I have a problem with two processes X_{1, t}, X_{2, t} \in \mathbb{R} which satisfy the SDEs
\begin{align} dX_{1, t} = f_1(t, X_{1, t}) \; dt + \sigma \; dW_t, \\ dX_{2, t} = f_2(t, X_{1, t}) \; dt + \sigma \; dW_t. \end{align}
The functions f_i are nonlinear. The two processes share the noise \sigma \; dW_t. I would like to simulate paths of X_{1, t} and X_{2, t} with the same noise realisations. I tried to do this with DifferentialEquations.jl and DiffEqNoiseProcess.jl, by defining W = WienerProcess(0., 0.), passing it as noise to SDEProblem and solving with the same seed. But when I check the noise realisation, I get different values.
See this MWE for f_1(t, x) = \sqrt{x} and f_2(t, x) = x, and \sigma = 1.
using DifferentialEquations
f₁(x, p, t) = sqrt(x)
f₂(x, p, t) = x
g(x, p, t) = 1.
W = WienerProcess(0., 0.)
x₀ = 10.
tspan = (0., 1.)
prob₁ = SDEProblem(f₁, g, x₀, tspan, noise = W)
prob₂ = SDEProblem(f₂, g, x₀, tspan, noise = W)
seed = 42
sol₁ = solve(prob₁; seed = seed, save_noise = true)
sol₂ = solve(prob₂; seed = seed, save_noise = true)
error = [sol₁.W(t)[1] - sol₂.W(t)[1] for t in 0:0.01:1]
Am I misunderstanding how the seed works? Is there a better way to achieve this?