# Ito Process paths over a Plot3D

I was wondering if its possible to draw the simulated paths of an Ito diffusion over the probability density function.

Ito Process:

proc[μ_, σ_] := ItoProcess[\[DifferentialD]x[t] == μ \[DifferentialD]t
+ σ \[DifferentialD]w[t], x[t], {x, 0}, t, w \[Distributed] WienerProcess[]]

paths = RandomFunction[proc[0.5, 1], {0, 10, 0.01}, 100]

plot = ListLinePlot[paths, AxesLabel -> {t, x},
PlotStyle -> Directive[Opacity[0.25], Thin],
PlotLabel -> Style[" μ=0.5, σ=1 ", Bold],
PlotRange -> All, ImageSize -> 450] And the transition probability function or Forward Kolmogorov is the following

Plot3D[E^(-((-x + x0 + t μ)^2/(2 t σ^2)))/(Sqrt[2 π] Sqrt[t σ^2])
/. {μ -> 0.5, σ -> 1, x0 -> 0}, {t, 0, 5}, {x, -5, 5}, AxesLabel -> Automatic] So, again my question is if it is possible to display the simulated paths over the canvas of the probability density function. It would be a very nice visual aid.

Using the code as in the OP, slightly modified:

proc[μ_, σ_] :=
ItoProcess[
\[DifferentialD]x[t] == μ \[DifferentialD]t + σ \[DifferentialD]w[t],
x[t], {x, 0}, t,
w \[Distributed] WienerProcess[]
]
paths = RandomFunction[proc[0.5, 1], {0, 10, 0.01}, 100];


Then, we define the function

f[x_, t_] = E^(-((-x + x0 + t μ)^2/(2 t σ^2)))/(Sqrt[2 π] Sqrt[t σ^2]) /. {μ -> 0.5, σ -> 1, x0 -> 0};


and construct sets of points tracing out the trajectories in 3D using

lsts = DeleteCases[
Table[{#1, #2, f[#2, #1]} & @@@ paths["Path", k], {k, 1, 20}],
{__, Indeterminate}, Infinity]


Finally, plot the function:

p1 = Plot3D[f[x, t],
{t, 0, 5}, {x, -5, 5},
AxesLabel -> Automatic, PlotStyle -> Opacity[0.3],
Mesh -> None, BoundaryStyle -> None];


and show this with the lines constructed from the sets of points:

Show[p1, Graphics3D[Line /@ lsts]]  To get different colors, you could do something like

Module[{i = 1},
Show[p1, Graphics3D[{Opacity[0.25], {Hue[i++/Length@lsts], Line@#} & /@ lsts}]]
] Play with it!

• Thank you so much. That looks awesome. Is there a chance you could write the tracing with regular notation. For me its a little hard to read with the Wolfram notation @@@ Jul 7 '16 at 23:19