# Plot A Function Of A Stochastic Process

I am trying to do something very simple in Mathematica 9. I want to play around with option pricing and for that I thought it best to use the new stochastic process functionality.

So, first of all I simulate one instance of a geometric brownian motion:

$$\frac{dX_t}{X_t} = \mu dt + \sigma dW_t\\ dW_t \sim N(0, 1)$$

Which in Mathematica is:

proc = ItoProcess[
\[DifferentialD]x[t]/
x[t] == \[Sigma] \[DifferentialD]w[t] + \[Mu] \[DifferentialD]t,
x[t],
{x, x0},
t,
w \[Distributed] WienerProcess[]];


And here's an example of what I get, when I plot it, assuming $X_0 = 100$.

So, okay, when I create a plot of a RandomFunction of the process, then I actually plot the TemporalData for $X_t$ and not $dX_t$. Cool, whatever.

But now I want to plot, say $f(dX_t)$ or $f(X_t)$, where I would like to define $f$ as I see fit. And this is where I hit a brick wall. I have tried looking for hints in the docs or for answers here, but there are no definitive ones or the ones that seem to work.

I also feel, that I'm missing something fundamental here. Could somebody kindly suggest an answer or the venue of inquiry?

-
For a TemporalData object td, Normal[td] gives the list of time-value pairs. You can apply your f to this list - e.g. f/@Normal[td][[All,2]]. –  kglr May 29 '14 at 18:42
If you have additional terms in your model, such as a density depend mean (or standard deviation), components with μ, or want to record the stochastic variable itself, I have answer at: mathematica.stackexchange.com/a/59470/8274. –  ambein Sep 12 '14 at 0:02

I'm not an expert on stochastic differential equations but I found the documentation clear enough.

Getting the output in terms of a function of your process variable x[t] is easy:

SeedRandom[1];
With[{σ = 1, μ = 1, x0 = 100},
proc = ItoProcess[
\[DifferentialD]x[t]/x[t] == σ \[DifferentialD]w[t] + μ \[DifferentialD]t,
Log10[x[t]],
{x, x0}, t, w \[Distributed] WienerProcess[]
];
rp = RandomFunction[proc, {0., 5., 0.01}]
]

ListLinePlot[rp]


As you can see I changed x[t] as the output expression into Log10[x[t]].

The last line in the syntax block of the ItoProcess help page names the object in this location the "output expression". In the examples you usually see x[t] there, but it doesn't have to be just that.

A different route you could have traveled was using the TemoralData object that is being generated.

So if you start with just x[t] as output expression (as you originally did):

SeedRandom[1];
With[{σ = 1, μ = 1, x0 = 100},
proc = ItoProcess[
\[DifferentialD]x[t]/x[t] == σ \[DifferentialD]w[t] + μ \[DifferentialD]t,
x[t], {x, x0}, t, w \[Distributed] WienerProcess[]
];
rp = RandomFunction[proc, {0., 5., 0.01}]
]
ListLinePlot[rp]


You can get the resulting path by getting the "Paths" properties from the TemporalData object:

rp["Paths"]

(*==>  {{{0., 1.9999999999999998}, {0.01, 2.0247187089520158},
{0.02, 2.0462858749445374}, {4.97, 3.811145935882086},
...
, {4.99, 3.7933881925620243}, {5., 3.826247092011493}}} *)


You could use the function that you wanted to apply on this output:

ListLinePlot[MapAt[Log10, rp["Paths"], {All, All, 2}]]


which yields the same result as when the function is directly used as the output expression.

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Thank you very much! I did feel, like I was missing something very obvious. –  jst May 29 '14 at 20:15