Solving an SDE with a Continuous Callback

General Usage
,
1

I am solving a stochastic differential equation with a ContinuousCallback. Every time the continuous callback is activated, I modify my state vector discontinuously.

All the randomness comes from a phase, which follows a simple Wiener process, i.e., the phase is not coupled to the remaining variables of the differential equation.

Nonetheless, if I use the default tolerances, I sometimes observe a discontinuous change in the phase when the ContinuousCallback is activated:

Random phase for high tolerances.
Random phase for high tolerances.919×489 13.7 KB

When I decrease the tolerances, I don’t see these spikes anymore. Nonetheless, this seems like a strange behavior.

Here is a snippet of my code:

function jump_condition_func(slv)
    return function (u, t, integrator)
        jump_condition(u, t, integrator, slv)
    end
end

function project_func!(slv, sys)
    return function (integrator)
        project!(integrator, slv, sys)
    end
end

jump_cb = ContinuousCallback(
    jump_condition_func(slv),
    project_func!(slv, sys),
    save_positions = (true, true),
)

prob = SDEProblem(eoms!, noise!, u₀, tspan, p)
sol = SDE.solve(
    prob,
    SDE.SRIW1(),
    abstol = slv.abstol,
    reltol = slv.reltol,
    callback = jump_cb,
    dtmax = slv.dtmax,
    maxiters = slv.maxiters,
    saveat = slv.saveat,
)

My questions are:

  1. Why does this happen? Is it because I save the dynamics before and after the jump?
  2. Can one even use a ContinuousCallback in combination with SDE solvers or does that lead to unexpected behavior? I know that there is also the option of using a jump diffusion process - is that preferred?
2

Update:

If I set saveat = [] (the default), then I also do not see the spikes…

3

it could be a saveat bug