Misunderstandig of default Brownian Motion/Wiener process about torchsde HOT 4 CLOSED

Hi, thanks for the note.

Is it Wiener process with zero mean and unit volatility?

That's loosely true up to the fact that W(t) may be multidimensional.

What is exactly batch_size?

You may interpret this as the number of particles that's being simulated which evolves as according to the SDE.

What is exactly brownian_size?

This is the dimension of the Brownian motion (it is not necessarily the size of the state). In most classic texts, the state dimension is denoted by d, and the dimension of the Brownian motion is denoted by m.

but if we set size=1, we see TypeError: int object is no iterable

We made it clear in the documentation that the input should be a tuple of int, as opposed to a single integer. If you set size=(1,), it should work, since (1,) is a tuple of one integer 1.

from torchsde.

@lxuechen , thank you so much for your reply!

I will test all the point about tuple of int and will comment it later. The clarification about batch_size is almost clear.

What about my question how to add the Wiener process with volatility != 1 to each equation of a set of coupled SDEs? Is the presented definition of g (it is given in my first comment) true or not?

from torchsde.

Note g should return a matrix of size (batch_size, state_size, brownian_size) when the noise_type is general. Again, batch_size is the number of particles (note one particle can be multi-dimensional). state_size is the dimension of each particle, and brownian_size is the size of the Brownian motion. Mathematically, during simulation, a matmul is performed behind for this g term, e.g., with Euler solver, you get X_{t+dt} = X_t + f(X_t, t) dt + matmul(g(X_t, t), W_{t + dt} - W_t).

Indirectly, this gives you a way to control for the volatility.

from torchsde.

Closing the issue for now. Free feel to reopen/followup if further clarification is needed.

from torchsde.