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Inclusion of absorbing boundary in continuous stochastic process simulation

Cameron Turner

Posted 3 months ago

While I'm now aware of the great functions Mathematica has for simulating stochastic processes on an unbounded domain, a common problem is a bounded domain with absorbing boundaries. In particular, once the process hits the boundary it stays there forever. Is there an elegant and efficient way to include absorbing boundaries into simulations such as the following. That is, to have the boundaries (a,b) be absorbing? \[Mu] = 1; \[Theta] = 1; \[Sigma] = 1; {a, b} = {0, 4}; numreps = 10; Tmax = 10; dt = .1; paths = Table[ RandomFunction[ ItoProcess[\[DifferentialD]x[ t] == \[Theta] (\[Mu] - x[t]) \[DifferentialD]t + \[Sigma] \[DifferentialD]w[t], x[t], {x, RandomReal[{a, b}]}, {t, 0}, w \[Distributed] WienerProcess[]], {0, Tmax, dt}], {i, numreps}]; ListLinePlot[paths, PlotTheme -> "Scientific", AspectRatio -> 1/GoldenRatio, PlotRange -> {Automatic, {a - 1, b + 1}}, GridLines -> {None, {a, b}}]

While I'm now aware of the great functions Mathematica has for simulating stochastic processes on an unbounded domain, a common problem is a bounded domain with absorbing boundaries. In particular, once the process hits the boundary it stays there forever.

Is there an elegant and efficient way to include absorbing boundaries into simulations such as the following. That is, to have the boundaries (a,b) be absorbing?

simulation output

\[Mu] = 1;
\[Theta] = 1;
\[Sigma] = 1;

{a, b} = {0, 4};

numreps = 10;
Tmax = 10;
dt = .1;


paths = Table[
   RandomFunction[
    ItoProcess[\[DifferentialD]x[
        t] == \[Theta] (\[Mu] - 
          x[t])  \[DifferentialD]t + \[Sigma] \[DifferentialD]w[t], 
     x[t], {x, RandomReal[{a, b}]}, {t, 0}, 
     w \[Distributed] WienerProcess[]], {0, Tmax, dt}], {i, numreps}];

ListLinePlot[paths, PlotTheme -> "Scientific", 
 AspectRatio -> 1/GoldenRatio, 
 PlotRange -> {Automatic, {a - 1, b + 1}}, 
 GridLines -> {None, {a, b}}]

POSTED BY: Cameron Turner

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Michael Rogers

Michael Rogers, Emory University

Posted 3 months ago

Multiplying the right-hand side of the SDE by `UnitStep[(x[t] - a)(b - x[t])]` causes `x[t]` to stop changing when it crosses the boundary, but it doesn't keep it from stepping over the boundary. However, `Clip[RandomFunction[..., {a, b}]` will correct the solution.

POSTED BY: Michael Rogers

Cameron Turner

Posted 3 months ago

This works quite nicely. I'm going to scale the simulation considerably, so I'm wondering about efficacy. This might be close the ideal of stopping the process on the boundary when it cross it, rather than editing the solution post-hoc. To be clear the updated code for Michael Rogers' idea: paths = Table[ Clip[RandomFunction[ ItoProcess[\[DifferentialD]x[ t] == (\[Theta] (\[Mu] - x[t]) \[DifferentialD]t + \[Sigma] \[DifferentialD]w[ t])*UnitStep[(x[t] - a) (b - x[t])], x[t], {x, RandomReal[{a, b}]}, {t, 0}, w \[Distributed] WienerProcess[]], {0, Tmax, dt}], {a, b}], {i, numreps}];

This works quite nicely. I'm going to scale the simulation considerably, so I'm wondering about efficacy. This might be close the ideal of stopping the process on the boundary when it cross it, rather than editing the solution post-hoc.

To be clear the updated code for Michael Rogers' idea:

paths = Table[  
   Clip[RandomFunction[
     ItoProcess[\[DifferentialD]x[
         t] == (\[Theta]  (\[Mu] - 
             x[t])   \[DifferentialD]t + \[Sigma]  \[DifferentialD]w[
             t])*UnitStep[(x[t] - a) (b - x[t])], 
      x[t], {x, RandomReal[{a, b}]}, {t, 0}, 
      w \[Distributed] WienerProcess[]], {0, Tmax, dt}], {a, b}], {i, 
    numreps}];

POSTED BY: Cameron Turner

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