Add `saveat` keyword argument

[albertomercurio]

Is your feature request related to a problem? Please describe.

The functions currently support the save_every_step keyword argument. This saves a Vector that can be more or less long. A saveat kwarg would be useful in such cases an SDE uses saveat and we want to extract the noise process only on that times, without necessary allocating memory for saving the entire process history.

Describe the solution you’d like

An additional keyword argument on all the AbstractNoiseProcesses, or at least starting with the WienerProcess and RealWienerProcess.

Describe alternatives you’ve considered

Just preallocating an empty vector, and updating it as long as the process evolves.

[ChrisRackauckas]

I agree it should be added. Just haven't gotten to it.

[albertomercurio]

Hello, do you think it is something easy to implement? Do you have a temporary workaround?

Moreover, it is a bit off-topic, but also related to this. Is there a direct way to extract the derivative of the noise $d W / d t$?

[ChrisRackauckas]

It wouldn't be difficult. You'd just add in https://github.com/SciML/DiffEqNoiseProcess.jl/blob/master/src/solve.jl#L28 something that checks through the saveat array and interpolates back. So you basically do this https://github.com/SciML/SimpleDiffEq.jl/blob/master/src/tsit5/gpuatsit5.jl#L67-L76, but for noise processes the interpolation is trivial because it's linear and already built into the type, so you'd just add something like that.

Is there a direct way to extract the derivative of the noise

That's undefined a.e.

[ChrisRackauckas]

Let me know if you plan to take a stab at it. It shouldn't take more than an hour. The main thing is writing test cases.