# How to use some other driving process than the WienerProcess?

According to the following reference page

http://reference.wolfram.com/language/ref/ItoProcess.html

The driving process dproc can be any process that can be converted to a standard Ito process

and

Processes proc that can be converted to standard ItoProcess form include OrnsteinUhlenbeckProcess, GeometricBrownianMotionProcess, StratonovichProcess, and ItoProcess.

However, it does not seem to work in my example:

proc = ItoProcess[\[DifferentialD]x[
t] == -x[t] \[DifferentialD]t + \[DifferentialD]w[t],
x[t], {x, 1}, t,
w \[Distributed] OrnsteinUhlenbeckProcess[0, 1, 1]]
RandomFunction[proc, {0., 5., 0.01}]
ListLinePlot[%, Filling -> Axis]


which resulted in the error:

RandomFunction::unsproc: The specification ItoProcess[[DifferentialD]x[t]==[DifferentialD]w[t]-[DifferentialD]t x[t],x[t],{x,1},t,w[Distributed]OrnsteinUhlenbeckProcess[0,1,1]] is not a random process recognized by the system.

Note that in defining the ItoProcess proc like above, w \[Distributed] WienerProcess[0,1] would work, but w \[Distributed] OrnsteinUhlenbeckProcess[0, 1, 1] does not.

• As I understand it, the usage is like proc = ItoProcess[GeometricBrownianMotionProcess[0, 1, 1]] path = RandomFunction[proc, {0., 2. Pi, 0.05}, 12, Method -> "StochasticRungeKutta"] ListLinePlot[path]  Mar 15, 2014 at 10:12
• @belisarius This piece of code did not successfully result in a graph. Otherwise, my question concerns the driving. In the definition of the ItoProcess, w \[Distributed] WienerProcess[] works, but w \[Distributed] OrnsteinUhlenbeckProcess[0, 1, 1] does not. Mar 15, 2014 at 11:14

## 1 Answer

You need to complete the specification of the driving process by providing an initial condition :

SeedRandom[6]
proc = ItoProcess[\[DifferentialD]x[t] == -x[t] \[DifferentialD]t +
\[DifferentialD]w[t], x[t], {x, 1}, t,
w \[Distributed] OrnsteinUhlenbeckProcess[0, 1, 1, 0]]
RandomFunction[proc, {0., 5., 0.01}]
ListLinePlot[%, Filling -> Axis]


• Thank you for your answer! Indeed I overlooked the issue with the initial condition. When one defines an equivalent to proc by a system of two SDE's (see the reference page of OrnsteinUhlenbeckProcess), the necessity of complete initial conditions is obvious. The resulting error message unfortunately was not helpful. Mar 16, 2014 at 9:29