Convergence of SDE solvers

General Usage
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1

I am wondering how to make sure that an SDE solver has converged on a trajectory level. In other words, I want to make sure that I have chosen abstol and reltol small enough.

What I would do in the deterministic case is that I would just solve the differential equation twice for different tolerances. If the results lie on top of each other, I can be sure that I have chosen the tolerances low enough.

In the stochastic case, however, this is not possible. I am new to these types of equations, but as far as I understand, stochastic solvers only have strong convergence, which means that the difference between two trajectories with different step sizes dt1 and dt2 vanishes as ~ |dt1 - dt2|^p.

I find this a bit unintuitive, as I don’t want my solution to depend on my choice of the time step (or the tolerances). To test this, I took an example from the tutorial and executed it with the same seed but different tolerances:

import StochasticDiffEq as SDE
α = 1
β = 1
u₀ = 1 / 2
f(u, p, t) = α * u
g(u, p, t) = β * u
tspan = (0.0, 1.0)
prob = SDE.SDEProblem(f, g, u₀, tspan)

sol = SDE.solve(prob, SDE.SRIW1(), seed = 123, abstol = 1e-7, reltol = 1e-7, maxiters = Int(1e10), saveat = 0.01)
sol_lower_tol = SDE.solve(prob, SDE.SRIW1(), seed = 123, abstol = 1e-8, reltol = 1e-8, maxiters = Int(1e10), saveat = 0.01)

Plotting the two solutions yields

Comparison of the solutions with different tolerances.
Comparison of the solutions with different tolerances.919×489 37.4 KB

Therefore my questions:

  1. Is it better to play with abstol and reltol or with dt and dtmax to ensure convergence?
  2. How can I be sure that I have chosen the correct solver and that my solver has converged?
  3. Is it not a problem that my solution depends on the solver parameters? Or is it just that the stochastics differ (i.e., that different random numbers are chosen) when I modify the tolerances?
2

Depends on the problem but generally yes.

I would instead state it as, for a chosen brownian motion, it converges like order p.

Those aren’t using the same Brownian motion. You’d want to use things like NoiseWrapper in order to refine the Brownian motion of a different solve.

It’s that the random process is completely different. The random seed only makes sure that your sequence of random numbers is the same. But the Brownian motion is a pairing of dt’s with random numbers, and so if you change the dts you just have a completely different Brownian motion.