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random - Boundary condition for stochastic differential equation


I have a simple stochastic differential equation (SDE) with white noise:


mysde[q_, I_, n0_, sd_] := 
  ItoProcess[

    \[DifferentialD]n[t] == 
      ( I - q n[t]) \[DifferentialD]t + sd \[DifferentialD]w[t], n[t], 
    {n, n0}, t, w \[Distributed] WienerProcess[]]

I want to impose a condition that n[t] cannot fall below zero. At below zero, or an arbitrarily low value of n[t], it is considered zero. I am not sure how to implement this in the ItoProcess function and/or in the numerical solution:


sol[q_, I_, n0_, sd_] := RandomFunction[mysde[q, I, n0, sd], {0, 100, 0.1}, 1]

The variable drops below zero many times, which is not a desired behaviour:


With[{q = 2, I = 0.1, n0 = 1, sd = 0.1}, 
  Show[ListLinePlot[sol[q, I, n0, sd]], ImageSize -> 400]]


What are some ways to implement such a condition?


I have tried various functions on n[t] within the ItoProcess[], to attempt to re-set negative values of n[t] to zero, like If[], Max[], Clip[], and Piecewise[] to no avail. Any reasons why these functions do not dynamically work within the solver?


Also is there a way to extract the values of the noise, w[t], in a particular simulated path of the temporal data timeseries object?


One potential solution, which is not confirmed to be correct, is to write the equation for the log of my variable, ln[t] and to use the Geometric Brownian motion random process formulation, obtaining:


logsde[q_, I_, ln0_, sd_] := 
  ItoProcess[
    \[DifferentialD]ln[t] == 
      ( I - q ln[t] - sd^2/2) \[DifferentialD]t + sd \[DifferentialD]w[t], ln[t], 
   {ln, ln0}, t, w \[Distributed] GeometricBrownianMotionProcess[I - q ln[t], sd, n0]]


logsol[q_, I_, ln0_, sd_] := RandomFunction[logsde[q, I, ln0, sd], {0, 100, 0.1}, 1]

Then I extract the simulated values, and exponentiate them to obtain the time series or path in the original scale:


td = With[{q = 0.1, I1 = 0.1, ln0 = 1, sd = .1}, 
  logsol[q, I1, ln0, sd]];
ListLinePlot[Exp[td["States"]]]

However I have lost the stochastic signal in this model; the simulation smooths out and looks continuous with no noise, except at the beginning of the simulation. This is obviously an incorrect formulation. What did I do here? How might I formulate this properly?


Are there other approaches?



So overall I have three questions:



  1. How to implement a boundary condition in ItoProcess[] for one of the state variables?

  2. How to recover the simulated values of the random variable? <- self-answered below

  3. How to properly specify a log-transformed version of the ItoProcess[]?




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