

Here we allowed the solver to automatically determine a starting `dt`. This estimate
at the beginning is conservative (small) to ensure accuracy. We can instead start
the method with a larger `dt` by passing in a value for the starting `dt`:
the method with a larger `dt` by passing it to `solve`:

```@example sde
sol = solve(prob, SRIW1(), dt = dt)
Expand Down