# ITO Process with random initial position

I am trying to define an ITO process with random initial state but its only drawing once an uses it for all paths. Here is the code:

proc[\[Mu]_, \[Sigma]_] :=
ItoProcess[\[DifferentialD]x[
t] == \[Mu] \[DifferentialD]t + \[Sigma] \[DifferentialD]w[t],
x[t], {x, RandomVariate[NormalDistribution[0, 2]]}, t,
w \[Distributed] WienerProcess[]]


Any suggestions how to make this work? Thank you in advance.

e

• I'll ponder this, my initial impression is no, since the process is evaluated before the ensemble is generated. Would simply generating a table of the random functions and then getting paths/etc. from those be a problem? – ciao May 30 '16 at 5:10
• Yes but I was hoping there would be a trick ... – Edv Beq May 30 '16 at 5:18
• See my (writing now) answer... – ciao May 30 '16 at 5:24

I don't believe this is directly possible - the process is evaluated before the random paths are generated. I'll ponder further, but if you want the same effect, try something like this:

ensemble1 = TemporalData[Table[RandomFunction[proc[1, 1], {0, 1}], 10]];


This will give you the same structure as

ensemble2 = RandomFunction[proc[1, 1], {0, 1},10]


that is, both examples give an ensemble of 10, but the former will give you the random starting positions:

Row[ListLinePlot[#, ImageSize -> 300] & /@ {ensemble1, ensemble2}]


• Great. If you think of better solution I would be very interested to see it. Thank you. – Edv Beq May 30 '16 at 5:34