# Regime Change Stochastic Process

I would like to simulate an Ito process in which the drift and diffusion terms change after hitting a boundary for the first time.

For example, a Geometric Brownian Motion X which has 0 drift and volatility X dZ_t until X reaches some boundary X-hat. Afterwards, there might be a different drift and volatility.

With my current knowledge of Mathematica, I only know how to simulate the first part, a standard GBM process. What are the next steps?

It is very possible that I'm using poor terminology due to inexperience, and so any comments or pointers are appreciated.

The code I'm using for the standard GBM is the following:

μ = 0;
σ = 0.5;
testfun = ItoProcess[{μ x, σ x}, {x, 1}, {t, 0}];
testdata = RandomFunction[testfun, {0, 1, 0.01}, 5];
ListLinePlot[testdata, AxesLabel -> {t, X[t]},
PlotLabel -> "Simulated Paths", PlotRange -> All]


Update: I have tried the proposed answer and found it does not work, as illustrated by my example below:

μ = 1;
σ = 0;
start = 0;
fun1 := If[x >= 0.5 || start > 0, start++;  -x, x];
testfun = ItoProcess[{fun1, σ x}, {x, 0.1}, {t, 0}];
testdata = RandomFunction[testfun, {0, 5, 0.01}, 1];
ListLinePlot[testdata]


This results in a process that "sticks" at the boundary, rather than one that decays afterwards. I believe the problem is that the start tracker is not working as intended.

Update #2:

I've added a means to simulate multiple paths

ClearAll[x, fun1, fun2, ff, gg]
μ = 1;
σ = 0.3;

fun1[x_, t_] :=
If[t < 0.1, ClearAll[ff, gg]; x,
If[
If[
x < 0.5,
gg = False; If[ff, gg = True]; False,
ff = True
] && ff || gg, -x, x]
]
fun2[x_, t_] :=
If[If[x < 0.5, gg = False; If[ff, gg = True]; False, ff = True] &&
ff || gg, σ x, 0]
testfun = ItoProcess[{fun1[x, t], fun2[x, t]}, {x, 0.1}, {t, 0}];
testdata = RandomFunction[testfun, {0, 5, 0.01}, 5];
ListLinePlot[testdata, PlotRange -> All]


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• Since you "know how to simulate the first part", please include that code (or a simplified version of it) as a starting point to provide context for readers. May 21, 2015 at 20:22
• Sure thing. I used the example in the ItoProcess documentation. The code has been edited into my original question. May 21, 2015 at 20:23
• Done, and thank you to Sjoerd for properly formatting the code. I realized right after adding that it didn't look right, but he beat me to the edit. May 21, 2015 at 20:26
• Is there to be a reversion of parameters if the boundary is re-crossed? In any case, there's no built-in (that I'm aware of) way to "parameterize" the parameters and/or modify them on the fly with some logical construct/test - you'll need to e.g. generate some data, scan it for a crossing, truncate it there, append data with new parameters from that point, rinse-n-repeat...
– ciao
May 21, 2015 at 20:51

Second trial.

The following seems to work. The only restriction so far is that it only works on the first path. I will be looking into that.

ClearAll[x, fun1, fun2, ff, gg]
μ = 1;
σ = 0.3;

fun1[x_, t_] :=
If[
If[x < 0.5,
gg = False; If[ff, gg = True]; False,
ff = True
] && ff || gg, -x, x
]
fun2[x_, t_] :=
If[
If[x < 0.5,
gg = False; If[ff, gg = True]; False,
ff = True
] && ff || gg, σ x, 0]
testfun = ItoProcess[{fun1[x, t], fun2[x, t]}, {x, 0.1}, {t, 0}];
testdata = RandomFunction[testfun, {0, 5, 0.01}, 1];
ListLinePlot[testdata, PlotRange -> All]


The trick of the baroque Boolean functions is that If returns unevaluated if it cannot determine the truth of its condition (and a third argument is missing). We need this because ItoProcess evaluates its first argument. Another thing that the above makes use of is that False && undefined and True || undefined both evaluate (to False and True, respectively).

• Color me surprised if this does what it appears to do, +1 on thought-provoking possibility...
– ciao
May 21, 2015 at 21:36
• I have tried a corner case and the start tracker does not appear to do the trick. May 21, 2015 at 23:26
• @Shffl what was the corner case? May 21, 2015 at 23:48
• I've updated my original question. I suppressed the diffusion component. May 22, 2015 at 0:12
• @Shffl see update May 22, 2015 at 23:06