WhenEvent in Stochastic Differential Equation

Question

Is there a way to add events with WhenEvent or similar when using ItoProcess?

Minimal example of my problem would be to change sign when diffusion hits a wall. (Post processing is no good - I need to change several things in the actual problem)

As per edit request a sample SDE:

$$ d x = \frac{\sin x}{1 - \cos x} dt + dW $$

with looping boundary. That is, with $x = \pi + \epsilon$ changing to $x = -\pi + \epsilon$ whenever $x > \pi$ and analog for $x < -\pi$

(Equation is well behaved within $(-\pi,\pi)$ domain)

Answer 1

This shows how it works:

proc = ItoProcess[\[DifferentialD]X[
     t] == -α WhenEvent[x[t] == 1., 
      x[t] -> -x[t]] Exp[-α t] Log[X0/K] X[
      t] \[DifferentialD]t + σ X[t] \[DifferentialD]W[t], 
  X[t], {X, X0}, t, W \[Distributed] WienerProcess[]]
Mean[proc[t]]
PDF[proc[t], x]
proc1 = proc /. {X0 -> 70, K -> 2, σ -> 0.1, α -> 0.3};
DiscretePlot[PDF[proc1[t], x], {x, 0, 10}]
CovarianceFunction[proc1, s, t]

The DiscretePlot is very time-consuming, so I shortened that. Definitely this does not work if the questions target to use RandomFunction.

Plot3D[CovarianceFunction[proc1, s, t], {s, 0, 5}, {t, 0, 5}, ColorFunction -> "Rainbow"]

The equation given in the question might not work because the denominator is singular at infinitely many points. Mathematica works on the Complexes.

In this methodolgoy Mod is an alternative.

proc4 = ItoProcess[\[DifferentialD]x[t] == 
   Sin[Mod[x[t], \[Pi]]]/(
     1 - Cos[Mod[x[t], \[Pi]]]) \[DifferentialD]t + \[DifferentialD]w[
      t], x[t], {x, 1}, t, w \[Distributed] WienerProcess[]]

(* ItoProcess[{{-( Sin[Mod[x[t], [Pi]]]/(-1 + Cos[Mod[x[t], [Pi]]]))}, {{1}}, x[t]}, {{x}, {1}}, {t, 0}] *)

RandomFunction[proc4, {0., 1., 0.01}]

Further extended:

    f[x_] := Piecewise[Table[{Sin[x - n*Pi - \[Epsilon]]/(1 - Cos[x - n*Pi - \[Epsilon]]), 
     -Pi + n*Pi + \[Epsilon] <= x <= Pi + n*Pi + \[Epsilon]}, {n, -5, 5}]]
Plot[f[x], {x, -6*Pi + \[Epsilon], 6*Pi + \[Epsilon]}]

works too with the RandomProcess representation and therefore complete. But still the problem remains that the function is singular for 2 n 𝜋, Element[n,Integers].

ListLinePlot[%, Filling -> Axis]

Using the Fourier make this ideally periodic. For example:

p = Table[
   f[x], {x, \[Epsilon], 
    6 \[Pi] + \[Epsilon], (6 \[Pi] + \[Epsilon])/999}];
s = Fourier[p];
ListPlot[Abs[InverseFourier@s], PlotRange -> Full]

And the use this as an InterpolatingFunction of x: Interpolation@InverseFourier@s.