The solve function's save_noise keyword defaults incorrectly to false and ignores saveat keyword

[henhen724]

Describe the bug 🐞
The docs describe the save_noise keyword for the solve function in following way:

save_noise: Determines whether the values of W are saved whenever the timeseries is saved. Defaults to true.

Firstly, when save_noise is not set, it defaults to false instead of true. As you can see in the example below, without save noise set it only saves the start and end points to the solution.

Secondly, when saveat is set to a time longer than the timestep, the noise process will save the noise at single every time step instead of when the timeseries is saved, which does not match the description above.

Expected behavior
Without setting save_noise, the solution object should contain a sol.W which is saved at the exact same timesteps as sol.t and sol.u.

Minimal Reproducible Examples 👇
Example 1: incorrectly defaults to false

using DifferentialEquations

function ou!(du, u, p, t)
    du[1] = -10.0 * u[1]
end

function σ_ou!(du, u, p, t)
    du[1] = 3.0
end

prob_ou = SDEProblem(ou!, σ_ou!, [1.0], (0.0, 10.0))
sol = solve(prob_ou, saveat=0.1);

println("Solution length: $(length(sol.u))  Noise Length: $(length(sol.W))")

Solution length: 101 Noise Length: 2

Example 2: saves every timestep instead of at the timeseries

using DifferentialEquations

function ou!(du, u, p, t)
    du[1] = -10.0 * u[1]
end

function σ_ou!(du, u, p, t)
    du[1] = 3.0
end

prob_ou = SDEProblem(ou!, σ_ou!, [1.0], (0.0, 10.0))
sol = solve(prob_ou, saveat=0.1, save_noise=true);

println("Solution length: $(length(sol.u))  Noise Length: $(length(sol.W))")

Solution length: 101 Noise Length: 24120

Environment (please complete the following information):

• Output of using Pkg; Pkg.status()

julia> using Pkg; Pkg.status()
Status `\\levlabserver2.stanford.edu\commondrive\Users\hshunt\LabNotebooks\BugEnv\Project.toml`
  [0c46a032] DifferentialEquations v7.13.0

• Output of using Pkg; Pkg.status(; mode = PKGMODE_MANIFEST)

julia> using Pkg; Pkg.status(; mode = PKGMODE_MANIFEST)
Status `\\levlabserver2.stanford.edu\commondrive\Users\hshunt\LabNotebooks\BugEnv\Manifest.toml`
  [47edcb42] ADTypes v1.6.1
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⌅ [fb77eaff] Sundials_jll v5.2.2+0
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Info Packages marked with ⌅ have new versions available but compatibility constraints restrict them from upgrading. To see why use `status --outdated -m`

• Output of versioninfo()

Julia Version 1.10.2
Commit bd47eca2c8 (2024-03-01 10:14 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Windows (x86_64-w64-mingw32)
  CPU: 8 × Intel(R) Core(TM) i7-6700K CPU @ 4.00GHz
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-15.0.7 (ORCJIT, skylake)
Threads: 1 default, 0 interactive, 1 GC (on 8 virtual cores)

[ChrisRackauckas]

save_noise: Determines whether the values of W are saved whenever the timeseries is saved. Defaults to true.

Yeah that's a typo in the docs. It's intended to default to false because most of the time it's not needed and it's expensive to save. That was a simple fix SciML/DiffEqDocs.jl#744.

Example 2: saves every timestep instead of at the timeseries

Yes that's not intended, moving to StocahsticDiffEq.jl