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There has been several entries on solving deterministic differential equations that include indicator functions. Stochastic differential equations may introduce new difficulties. Namely, does the usual technique used in the deterministic situation, of dividing the domain into intervals, still apply?

Take the following SDE: $$\mathrm{d}X_t= (\mu\mathrm{d}t+\sigma \mathrm{d}W_t)\theta(X_t)$$

Here, $\theta$ is the unit step function, $W_t$ the Weiner process, and $\mu$ and $\sigma$ Real-valued constants.

Can a general solution be produced?

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  • $\begingroup$ What did you try yourself? $\endgroup$
    – user64494
    Commented Sep 3 at 13:32
  • $\begingroup$ Putting an integral sign on both sides and thinking maybe that was problematic already $\endgroup$
    – CRTmonitor
    Commented Sep 3 at 13:37

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The following works to me.

proc = StratonovichProcess[\[DifferentialD]x[t] == 
 UnitStep[x[t]] *\[DifferentialD]t + 
UnitStep[x[t]] \[DifferentialD]w[t], x[t], {x, 1}, t, 
  w \[Distributed] WienerProcess[]];
RandomFunction[proc, {0., 5., 0.01}]
ListLinePlot[%]

enter image description here

I use UnitStep instead of HeavisideTheta since the latter is an attempt to implement a generalized function, not a usual function, in Mathematica.

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  • $\begingroup$ The result with proc = StratonovichProcess[\[DifferentialD]x[t] == HeavisideTheta[x[t]] *\[DifferentialD]t + HeavisideTheta[x[t]] \[DifferentialD]w[t], x[t], {x, 1}, t, w \[Distributed] WienerProcess[]] is the same up to randomness. $\endgroup$
    – user64494
    Commented Sep 3 at 13:58

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